Introduction
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Separable Equations
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Homogeneous Equations
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Homogeneous after Substitution
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Exact Equations
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Integration Factor
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Linear Equations
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Ricatti Equations
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Existence and Uniqueness
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Bernoulli Equations
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Clairaut Equation
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[{"Name":"Introduction","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction","Duration":"23m 8s","ChapterTopicVideoID":7563,"CourseChapterTopicPlaylistID":4217,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"We\u0027re beginning a new subject, differential equations."},{"Start":"00:03.240 ","End":"00:04.410","Text":"This is an introduction."},{"Start":"00:04.410 ","End":"00:06.375","Text":"Let\u0027s start with the definition."},{"Start":"00:06.375 ","End":"00:10.800","Text":"Differential equation is an equation,"},{"Start":"00:10.800 ","End":"00:15.585","Text":"first of all, which there appears at least 1 derivative of an unknown function."},{"Start":"00:15.585 ","End":"00:19.970","Text":"That unknown function could be y as a function of x,"},{"Start":"00:19.970 ","End":"00:24.315","Text":"in which case, we could get some examples."},{"Start":"00:24.315 ","End":"00:26.535","Text":"Here\u0027s 3 examples."},{"Start":"00:26.535 ","End":"00:30.855","Text":"Notice that in each of them we have y\u0027,"},{"Start":"00:30.855 ","End":"00:34.655","Text":"the derivative of y: y as some function of x."},{"Start":"00:34.655 ","End":"00:36.950","Text":"Here we have a derivative of y,"},{"Start":"00:36.950 ","End":"00:39.575","Text":"and here we have y as a derivative."},{"Start":"00:39.575 ","End":"00:43.835","Text":"But it\u0027s also possible for a higher-order derivative."},{"Start":"00:43.835 ","End":"00:45.500","Text":"Here\u0027s some more examples."},{"Start":"00:45.500 ","End":"00:52.155","Text":"Here we have y\u0027\u0027, and in this one also we have y\u0027\u0027,"},{"Start":"00:52.155 ","End":"00:57.950","Text":"and here we have y\u0027\u0027\u0027, third derivative."},{"Start":"00:57.950 ","End":"01:00.395","Text":"Not all derivatives have to be present."},{"Start":"01:00.395 ","End":"01:05.060","Text":"Y\u0027 could be missing here, y is missing."},{"Start":"01:05.060 ","End":"01:10.940","Text":"But as long as at least 1 derivative or higher-order derivative appears,"},{"Start":"01:10.940 ","End":"01:12.650","Text":"then that\u0027s a differential equation,"},{"Start":"01:12.650 ","End":"01:14.990","Text":"and it has to be an equation."},{"Start":"01:14.990 ","End":"01:18.080","Text":"Now, the obvious thing to do with an equation"},{"Start":"01:18.080 ","End":"01:20.585","Text":"is to solve it and that\u0027s what we\u0027re going to do here."},{"Start":"01:20.585 ","End":"01:25.740","Text":"We\u0027re going to talk about solution of a differential equation."},{"Start":"01:27.230 ","End":"01:32.240","Text":"Solution is a function which satisfies the equation just like with numbers."},{"Start":"01:32.240 ","End":"01:37.590","Text":"You can tell if a number satisfies a regular equation if you substitute it and it works,"},{"Start":"01:37.590 ","End":"01:39.830","Text":"and the same thing with functions."},{"Start":"01:39.830 ","End":"01:43.790","Text":"A function means if you substitute it into the equation, it\u0027ll work."},{"Start":"01:43.790 ","End":"01:46.650","Text":"Let\u0027s take, for example,"},{"Start":"01:46.690 ","End":"01:51.200","Text":"this equation, which is taken from here, number 2."},{"Start":"01:51.200 ","End":"02:00.365","Text":"I\u0027m proposing that the function y=x^2 is a solution to that equation."},{"Start":"02:00.365 ","End":"02:02.075","Text":"This equation in word,"},{"Start":"02:02.075 ","End":"02:04.340","Text":"it says, we want a function,"},{"Start":"02:04.340 ","End":"02:06.950","Text":"such that if you multiply by its derivative,"},{"Start":"02:06.950 ","End":"02:08.090","Text":"we get 2x^3,"},{"Start":"02:08.090 ","End":"02:11.460","Text":"but let\u0027s see that this indeed is a solution."},{"Start":"02:11.500 ","End":"02:16.475","Text":"If I put y=x^2,"},{"Start":"02:16.475 ","End":"02:20.460","Text":"here I have x^2\u0027 equal,"},{"Start":"02:20.460 ","End":"02:23.840","Text":"I should put question mark here,"},{"Start":"02:23.840 ","End":"02:25.190","Text":"because we don\u0027t know that it\u0027s equal."},{"Start":"02:25.190 ","End":"02:27.660","Text":"That\u0027s what we\u0027re going to verify."},{"Start":"02:28.000 ","End":"02:31.445","Text":"The derivative of x^2 is 2x,"},{"Start":"02:31.445 ","End":"02:35.690","Text":"and if you multiply x^2 by 2x,"},{"Start":"02:35.690 ","End":"02:38.825","Text":"then you indeed get 2x^3."},{"Start":"02:38.825 ","End":"02:44.270","Text":"So x^2 is a solution to this equation."},{"Start":"02:44.270 ","End":"02:47.210","Text":"Now let\u0027s take another example."},{"Start":"02:47.210 ","End":"02:49.970","Text":"We have this equation."},{"Start":"02:49.970 ","End":"02:51.925","Text":"It was one of the above."},{"Start":"02:51.925 ","End":"02:56.310","Text":"I think this was number 4 or 3."},{"Start":"02:56.310 ","End":"02:58.725","Text":"No, it\u0027s equation 3 above, never mind."},{"Start":"02:58.725 ","End":"03:04.535","Text":"Let\u0027s see that this is indeed a solution as proposed. We want to check."},{"Start":"03:04.535 ","End":"03:07.460","Text":"It should really be equals with a question mark."},{"Start":"03:07.460 ","End":"03:09.050","Text":"That\u0027s what we\u0027re going to check."},{"Start":"03:09.050 ","End":"03:16.025","Text":"That when I put 2 plus e to the minus x^2 in place of y everywhere, that it will work."},{"Start":"03:16.025 ","End":"03:18.355","Text":"Y is 2 plus this."},{"Start":"03:18.355 ","End":"03:21.685","Text":"Here I also replace y by this."},{"Start":"03:21.685 ","End":"03:24.065","Text":"We need to do a derivative here."},{"Start":"03:24.065 ","End":"03:29.760","Text":"The derivative of this is just the derivative of this part,"},{"Start":"03:29.760 ","End":"03:33.335","Text":"which is minus 2x as the inner derivative and then"},{"Start":"03:33.335 ","End":"03:37.450","Text":"e to the minus x^2 plus this, plus this."},{"Start":"03:37.450 ","End":"03:40.030","Text":"Then we\u0027re still checking, is it equal to?"},{"Start":"03:40.030 ","End":"03:42.410","Text":"Notice that this cancels with this."},{"Start":"03:42.410 ","End":"03:43.730","Text":"So we\u0027re left with just this,"},{"Start":"03:43.730 ","End":"03:45.650","Text":"which is true and so yes,"},{"Start":"03:45.650 ","End":"03:49.415","Text":"this is indeed a solution."},{"Start":"03:49.415 ","End":"03:51.800","Text":"Continuing with the examples,"},{"Start":"03:51.800 ","End":"03:53.494","Text":"and I\u0027m going to do quite a few examples."},{"Start":"03:53.494 ","End":"03:57.215","Text":"This was our equation number 4, I believe."},{"Start":"03:57.215 ","End":"04:01.640","Text":"We\u0027re proposing that y=x^3 is a solution to this equation."},{"Start":"04:01.640 ","End":"04:03.620","Text":"Let\u0027s verify this."},{"Start":"04:03.620 ","End":"04:07.595","Text":"We put in place of y, x^3."},{"Start":"04:07.595 ","End":"04:09.290","Text":"Notice that this one is different than"},{"Start":"04:09.290 ","End":"04:13.610","Text":"the previous couple that we did because it has a second derivative."},{"Start":"04:13.610 ","End":"04:16.760","Text":"But other than that, no essential difference."},{"Start":"04:16.760 ","End":"04:22.400","Text":"So what we do is we do the second derivative of this."},{"Start":"04:22.400 ","End":"04:28.055","Text":"Well, we would do the first derivative first and get from here to here, 3x^2,"},{"Start":"04:28.055 ","End":"04:29.750","Text":"and now that I have the first derivative,"},{"Start":"04:29.750 ","End":"04:32.420","Text":"I can differentiate this and get the 6x,"},{"Start":"04:32.420 ","End":"04:34.205","Text":"then I substitute it all."},{"Start":"04:34.205 ","End":"04:36.970","Text":"Let\u0027s see if this really works."},{"Start":"04:36.970 ","End":"04:42.239","Text":"What we get is here minus 3x^3 here plus 3x^3, and that cancels."},{"Start":"04:42.239 ","End":"04:46.025","Text":"So we\u0027re down to 6x=6x, and that works."},{"Start":"04:46.025 ","End":"04:48.890","Text":"Now let\u0027s go to yet another example."},{"Start":"04:48.890 ","End":"04:52.310","Text":"This I think was example number 6 above,"},{"Start":"04:52.310 ","End":"04:57.550","Text":"which had a third derivative in it."},{"Start":"04:57.550 ","End":"05:00.765","Text":"This is being proposed as a solution."},{"Start":"05:00.765 ","End":"05:03.790","Text":"Let\u0027s check that indeed it is."},{"Start":"05:04.470 ","End":"05:09.760","Text":"We\u0027re asking, is this equal to 0."},{"Start":"05:09.760 ","End":"05:14.325","Text":"Let\u0027s see. What do I have here?"},{"Start":"05:14.325 ","End":"05:15.660","Text":"I substituted."},{"Start":"05:15.660 ","End":"05:18.120","Text":"I substituted y\u0027 first."},{"Start":"05:18.120 ","End":"05:20.740","Text":"There is no y. There\u0027s only derivatives."},{"Start":"05:20.740 ","End":"05:27.940","Text":"Y\u0027 is this because a derivative of this is minus e to the minus x,"},{"Start":"05:27.940 ","End":"05:29.785","Text":"and here 3e to the 3x."},{"Start":"05:29.785 ","End":"05:34.050","Text":"Then I can use this to get y\u0027\u0027 here."},{"Start":"05:34.050 ","End":"05:35.980","Text":"By differentiating this, you get this."},{"Start":"05:35.980 ","End":"05:39.010","Text":"By differentiating this, you get this,"},{"Start":"05:39.010 ","End":"05:41.010","Text":"that\u0027s the third derivative."},{"Start":"05:41.010 ","End":"05:44.380","Text":"After substituting, we want to see if we get 0."},{"Start":"05:44.380 ","End":"05:47.540","Text":"Simplify by opening the brackets."},{"Start":"05:47.540 ","End":"05:51.980","Text":"I\u0027m not going to go into all the details, it\u0027s here."},{"Start":"05:51.980 ","End":"05:54.665","Text":"After we cancel everything out,"},{"Start":"05:54.665 ","End":"05:56.965","Text":"we get, for example,"},{"Start":"05:56.965 ","End":"05:58.560","Text":"27e to the 3x."},{"Start":"05:58.560 ","End":"06:00.270","Text":"Then we have minus 18 minus 9,"},{"Start":"06:00.270 ","End":"06:04.865","Text":"that cancels, and so 1e to the minus x also cancels."},{"Start":"06:04.865 ","End":"06:07.280","Text":"We have minus 1, minus 1 plus 3."},{"Start":"06:07.280 ","End":"06:10.910","Text":"Eventually you just get 0=0, which is true."},{"Start":"06:10.910 ","End":"06:14.075","Text":"Let\u0027s do yet another example."},{"Start":"06:14.075 ","End":"06:16.970","Text":"This time, just for a change,"},{"Start":"06:16.970 ","End":"06:20.780","Text":"we have x as a function of t, which can happen."},{"Start":"06:20.780 ","End":"06:24.125","Text":"It doesn\u0027t always have to be y as a function of x."},{"Start":"06:24.125 ","End":"06:26.930","Text":"A variable can be a function of another variable."},{"Start":"06:26.930 ","End":"06:28.820","Text":"Here is the equation."},{"Start":"06:28.820 ","End":"06:31.135","Text":"It\u0027s not one of the ones above,"},{"Start":"06:31.135 ","End":"06:34.485","Text":"x\u0027 plus x equals 2 plus 2t."},{"Start":"06:34.485 ","End":"06:40.490","Text":"Here the prime means derivative with respect to t. Here\u0027s a proposed solution for x."},{"Start":"06:40.490 ","End":"06:42.740","Text":"Let\u0027s see if this really is."},{"Start":"06:42.740 ","End":"06:48.770","Text":"We want to know if when I substitute this into here, it really works."},{"Start":"06:48.770 ","End":"06:50.945","Text":"The question mark here is what we\u0027re checking."},{"Start":"06:50.945 ","End":"06:53.750","Text":"This is x, just copied from here."},{"Start":"06:53.750 ","End":"06:56.370","Text":"This is the derivative,"},{"Start":"06:58.910 ","End":"07:01.610","Text":"well, not the derivative, I just put prime."},{"Start":"07:01.610 ","End":"07:03.650","Text":"We actually have to do the derivative."},{"Start":"07:03.650 ","End":"07:06.800","Text":"From 2t we get 2,"},{"Start":"07:06.800 ","End":"07:07.850","Text":"from e to the minus t,"},{"Start":"07:07.850 ","End":"07:11.095","Text":"we get minus e to the minus t. This is copied."},{"Start":"07:11.095 ","End":"07:14.330","Text":"The left-hand side simplifies because e to"},{"Start":"07:14.330 ","End":"07:17.930","Text":"the minus t cancels and we\u0027re left with equality,"},{"Start":"07:17.930 ","End":"07:21.995","Text":"and so, yes, this is a solution of this."},{"Start":"07:21.995 ","End":"07:25.835","Text":"I think there was about 5 examples and I think that should do."},{"Start":"07:25.835 ","End":"07:28.565","Text":"Now we come to another definition."},{"Start":"07:28.565 ","End":"07:33.290","Text":"We classify differential equations according to their order,"},{"Start":"07:33.290 ","End":"07:35.500","Text":"and I\u0027ll give the definition,"},{"Start":"07:35.500 ","End":"07:38.090","Text":"that the order of a differential equation is"},{"Start":"07:38.090 ","End":"07:40.835","Text":"the highest order of a derivative in the equation,"},{"Start":"07:40.835 ","End":"07:42.590","Text":"and best to explain by example."},{"Start":"07:42.590 ","End":"07:47.480","Text":"Let\u0027s take the previous examples or 6 examples from above."},{"Start":"07:47.480 ","End":"07:50.155","Text":"Notice that in the first 3,"},{"Start":"07:50.155 ","End":"07:52.085","Text":"I have y\u0027,"},{"Start":"07:52.085 ","End":"07:54.270","Text":"but I don\u0027t have y\u0027\u0027."},{"Start":"07:54.270 ","End":"07:57.510","Text":"So 1-3 are of the first order."},{"Start":"07:57.510 ","End":"07:59.265","Text":"Instead of saying the order equals 1,"},{"Start":"07:59.265 ","End":"08:02.030","Text":"we say that these are of the first order,"},{"Start":"08:02.030 ","End":"08:06.135","Text":"so that 4 and 5 have a second derivative,"},{"Start":"08:06.135 ","End":"08:09.840","Text":"but now higher, so they would be of the second order."},{"Start":"08:09.840 ","End":"08:12.850","Text":"This is differential equation of the second order."},{"Start":"08:12.850 ","End":"08:17.180","Text":"The last one is a differential equation of the third order."},{"Start":"08:17.180 ","End":"08:18.500","Text":"If we ask what is the order,"},{"Start":"08:18.500 ","End":"08:19.670","Text":"it\u0027s equal to 3."},{"Start":"08:19.670 ","End":"08:21.890","Text":"Those are different ways of saying it."},{"Start":"08:21.890 ","End":"08:32.839","Text":"Now, we can also generalize this slightly."},{"Start":"08:32.839 ","End":"08:35.500","Text":"We bring everything to the left hand side."},{"Start":"08:35.500 ","End":"08:39.010","Text":"It\u0027s some expression with x,"},{"Start":"08:39.010 ","End":"08:44.440","Text":"y and y\u0027 some equation involving the answer equals 0,"},{"Start":"08:44.440 ","End":"08:46.765","Text":"but y\u0027 has to be present."},{"Start":"08:46.765 ","End":"08:49.225","Text":"Otherwise it\u0027s not a differential equation."},{"Start":"08:49.225 ","End":"08:55.990","Text":"A second order would be some expression containing x,"},{"Start":"08:55.990 ","End":"08:59.185","Text":"y, y\u0027, and y\"equals 0."},{"Start":"08:59.185 ","End":"09:00.865","Text":"Not all have to be present,"},{"Start":"09:00.865 ","End":"09:05.620","Text":"but y\" has to be present for it to be of the second order."},{"Start":"09:05.620 ","End":"09:11.030","Text":"You can generalize to third order and so on and so on."},{"Start":"09:11.030 ","End":"09:14.260","Text":"Now, I want to say something about notation."},{"Start":"09:14.260 ","End":"09:18.430","Text":"Remember that there are 2 forms of writing a derivative."},{"Start":"09:18.430 ","End":"09:22.750","Text":"There\u0027s, it\u0027s known as the Newton or Leibniz notations."},{"Start":"09:22.750 ","End":"09:29.440","Text":"Newton wrote prime and Leibniz wrote dy / dx."},{"Start":"09:29.440 ","End":"09:31.570","Text":"These 2 not an equality."},{"Start":"09:31.570 ","End":"09:34.240","Text":"This is like the Newton form and the Leibniz form,"},{"Start":"09:34.240 ","End":"09:36.385","Text":"people don\u0027t often remember the names,"},{"Start":"09:36.385 ","End":"09:40.675","Text":"but does the prime and this is a d by d way."},{"Start":"09:40.675 ","End":"09:43.480","Text":"y\" is written this way in"},{"Start":"09:43.480 ","End":"09:48.100","Text":"the Leibniz notation y\u0027\u0027\u0027 and when we get tired of writing prime,"},{"Start":"09:48.100 ","End":"09:49.405","Text":"prime, prime, prime,"},{"Start":"09:49.405 ","End":"09:52.765","Text":"we just use a number inside a bracket."},{"Start":"09:52.765 ","End":"09:55.450","Text":"Like if this was the fifth derivative,"},{"Start":"09:55.450 ","End":"10:01.600","Text":"we would put 5 in brackets and this is the other way of writing it."},{"Start":"10:01.600 ","End":"10:05.170","Text":"What we get is a different way we take"},{"Start":"10:05.170 ","End":"10:10.870","Text":"our same 6 examples and I could rewrite them in the other notation."},{"Start":"10:10.870 ","End":"10:13.750","Text":"Instead of y\u0027, I\u0027d have dy / dx,"},{"Start":"10:13.750 ","End":"10:16.120","Text":"instead of y\","},{"Start":"10:16.120 ","End":"10:19.240","Text":"d2y / dx^2,"},{"Start":"10:19.240 ","End":"10:21.325","Text":"and so on and so on."},{"Start":"10:21.325 ","End":"10:25.015","Text":"Just reminding you that there\u0027s another notation for"},{"Start":"10:25.015 ","End":"10:29.785","Text":"derivatives and some books, professors prefer."},{"Start":"10:29.785 ","End":"10:34.345","Text":"We\u0027re going to use both depending on what suits us."},{"Start":"10:34.345 ","End":"10:38.470","Text":"Now, another remark, a continuation of the above,"},{"Start":"10:38.470 ","End":"10:41.470","Text":"is that if we have a first order equation,"},{"Start":"10:41.470 ","End":"10:44.470","Text":"meaning just y\u0027 no y\", and so on."},{"Start":"10:44.470 ","End":"10:50.960","Text":"If we\u0027re using the dy by dx notation,"},{"Start":"10:54.390 ","End":"10:59.170","Text":"I\u0027ll illustrate, we use a common denominator and possibly rearrange terms."},{"Start":"10:59.170 ","End":"11:01.090","Text":"I\u0027ll show you what I mean."},{"Start":"11:01.090 ","End":"11:05.830","Text":"If I have the equation with the dy / dx = x / y,"},{"Start":"11:05.830 ","End":"11:09.220","Text":"what we do often is use a common denominator."},{"Start":"11:09.220 ","End":"11:17.170","Text":"I could say that x times dx = y times dy."},{"Start":"11:17.170 ","End":"11:21.610","Text":"That would be okay and commonly"},{"Start":"11:21.610 ","End":"11:26.245","Text":"we not necessarily bring everything to the left and make it equal to 0,"},{"Start":"11:26.245 ","End":"11:28.045","Text":"but you don\u0027t have to do that."},{"Start":"11:28.045 ","End":"11:35.635","Text":"But it\u0027s common to use dx\u0027s and dy\u0027s and not use the fraction form."},{"Start":"11:35.635 ","End":"11:42.040","Text":"If for example, we had another example like this,"},{"Start":"11:42.040 ","End":"11:47.320","Text":"multiply both sides by dx and get ydy equals 2x^3dx,"},{"Start":"11:47.320 ","End":"11:51.430","Text":"or bring everything to one side and make it equal to 0."},{"Start":"11:51.430 ","End":"11:57.625","Text":"Last example that we had above in the new notation or in the Leibniz notation."},{"Start":"11:57.625 ","End":"12:02.720","Text":"If you get this, you multiply both sides by dx."},{"Start":"12:03.090 ","End":"12:06.850","Text":"Or possibly you might want to move this to the other side first and"},{"Start":"12:06.850 ","End":"12:10.240","Text":"multiply it by dx and after rearranging,"},{"Start":"12:10.240 ","End":"12:12.610","Text":"you could get that this equals 0,"},{"Start":"12:12.610 ","End":"12:15.925","Text":"or you could just leave the dy on the other side."},{"Start":"12:15.925 ","End":"12:19.340","Text":"That\u0027s something that\u0027s commonly done."},{"Start":"12:19.710 ","End":"12:24.220","Text":"Now another term to learn,"},{"Start":"12:24.220 ","End":"12:30.320","Text":"very important, something called an initial condition for a differential equation."},{"Start":"12:31.500 ","End":"12:36.190","Text":"It turns out that a differential equation can have many solutions,"},{"Start":"12:36.190 ","End":"12:37.615","Text":"I didn\u0027t show this above."},{"Start":"12:37.615 ","End":"12:39.745","Text":"But often it has many solutions."},{"Start":"12:39.745 ","End":"12:42.940","Text":"Pretty much like when we did an indefinite integral,"},{"Start":"12:42.940 ","End":"12:47.560","Text":"we had a constant and then we needed maybe a pair of values,"},{"Start":"12:47.560 ","End":"12:49.660","Text":"x, y to find that constant."},{"Start":"12:49.660 ","End":"12:52.435","Text":"A similar thing exists for differential equations."},{"Start":"12:52.435 ","End":"12:55.795","Text":"Because there are many solutions to a given equation,"},{"Start":"12:55.795 ","End":"12:59.470","Text":"we give what is called an initial condition."},{"Start":"12:59.470 ","End":"13:05.230","Text":"An initial condition says that when x=a,"},{"Start":"13:05.230 ","End":"13:06.855","Text":"y= b, in other words,"},{"Start":"13:06.855 ","End":"13:09.835","Text":"y(a)=b, that\u0027s a common initial condition."},{"Start":"13:09.835 ","End":"13:12.880","Text":"But it could also be with a higher order derivative."},{"Start":"13:12.880 ","End":"13:14.095","Text":"We could say the second,"},{"Start":"13:14.095 ","End":"13:15.655","Text":"say n was 2 here."},{"Start":"13:15.655 ","End":"13:21.795","Text":"The second derivative of y at the point a is equal to b."},{"Start":"13:21.795 ","End":"13:25.500","Text":"We pretty much only going to see this form because we"},{"Start":"13:25.500 ","End":"13:30.820","Text":"only going to be talking about first order differential equations in this section."},{"Start":"13:31.560 ","End":"13:34.195","Text":"That\u0027s an initial condition,"},{"Start":"13:34.195 ","End":"13:35.980","Text":"typically looks like this,"},{"Start":"13:35.980 ","End":"13:39.850","Text":"but could be y\", y\u0027\u0027\u0027, or any order."},{"Start":"13:39.850 ","End":"13:43.465","Text":"I\u0027ve noticed that many places on the Internet,"},{"Start":"13:43.465 ","End":"13:48.670","Text":"to some different professors like the term the initial value problem,"},{"Start":"13:48.670 ","End":"13:52.630","Text":"which means that when you have a differential equation with initial conditions,"},{"Start":"13:52.630 ","End":"13:57.370","Text":"the whole package is called an initial value problem, IVP."},{"Start":"13:57.370 ","End":"13:58.840","Text":"I don\u0027t particularly like it,"},{"Start":"13:58.840 ","End":"14:01.780","Text":"so I will just say differential equation with"},{"Start":"14:01.780 ","End":"14:05.710","Text":"initial conditions and not use the shortcuts."},{"Start":"14:05.710 ","End":"14:08.695","Text":"I certainly won\u0027t use IVP."},{"Start":"14:08.695 ","End":"14:11.590","Text":"But you might encounter this,"},{"Start":"14:11.590 ","End":"14:13.850","Text":"so I\u0027m bringing it."},{"Start":"14:13.860 ","End":"14:17.380","Text":"Now, with different initial conditions,"},{"Start":"14:17.380 ","End":"14:18.670","Text":"even on the same equation,"},{"Start":"14:18.670 ","End":"14:21.640","Text":"you could get different solutions and I\u0027ll illustrate this."},{"Start":"14:21.640 ","End":"14:29.050","Text":"Suppose we took the equation above y\u0027 times x equals y."},{"Start":"14:29.050 ","End":"14:39.040","Text":"Then if I let y equals 2x,"},{"Start":"14:39.040 ","End":"14:42.655","Text":"that would be a solution."},{"Start":"14:42.655 ","End":"14:47.560","Text":"I should have written like y(x) in full,"},{"Start":"14:47.560 ","End":"14:49.120","Text":"because y is a function of x,"},{"Start":"14:49.120 ","End":"14:52.150","Text":"so y(x) is 2x,"},{"Start":"14:52.150 ","End":"14:58.510","Text":"and y\u0027(x) is just 2."},{"Start":"14:58.510 ","End":"15:02.380","Text":"It\u0027s the derivative, but we don\u0027t usually write the of x."},{"Start":"15:02.380 ","End":"15:06.445","Text":"But here, when I want to check the initial condition,"},{"Start":"15:06.445 ","End":"15:10.135","Text":"I can see that when I put x equals 1,"},{"Start":"15:10.135 ","End":"15:12.820","Text":"y does indeed equal 2,"},{"Start":"15:12.820 ","End":"15:16.855","Text":"so this condition is met and"},{"Start":"15:16.855 ","End":"15:23.725","Text":"the equation is also met because y\u0027 we saw is 2,"},{"Start":"15:23.725 ","End":"15:27.220","Text":"and 2 times x is y."},{"Start":"15:27.220 ","End":"15:30.010","Text":"Because y is 2x."},{"Start":"15:30.010 ","End":"15:32.380","Text":"This equation also works."},{"Start":"15:32.380 ","End":"15:40.300","Text":"This satisfies both the equation and the initial condition."},{"Start":"15:40.300 ","End":"15:44.035","Text":"On the other hand, in a different example,"},{"Start":"15:44.035 ","End":"15:47.155","Text":"the same equation as above,"},{"Start":"15:47.155 ","End":"15:49.555","Text":"but with a different initial condition,"},{"Start":"15:49.555 ","End":"15:51.445","Text":"has a different solution."},{"Start":"15:51.445 ","End":"15:54.500","Text":"Let\u0027s check if y equals 5x,"},{"Start":"15:54.690 ","End":"16:00.583","Text":"then y\u0027 is equal to 5."},{"Start":"16:00.583 ","End":"16:04.595","Text":"Then we still have y(2)=10."},{"Start":"16:04.595 ","End":"16:07.385","Text":"We say y(2), you don\u0027t see anywhere to substitute."},{"Start":"16:07.385 ","End":"16:10.655","Text":"This is really y(x),"},{"Start":"16:10.655 ","End":"16:12.695","Text":"which means that when x is 2,"},{"Start":"16:12.695 ","End":"16:14.915","Text":"y=5 times 2 is 10."},{"Start":"16:14.915 ","End":"16:16.910","Text":"The initial condition is met."},{"Start":"16:16.910 ","End":"16:21.110","Text":"But the equation is also satisfied because y\u0027 is 5,"},{"Start":"16:21.110 ","End":"16:24.005","Text":"and y is 5x,"},{"Start":"16:24.005 ","End":"16:27.360","Text":"and 5 times x is indeed 5x."},{"Start":"16:28.660 ","End":"16:32.300","Text":"If I just gave you without an initial condition,"},{"Start":"16:32.300 ","End":"16:35.435","Text":"if I just gave you this equation which is here and here,"},{"Start":"16:35.435 ","End":"16:39.740","Text":"here\u0027s 2 solutions, y=2x and y=5x."},{"Start":"16:39.740 ","End":"16:44.810","Text":"That\u0027s why we need an initial condition to narrow it down to specify it."},{"Start":"16:44.810 ","End":"16:46.775","Text":"That\u0027s just in a nutshell."},{"Start":"16:46.775 ","End":"16:49.595","Text":"We\u0027ll see this later."},{"Start":"16:49.595 ","End":"16:51.290","Text":"Let me just for emphasis,"},{"Start":"16:51.290 ","End":"16:56.540","Text":"go over again the initial condition is this."},{"Start":"16:56.540 ","End":"17:00.050","Text":"As I said though it could be y\" or triple prime."},{"Start":"17:00.050 ","End":"17:03.500","Text":"But in our case we\u0027ll just have this form."},{"Start":"17:03.500 ","End":"17:09.335","Text":"Here\u0027s our initial condition here and here\u0027s our initial condition here."},{"Start":"17:09.335 ","End":"17:11.000","Text":"When we do initial conditions,"},{"Start":"17:11.000 ","End":"17:14.495","Text":"we have to remember that if I give you y,"},{"Start":"17:14.495 ","End":"17:21.125","Text":"y is a function of x. I can substitute when I find a solution,"},{"Start":"17:21.125 ","End":"17:23.900","Text":"and this means substitute x=1,"},{"Start":"17:23.900 ","End":"17:25.565","Text":"and see what y gives."},{"Start":"17:25.565 ","End":"17:30.110","Text":"I\u0027m just going over again to make sure you really know what the initial condition is."},{"Start":"17:30.110 ","End":"17:33.120","Text":"Now let\u0027s get onto another concept."},{"Start":"17:33.220 ","End":"17:37.550","Text":"Another classification of differential equations is that they can"},{"Start":"17:37.550 ","End":"17:42.590","Text":"be ordinary and they can be partial."},{"Start":"17:42.590 ","End":"17:45.005","Text":"In all our examples,"},{"Start":"17:45.005 ","End":"17:48.845","Text":"we had y as a function of a single variable x."},{"Start":"17:48.845 ","End":"17:51.920","Text":"We have an ordinary differential equation if"},{"Start":"17:51.920 ","End":"17:55.235","Text":"the unknown function is a function of one variable,"},{"Start":"17:55.235 ","End":"17:58.385","Text":"which is all the examples that we had so far."},{"Start":"17:58.385 ","End":"18:00.380","Text":"Y was just a function of x."},{"Start":"18:00.380 ","End":"18:03.365","Text":"But sometimes we have an unknown function."},{"Start":"18:03.365 ","End":"18:06.410","Text":"Maybe z is a function of x and y."},{"Start":"18:06.410 ","End":"18:08.690","Text":"We have a function of more than one variable,"},{"Start":"18:08.690 ","End":"18:11.705","Text":"it\u0027s called a partial differential equation."},{"Start":"18:11.705 ","End":"18:14.610","Text":"They have abbreviations."},{"Start":"18:14.940 ","End":"18:20.620","Text":"Very commonly, I\u0027ll say ODE for ordinary differential equations and"},{"Start":"18:20.620 ","End":"18:26.750","Text":"partial differential equations called PDE although won\u0027t be dealing with those here."},{"Start":"18:26.750 ","End":"18:31.520","Text":"Like I said, all the equations we\u0027ve seen so far are ordinary."},{"Start":"18:31.520 ","End":"18:35.045","Text":"Y is a function of a single variable x."},{"Start":"18:35.045 ","End":"18:38.390","Text":"All of these doesn\u0027t matter what order of derivative we take,"},{"Start":"18:38.390 ","End":"18:42.050","Text":"y is still just a function of a single variable x."},{"Start":"18:42.050 ","End":"18:46.475","Text":"But here\u0027s an example of a partial differential equation."},{"Start":"18:46.475 ","End":"18:48.980","Text":"In this case f,"},{"Start":"18:48.980 ","End":"18:51.470","Text":"we have to think of it as a function of x and"},{"Start":"18:51.470 ","End":"18:58.040","Text":"y. I\u0027ll write that f is a function of x and y."},{"Start":"18:58.040 ","End":"19:02.900","Text":"Then this makes sense to say that x times the partial derivative of f with"},{"Start":"19:02.900 ","End":"19:08.137","Text":"respect to x plus y times partial derivative equal 2xy."},{"Start":"19:08.137 ","End":"19:12.800","Text":"Why? This is a partial differential equation because you see partial derivatives."},{"Start":"19:12.800 ","End":"19:15.740","Text":"Of course, with PDE is also a solution,"},{"Start":"19:15.740 ","End":"19:20.015","Text":"is a function f which satisfies the equation."},{"Start":"19:20.015 ","End":"19:25.145","Text":"Here\u0027s a proposed solution to the above equation."},{"Start":"19:25.145 ","End":"19:27.515","Text":"F(x, y)=xy."},{"Start":"19:27.515 ","End":"19:32.190","Text":"Let\u0027s check that this function satisfies this equation."},{"Start":"19:32.860 ","End":"19:37.025","Text":"Wherever I see f in the equation,"},{"Start":"19:37.025 ","End":"19:40.445","Text":"here, I just substitute xy."},{"Start":"19:40.445 ","End":"19:41.810","Text":"Here I have xy,"},{"Start":"19:41.810 ","End":"19:43.625","Text":"and here I have xy."},{"Start":"19:43.625 ","End":"19:47.780","Text":"I added a prime here we don\u0027t usually just put a subscript x,"},{"Start":"19:47.780 ","End":"19:49.040","Text":"doesn\u0027t say prime here,"},{"Start":"19:49.040 ","End":"19:51.630","Text":"but you could have written prime or prime."},{"Start":"19:56.650 ","End":"20:02.119","Text":"The derivative of xy with respect to x is just y,"},{"Start":"20:02.119 ","End":"20:06.740","Text":"because y is a constant."},{"Start":"20:06.740 ","End":"20:08.750","Text":"Constant times x derivative is y,"},{"Start":"20:08.750 ","End":"20:10.190","Text":"on the other hand here,"},{"Start":"20:10.190 ","End":"20:12.890","Text":"differentiating with respect to y,"},{"Start":"20:12.890 ","End":"20:15.710","Text":"we treat x like a constant."},{"Start":"20:15.710 ","End":"20:19.460","Text":"The answer is just x."},{"Start":"20:19.460 ","End":"20:22.205","Text":"We get x times y + y times x,"},{"Start":"20:22.205 ","End":"20:24.845","Text":"it is equal to 2xy."},{"Start":"20:24.845 ","End":"20:32.400","Text":"That works, I should\u0027ve said."},{"Start":"20:32.400 ","End":"20:35.080","Text":"Last remark, in this course,"},{"Start":"20:35.080 ","End":"20:38.770","Text":"we\u0027re going to only be working with ordinary differential equations,"},{"Start":"20:38.770 ","End":"20:41.770","Text":"no partial differential equations here."},{"Start":"20:41.770 ","End":"20:45.050","Text":"That\u0027s much more complicated."},{"Start":"20:47.620 ","End":"20:53.910","Text":"Let\u0027s just say another couple of things about differential equations."},{"Start":"20:53.910 ","End":"20:59.590","Text":"Uses. What do we use differential equations for?"},{"Start":"20:59.590 ","End":"21:03.250","Text":"Well, differential equation is a mathematical equation connecting"},{"Start":"21:03.250 ","End":"21:07.520","Text":"some function with its derivatives."},{"Start":"21:07.520 ","End":"21:12.365","Text":"In the real-world, the unknown function usually describes the physical quantity."},{"Start":"21:12.365 ","End":"21:16.700","Text":"Derivative represents its rate of change."},{"Start":"21:16.700 ","End":"21:20.469","Text":"The equation describes the relation between the two."},{"Start":"21:20.469 ","End":"21:24.200","Text":"Because this type of relation is very common,"},{"Start":"21:24.200 ","End":"21:28.490","Text":"differential equations play a central role in many fields such as engineering,"},{"Start":"21:28.490 ","End":"21:34.295","Text":"physics, economics, biology, and computer science and of course many more."},{"Start":"21:34.295 ","End":"21:37.520","Text":"The last thing I want to say about the course"},{"Start":"21:37.520 ","End":"21:41.120","Text":"is what is the prerequisite knowledge for this course."},{"Start":"21:41.120 ","End":"21:45.770","Text":"You certainly need to know integration methods."},{"Start":"21:45.770 ","End":"21:48.710","Text":"I mean, you can expect that if we\u0027re solving something"},{"Start":"21:48.710 ","End":"21:51.380","Text":"involving a derivative that will have to do an integration,"},{"Start":"21:51.380 ","End":"21:55.220","Text":"so you should know all the basic integration methods such as substitution,"},{"Start":"21:55.220 ","End":"21:58.010","Text":"integration by parts, etc."},{"Start":"21:58.010 ","End":"22:02.015","Text":"Then depending on what you\u0027re studying, what you\u0027re learning,"},{"Start":"22:02.015 ","End":"22:07.280","Text":"you will need possibly some extra,"},{"Start":"22:07.280 ","End":"22:09.230","Text":"well, mostly if you\u0027re in this course,"},{"Start":"22:09.230 ","End":"22:11.555","Text":"you will be learning exact equations."},{"Start":"22:11.555 ","End":"22:13.490","Text":"This is pretty much for everyone."},{"Start":"22:13.490 ","End":"22:16.910","Text":"You need to know what partial derivatives are."},{"Start":"22:16.910 ","End":"22:19.910","Text":"We\u0027re not going to be doing partial differential equations,"},{"Start":"22:19.910 ","End":"22:22.805","Text":"but you still need to know what partial derivatives are."},{"Start":"22:22.805 ","End":"22:28.940","Text":"Then certain students will be studying systems of equations."},{"Start":"22:28.940 ","End":"22:32.210","Text":"Only a few of you will probably be learning these."},{"Start":"22:32.210 ","End":"22:34.790","Text":"But if you are, you\u0027ll need to know from"},{"Start":"22:34.790 ","End":"22:39.395","Text":"linear algebra the concept of eigenvalues and eigenvectors."},{"Start":"22:39.395 ","End":"22:45.860","Text":"If you\u0027re learning how to solve differential equations with series,"},{"Start":"22:45.860 ","End":"22:48.830","Text":"Maclaurin series, Taylor series,"},{"Start":"22:48.830 ","End":"22:51.635","Text":"then you should also be familiar with"},{"Start":"22:51.635 ","End":"23:00.210","Text":"Maclaurin series for power series solutions."},{"Start":"23:00.580 ","End":"23:08.490","Text":"That\u0027s all I want to say for the introduction. We\u0027re done here."}],"Thumbnail":null,"ID":7636}],"ID":4217},{"Name":"Separable Equations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Separable Equations","Duration":"4m 31s","ChapterTopicVideoID":7564,"CourseChapterTopicPlaylistID":4218,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.280","Text":"In this clip, we\u0027re going to talk about type of"},{"Start":"00:02.280 ","End":"00:05.714","Text":"differential equation called a separable differential equation."},{"Start":"00:05.714 ","End":"00:09.255","Text":"This belongs to ordinary differential equations,"},{"Start":"00:09.255 ","End":"00:12.179","Text":"first order. Here\u0027s the definition."},{"Start":"00:12.179 ","End":"00:15.660","Text":"A separable differential equation is one that can be brought to"},{"Start":"00:15.660 ","End":"00:20.295","Text":"the form some function of xdx equals another function of ydy."},{"Start":"00:20.295 ","End":"00:22.605","Text":"When we get it to this form,"},{"Start":"00:22.605 ","End":"00:26.205","Text":"then the obvious thing to do is to put an integral sign in front of each,"},{"Start":"00:26.205 ","End":"00:28.710","Text":"and that will give us the solution."},{"Start":"00:28.710 ","End":"00:29.925","Text":"I\u0027ll give an example."},{"Start":"00:29.925 ","End":"00:34.830","Text":"Here\u0027s a differential equation y\u0027 equals x/y,"},{"Start":"00:34.830 ","End":"00:38.685","Text":"of course, we have the condition, exclude y=0."},{"Start":"00:38.685 ","End":"00:41.150","Text":"This does not look like this,"},{"Start":"00:41.150 ","End":"00:45.170","Text":"but notice that it said that can be brought to the form."},{"Start":"00:45.170 ","End":"00:49.850","Text":"It means with a little bit of work if we can get it to look like this."},{"Start":"00:49.850 ","End":"00:53.675","Text":"Such equations are always written in Leibniz\u0027s notation,"},{"Start":"00:53.675 ","End":"00:56.390","Text":"not y\u0027 but dy/dx."},{"Start":"00:56.390 ","End":"00:57.590","Text":"That\u0027s the first thing to do."},{"Start":"00:57.590 ","End":"01:05.255","Text":"Then of course we can cross multiply and we get ydy equals xdx."},{"Start":"01:05.255 ","End":"01:07.820","Text":"In fact, actually, it\u0027s more common to put the ys on"},{"Start":"01:07.820 ","End":"01:10.940","Text":"the left and the xs on the right because it doesn\u0027t matter really."},{"Start":"01:10.940 ","End":"01:12.440","Text":"After we\u0027ve got that,"},{"Start":"01:12.440 ","End":"01:15.995","Text":"we then can just put the integral sign in front of each,"},{"Start":"01:15.995 ","End":"01:18.610","Text":"and now we have to start doing integration."},{"Start":"01:18.610 ","End":"01:25.085","Text":"The integral of y with respect to y is just 1/2y^2."},{"Start":"01:25.085 ","End":"01:28.190","Text":"The integral of x with respect to x is 1/2x^2."},{"Start":"01:28.190 ","End":"01:30.845","Text":"But we really need this constant of integration."},{"Start":"01:30.845 ","End":"01:32.090","Text":"Now in a sense,"},{"Start":"01:32.090 ","End":"01:34.040","Text":"you can say at this point that we\u0027ve solved"},{"Start":"01:34.040 ","End":"01:37.460","Text":"the differential equation because there\u0027s no more derivatives."},{"Start":"01:37.460 ","End":"01:39.400","Text":"There is no y\u0027 anymore,"},{"Start":"01:39.400 ","End":"01:40.985","Text":"or dy/dx or anything."},{"Start":"01:40.985 ","End":"01:44.195","Text":"But we don\u0027t have y as the function of x."},{"Start":"01:44.195 ","End":"01:45.725","Text":"This is an implicit form."},{"Start":"01:45.725 ","End":"01:47.510","Text":"Sometimes that\u0027s all we can do,"},{"Start":"01:47.510 ","End":"01:51.605","Text":"but usually we prefer to get y as a function of x."},{"Start":"01:51.605 ","End":"01:56.960","Text":"Ask your professor whether he\u0027s teaching you that if it\u0027s possible,"},{"Start":"01:56.960 ","End":"02:01.625","Text":"it\u0027s okay to leave the solution like this or whether you\u0027re expected to extract y."},{"Start":"02:01.625 ","End":"02:04.054","Text":"In this case, we can, and let\u0027s continue."},{"Start":"02:04.054 ","End":"02:06.980","Text":"What we can do is multiply both sides by 2"},{"Start":"02:06.980 ","End":"02:10.790","Text":"and because 2c is just as much a general constant c,"},{"Start":"02:10.790 ","End":"02:13.534","Text":"then use another letter k for a constant."},{"Start":"02:13.534 ","End":"02:16.000","Text":"The last thing is to take the square root of both sides,"},{"Start":"02:16.000 ","End":"02:18.259","Text":"but remember, we need plus or minus."},{"Start":"02:18.259 ","End":"02:19.925","Text":"This is the general solution."},{"Start":"02:19.925 ","End":"02:22.939","Text":"In fact, there\u0027s like twice infinity solutions,"},{"Start":"02:22.939 ","End":"02:25.875","Text":"each value of k and for each value of k,"},{"Start":"02:25.875 ","End":"02:28.040","Text":"we can still take a plus or a minus."},{"Start":"02:28.040 ","End":"02:30.275","Text":"There\u0027s many solutions in one here."},{"Start":"02:30.275 ","End":"02:32.390","Text":"I could say we\u0027re done, but optionally,"},{"Start":"02:32.390 ","End":"02:33.440","Text":"and if you have the time,"},{"Start":"02:33.440 ","End":"02:36.095","Text":"it\u0027s a good idea to verify the solution."},{"Start":"02:36.095 ","End":"02:38.540","Text":"Here it is again, now let\u0027s verify."},{"Start":"02:38.540 ","End":"02:41.720","Text":"I will want to check whether this equation holds."},{"Start":"02:41.720 ","End":"02:44.570","Text":"I\u0027ll just copy the differential equation."},{"Start":"02:44.570 ","End":"02:49.085","Text":"I need to put a question mark here because we\u0027re now doing the verify part."},{"Start":"02:49.085 ","End":"02:50.630","Text":"The first step, we\u0027ll just leave it with"},{"Start":"02:50.630 ","End":"02:53.255","Text":"prime and then we\u0027ll do the actual derivative in the next step."},{"Start":"02:53.255 ","End":"02:56.825","Text":"I want to know whether this equals and here I put the x from here,"},{"Start":"02:56.825 ","End":"02:58.700","Text":"and again I copy the y here."},{"Start":"02:58.700 ","End":"03:01.760","Text":"From here to here it was just replacing y in"},{"Start":"03:01.760 ","End":"03:05.660","Text":"each case by what it\u0027s equal to with the k and with the plus or minus,"},{"Start":"03:05.660 ","End":"03:09.850","Text":"we might do the general solution and then we can do the differentiation."},{"Start":"03:09.850 ","End":"03:11.720","Text":"This is what we get."},{"Start":"03:11.720 ","End":"03:13.940","Text":"If you\u0027re not sure how I got from here to here,"},{"Start":"03:13.940 ","End":"03:19.550","Text":"let me remind you of a general rule that if I have the square root of some function of x,"},{"Start":"03:19.550 ","End":"03:21.094","Text":"let\u0027s call it box,"},{"Start":"03:21.094 ","End":"03:23.244","Text":"and I want to differentiate it,"},{"Start":"03:23.244 ","End":"03:25.655","Text":"then if it was just square root of x,"},{"Start":"03:25.655 ","End":"03:31.160","Text":"it would be 1 over twice the square root of x only it\u0027s not x,"},{"Start":"03:31.160 ","End":"03:34.550","Text":"it\u0027s box, which is some function of x."},{"Start":"03:34.550 ","End":"03:37.010","Text":"We need to look what\u0027s called the inner derivative."},{"Start":"03:37.010 ","End":"03:38.845","Text":"We need (box)\u0027."},{"Start":"03:38.845 ","End":"03:41.870","Text":"This is the general rule for square roots."},{"Start":"03:41.870 ","End":"03:44.870","Text":"Sometimes you put 1 over and the box to the side."},{"Start":"03:44.870 ","End":"03:51.450","Text":"You could also write it as 1 over twice square root of box, (box)\u0027."},{"Start":"03:51.450 ","End":"03:53.775","Text":"I prefer to put it on the top doesn\u0027t matter."},{"Start":"03:53.775 ","End":"03:57.125","Text":"Here we get 1 over twice the square root."},{"Start":"03:57.125 ","End":"03:58.610","Text":"The plus or minus stays."},{"Start":"03:58.610 ","End":"04:02.340","Text":"Let\u0027s see the plus 1 or minus 1 is the constant times and"},{"Start":"04:02.340 ","End":"04:06.330","Text":"then the derivative of x^2 plus k. It doesn\u0027t matter what k is."},{"Start":"04:06.330 ","End":"04:09.240","Text":"This is going to be 2x, that\u0027s this (box)\u0027."},{"Start":"04:09.240 ","End":"04:10.620","Text":"Here I just copied."},{"Start":"04:10.620 ","End":"04:14.510","Text":"Now if I just arrange this canceling the 2s and putting the x on top,"},{"Start":"04:14.510 ","End":"04:17.150","Text":"then you see that these really are equal,"},{"Start":"04:17.150 ","End":"04:18.614","Text":"2 with the 2,"},{"Start":"04:18.614 ","End":"04:20.570","Text":"the x goes here,"},{"Start":"04:20.570 ","End":"04:23.003","Text":"just like it would have been better to use this form,"},{"Start":"04:23.003 ","End":"04:24.649","Text":"and then we have equality."},{"Start":"04:24.649 ","End":"04:27.410","Text":"This really is a solution."},{"Start":"04:27.410 ","End":"04:32.070","Text":"That\u0027s all. You\u0027ll have plenty more in the solved exercises."}],"Thumbnail":null,"ID":7637},{"Watched":false,"Name":"Exercise 1","Duration":"57s","ChapterTopicVideoID":7574,"CourseChapterTopicPlaylistID":4218,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.865","Text":"We have here a differential equation to solve."},{"Start":"00:02.865 ","End":"00:06.780","Text":"Note that y in the denominator is not a problem because y is"},{"Start":"00:06.780 ","End":"00:10.875","Text":"not 0 and this looks like a case for separation of variables."},{"Start":"00:10.875 ","End":"00:13.860","Text":"Well, first I\u0027ll copy it."},{"Start":"00:13.860 ","End":"00:18.000","Text":"If we cross-multiply, we\u0027ll get y dy equals x^2 dx."},{"Start":"00:18.000 ","End":"00:22.380","Text":"Now, put an integral sign in front of each and this is what we get."},{"Start":"00:22.380 ","End":"00:24.665","Text":"These are both easy integrals to do."},{"Start":"00:24.665 ","End":"00:26.105","Text":"This is y^2/2,"},{"Start":"00:26.105 ","End":"00:28.115","Text":"this is x^3/3,"},{"Start":"00:28.115 ","End":"00:30.905","Text":"and then we have the constant of integration."},{"Start":"00:30.905 ","End":"00:33.110","Text":"Let\u0027s multiply both sides by 2."},{"Start":"00:33.110 ","End":"00:34.730","Text":"We have a constant 2c."},{"Start":"00:34.730 ","End":"00:39.590","Text":"2c is just as much a constant as c so let\u0027s rename the 2c to k, let\u0027s say."},{"Start":"00:39.590 ","End":"00:42.575","Text":"Now, we need to take the square root of both sides."},{"Start":"00:42.575 ","End":"00:49.710","Text":"We get y is plus or minus square root of 2/3x^3 plus k. Actually,"},{"Start":"00:49.710 ","End":"00:51.360","Text":"this is 2 separate solutions."},{"Start":"00:51.360 ","End":"00:55.260","Text":"You could say one solution is with the plus and one solution is with the minus."},{"Start":"00:55.260 ","End":"00:58.060","Text":"In any case, we are done."}],"Thumbnail":null,"ID":7647},{"Watched":false,"Name":"Exercise 2","Duration":"2m 35s","ChapterTopicVideoID":7575,"CourseChapterTopicPlaylistID":4218,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.250","Text":"Here we have a differential equation,"},{"Start":"00:02.250 ","End":"00:06.270","Text":"and it looks like it\u0027s a case for separation of variables."},{"Start":"00:06.270 ","End":"00:10.575","Text":"I\u0027m just rewriting it but instead of y prime, we write dy/dx."},{"Start":"00:10.575 ","End":"00:14.190","Text":"Now, if I bring the dx over to the right-hand side,"},{"Start":"00:14.190 ","End":"00:15.330","Text":"this is what I get,"},{"Start":"00:15.330 ","End":"00:19.860","Text":"and now I can divide by y^2 times 1 minus x,"},{"Start":"00:19.860 ","End":"00:22.230","Text":"and this is what I get."},{"Start":"00:22.230 ","End":"00:25.080","Text":"But pay attention that we have"},{"Start":"00:25.080 ","End":"00:29.430","Text":"denominators here and that y should not be 0 and x should not be 1."},{"Start":"00:29.430 ","End":"00:35.235","Text":"I\u0027ll come back later to this condition and see if this was some restriction or not."},{"Start":"00:35.235 ","End":"00:38.745","Text":"Just put an integral sign in front of each,"},{"Start":"00:38.745 ","End":"00:44.150","Text":"and remember the formula that the integral of function and"},{"Start":"00:44.150 ","End":"00:45.890","Text":"the denominator and its derivative on"},{"Start":"00:45.890 ","End":"00:49.655","Text":"the numerator gives us a natural log of the denominator."},{"Start":"00:49.655 ","End":"00:51.380","Text":"With this in mind,"},{"Start":"00:51.380 ","End":"00:54.320","Text":"my aim is to get this into this form."},{"Start":"00:54.320 ","End":"00:57.650","Text":"But the derivative of the denominator is not the numerator,"},{"Start":"00:57.650 ","End":"01:01.565","Text":"the derivative is minus 1 so with a little trick here,"},{"Start":"01:01.565 ","End":"01:06.485","Text":"I can put a minus here and a minus here."},{"Start":"01:06.485 ","End":"01:12.305","Text":"At the same time, I also put y^minus 2 instead of the denominator."},{"Start":"01:12.305 ","End":"01:14.960","Text":"Now we can do the integration."},{"Start":"01:14.960 ","End":"01:18.215","Text":"This becomes y^minus 1 over minus 1,"},{"Start":"01:18.215 ","End":"01:20.015","Text":"and this becomes this minus,"},{"Start":"01:20.015 ","End":"01:21.364","Text":"and using this formula,"},{"Start":"01:21.364 ","End":"01:26.065","Text":"natural log of 1 minus x and the constant of integration."},{"Start":"01:26.065 ","End":"01:28.850","Text":"Let me just say, this is actually a solution to"},{"Start":"01:28.850 ","End":"01:31.400","Text":"the differential equation because there\u0027s no more derivatives,"},{"Start":"01:31.400 ","End":"01:34.640","Text":"but it\u0027s quite common to isolate y also,"},{"Start":"01:34.640 ","End":"01:36.980","Text":"and if we can do that, then so much the better."},{"Start":"01:36.980 ","End":"01:42.785","Text":"Bringing the minus to the other side and then taking the reciprocal of both sides,"},{"Start":"01:42.785 ","End":"01:44.510","Text":"y is still not 0,"},{"Start":"01:44.510 ","End":"01:46.505","Text":"then reciprocal of both sides,"},{"Start":"01:46.505 ","End":"01:47.705","Text":"that was just rewriting,"},{"Start":"01:47.705 ","End":"01:49.240","Text":"and this is the reciprocal."},{"Start":"01:49.240 ","End":"01:53.190","Text":"This is essentially the end of the exercise,"},{"Start":"01:53.190 ","End":"01:56.490","Text":"but remember we had some conditions before,"},{"Start":"01:56.490 ","End":"01:57.810","Text":"let me go back to it,"},{"Start":"01:57.810 ","End":"02:03.740","Text":"that we insisted that y not be 0 and x not equal to 1 is just part of the domain."},{"Start":"02:03.740 ","End":"02:06.380","Text":"But let\u0027s look at the y not equal to 0."},{"Start":"02:06.380 ","End":"02:08.449","Text":"What if y were equal to 0?"},{"Start":"02:08.449 ","End":"02:10.310","Text":"Let\u0027s check in the original equation,"},{"Start":"02:10.310 ","End":"02:11.510","Text":"it might just work."},{"Start":"02:11.510 ","End":"02:14.630","Text":"But look, if y is the function 0,"},{"Start":"02:14.630 ","End":"02:16.625","Text":"then y^2 is always 0,"},{"Start":"02:16.625 ","End":"02:18.079","Text":"and since it\u0027s a constant,"},{"Start":"02:18.079 ","End":"02:19.970","Text":"y prime is also 0."},{"Start":"02:19.970 ","End":"02:22.610","Text":"We get 0=0, which is true,"},{"Start":"02:22.610 ","End":"02:25.820","Text":"so y=0 is also a solution."},{"Start":"02:25.820 ","End":"02:29.060","Text":"Actually, we have a second solution,"},{"Start":"02:29.060 ","End":"02:31.115","Text":"y=0, I just marked with an asterisk,"},{"Start":"02:31.115 ","End":"02:32.359","Text":"but there are 2 solutions."},{"Start":"02:32.359 ","End":"02:33.425","Text":"This is 1 of them,"},{"Start":"02:33.425 ","End":"02:36.540","Text":"and this is the other, and now we\u0027re done."}],"Thumbnail":null,"ID":7648},{"Watched":false,"Name":"Exercise 3","Duration":"2m 4s","ChapterTopicVideoID":7576,"CourseChapterTopicPlaylistID":4218,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"Here we have a differential equation to solve,"},{"Start":"00:02.850 ","End":"00:06.435","Text":"and it looks like a case for separation of variables."},{"Start":"00:06.435 ","End":"00:08.355","Text":"First, let me copy it,"},{"Start":"00:08.355 ","End":"00:11.640","Text":"but I\u0027m going to put dy over dx instead of y\u0027"},{"Start":"00:11.640 ","End":"00:15.300","Text":"because in separation of variables we need it in this form."},{"Start":"00:15.300 ","End":"00:19.035","Text":"First thing to do is to bring this over to the other side."},{"Start":"00:19.035 ","End":"00:20.940","Text":"Next thing we can get rid of fractions,"},{"Start":"00:20.940 ","End":"00:23.310","Text":"bringing the dx over to the other side."},{"Start":"00:23.310 ","End":"00:24.375","Text":"There it is,"},{"Start":"00:24.375 ","End":"00:26.445","Text":"and the next step will be,"},{"Start":"00:26.445 ","End":"00:27.540","Text":"let me just scroll a bit,"},{"Start":"00:27.540 ","End":"00:32.220","Text":"to divide 1 plus y^2 goes into the denominator here,"},{"Start":"00:32.220 ","End":"00:35.280","Text":"and the square root of 1 plus x^2 goes here."},{"Start":"00:35.280 ","End":"00:37.860","Text":"Now we have a full separation of variables,"},{"Start":"00:37.860 ","End":"00:38.910","Text":"y on the left,"},{"Start":"00:38.910 ","End":"00:40.225","Text":"x on the right."},{"Start":"00:40.225 ","End":"00:42.485","Text":"Put an integral sign in front of each,"},{"Start":"00:42.485 ","End":"00:43.835","Text":"and that\u0027s how we solve it."},{"Start":"00:43.835 ","End":"00:46.115","Text":"Now we have some integrals to compute."},{"Start":"00:46.115 ","End":"00:47.990","Text":"Now, you could do it with substitution,"},{"Start":"00:47.990 ","End":"00:52.595","Text":"but it\u0027s easier if you\u0027re allowed to use the table of integrals."},{"Start":"00:52.595 ","End":"00:58.010","Text":"Then we notice that what we have is we pretty much have this formula where"},{"Start":"00:58.010 ","End":"00:59.750","Text":"if you have a square root of something on"},{"Start":"00:59.750 ","End":"01:03.350","Text":"the denominator and you have the derivative on the numerator,"},{"Start":"01:03.350 ","End":"01:05.825","Text":"then the answer is twice the square root."},{"Start":"01:05.825 ","End":"01:11.180","Text":"Now, we don\u0027t exactly have the derivative of f. If f is 1 plus y^2,"},{"Start":"01:11.180 ","End":"01:12.860","Text":"we would need 2y."},{"Start":"01:12.860 ","End":"01:16.220","Text":"Here, we would need 2x and we have minus x,"},{"Start":"01:16.220 ","End":"01:20.870","Text":"but these are constants that we know the usual tricks and how to adjust that."},{"Start":"01:20.870 ","End":"01:24.800","Text":"What we do is we just put a constant here to make it 2y,"},{"Start":"01:24.800 ","End":"01:26.090","Text":"so it is the derivative."},{"Start":"01:26.090 ","End":"01:28.625","Text":"But then your compensate by putting a 2 down here."},{"Start":"01:28.625 ","End":"01:29.960","Text":"Now we need a 2 here,"},{"Start":"01:29.960 ","End":"01:34.820","Text":"so we put a 2 in the denominator and an extra minus to compensate for the minus here."},{"Start":"01:34.820 ","End":"01:40.970","Text":"Now we can use this formula and what we get is that twice the square root,"},{"Start":"01:40.970 ","End":"01:42.500","Text":"so here\u0027s twice the square root,"},{"Start":"01:42.500 ","End":"01:43.700","Text":"but we have the 1/2 still."},{"Start":"01:43.700 ","End":"01:45.350","Text":"Here we have twice the square root,"},{"Start":"01:45.350 ","End":"01:47.060","Text":"but we still have the minus 1/2."},{"Start":"01:47.060 ","End":"01:49.055","Text":"The 2 with the 2 cancels,"},{"Start":"01:49.055 ","End":"01:54.830","Text":"and so what we get is just that 1 plus y^2 into the square root equals this."},{"Start":"01:54.830 ","End":"01:58.625","Text":"Now, this is actually a solution to the differential equation."},{"Start":"01:58.625 ","End":"02:02.210","Text":"Sometimes we continue and I\u0027ll try and isolate y in terms of x."},{"Start":"02:02.210 ","End":"02:03.290","Text":"But I think in this case,"},{"Start":"02:03.290 ","End":"02:05.670","Text":"we\u0027ll stop right here."}],"Thumbnail":null,"ID":7649},{"Watched":false,"Name":"Exercise 4","Duration":"2m 49s","ChapterTopicVideoID":7565,"CourseChapterTopicPlaylistID":4218,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.110","Text":"In this exercise, we have a differential equation to solve but notice"},{"Start":"00:04.110 ","End":"00:05.580","Text":"that we also have what is called"},{"Start":"00:05.580 ","End":"00:08.715","Text":"an initial condition that will help us find the constant."},{"Start":"00:08.715 ","End":"00:13.335","Text":"I\u0027m going to go for this by the separation of variables method."},{"Start":"00:13.335 ","End":"00:17.580","Text":"I\u0027m going to first of all, multiply by dx to get rid of fractions."},{"Start":"00:17.580 ","End":"00:22.005","Text":"Now let\u0027s bring all the ys to the left and the xs to the right."},{"Start":"00:22.005 ","End":"00:23.555","Text":"But when I do this,"},{"Start":"00:23.555 ","End":"00:27.260","Text":"I get a y in the denominator here and an x minus 1 in"},{"Start":"00:27.260 ","End":"00:31.745","Text":"the denominator here I have to attach conditions and we\u0027ll return to these later."},{"Start":"00:31.745 ","End":"00:36.230","Text":"Let\u0027s continue meanwhile and just put the integral sign in front of each."},{"Start":"00:36.230 ","End":"00:38.450","Text":"But of course, I can take the quarter in front of"},{"Start":"00:38.450 ","End":"00:42.620","Text":"the integral sign and now the actual integration,"},{"Start":"00:42.620 ","End":"00:45.350","Text":"we have the natural log here and here."},{"Start":"00:45.350 ","End":"00:48.575","Text":"Here with a quarter and here with plus constant."},{"Start":"00:48.575 ","End":"00:53.300","Text":"Now, we have an initial condition that y(2) is equal to 1."},{"Start":"00:53.300 ","End":"00:57.860","Text":"What this means is that when x is 2, y is 1."},{"Start":"00:57.860 ","End":"01:02.360","Text":"If I replace y by 1 and x by 2 in this equation,"},{"Start":"01:02.360 ","End":"01:08.015","Text":"what I get is the following and if we evaluate this,"},{"Start":"01:08.015 ","End":"01:12.420","Text":"what we\u0027ll get is that natural log of 1 is 0."},{"Start":"01:12.420 ","End":"01:16.650","Text":"Natural log of 2 minus 1 is also natural log of 1, which is 0."},{"Start":"01:16.650 ","End":"01:20.535","Text":"We get 0 is 0 plus c. Of course c equals 0,"},{"Start":"01:20.535 ","End":"01:22.890","Text":"which means that, well,"},{"Start":"01:22.890 ","End":"01:24.930","Text":"we put c is 0, the c disappears."},{"Start":"01:24.930 ","End":"01:31.910","Text":"This is actually the solution to our differential equation and we haven\u0027t taken y aside,"},{"Start":"01:31.910 ","End":"01:37.640","Text":"but there is a debt that I owe and that is about the assumptions we made over here,"},{"Start":"01:37.640 ","End":"01:41.719","Text":"that x is not equal to 1 and y is not equal to 0."},{"Start":"01:41.719 ","End":"01:44.375","Text":"Well, the x not equal to 1 bit."},{"Start":"01:44.375 ","End":"01:46.055","Text":"This is fair enough,"},{"Start":"01:46.055 ","End":"01:48.830","Text":"we\u0027ll keep this to the end because we definitely don\u0027t"},{"Start":"01:48.830 ","End":"01:52.520","Text":"want x to be 1 so just x is not equal to 1."},{"Start":"01:52.520 ","End":"01:54.020","Text":"It\u0027s a restriction on the domain."},{"Start":"01:54.020 ","End":"01:57.800","Text":"If x is 1, then we get natural log of 0, which is undefined."},{"Start":"01:57.800 ","End":"02:01.895","Text":"But how about the other condition that y is not equal to 0?"},{"Start":"02:01.895 ","End":"02:05.420","Text":"It\u0027s conceivable that y equals 0 is a solution."},{"Start":"02:05.420 ","End":"02:08.180","Text":"Let\u0027s see, we could go to this stage."},{"Start":"02:08.180 ","End":"02:11.540","Text":"I prefer to go right back to the source and check if"},{"Start":"02:11.540 ","End":"02:15.125","Text":"y equals 0 will fulfill this differential equation."},{"Start":"02:15.125 ","End":"02:19.220","Text":"Well, sure it fulfills this differential equation because if y is 0,"},{"Start":"02:19.220 ","End":"02:21.890","Text":"the right-hand side is 0 and if y is 0,"},{"Start":"02:21.890 ","End":"02:23.180","Text":"which is a constant function,"},{"Start":"02:23.180 ","End":"02:26.000","Text":"then dy over dx is 0 so we get 0 equals 0."},{"Start":"02:26.000 ","End":"02:27.170","Text":"This is all very well,"},{"Start":"02:27.170 ","End":"02:32.645","Text":"but the thing is it does not meet the initial condition because if y is 0,"},{"Start":"02:32.645 ","End":"02:35.120","Text":"y of 2 is also 0 not 1."},{"Start":"02:35.120 ","End":"02:37.235","Text":"The 0 disappears so in other words,"},{"Start":"02:37.235 ","End":"02:39.979","Text":"not really a restriction because of the initial condition."},{"Start":"02:39.979 ","End":"02:43.190","Text":"We\u0027re left with the fact with the solution that"},{"Start":"02:43.190 ","End":"02:46.820","Text":"this is it with this restriction that x is not equal to 1,"},{"Start":"02:46.820 ","End":"02:50.280","Text":"which is only natural. We\u0027re done."}],"Thumbnail":null,"ID":7638},{"Watched":false,"Name":"Exercise 5","Duration":"3m 33s","ChapterTopicVideoID":7566,"CourseChapterTopicPlaylistID":4218,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"In this exercise, we\u0027re given a differential equation with"},{"Start":"00:03.210 ","End":"00:05.760","Text":"an initial condition and we\u0027re going to solve"},{"Start":"00:05.760 ","End":"00:08.415","Text":"it using the separation of variables method,"},{"Start":"00:08.415 ","End":"00:10.320","Text":"meaning we want to get everything with y on"},{"Start":"00:10.320 ","End":"00:13.395","Text":"the left side and everything with x on the right."},{"Start":"00:13.395 ","End":"00:15.300","Text":"But here it looks a little mixed up,"},{"Start":"00:15.300 ","End":"00:16.530","Text":"x\u0027s and y\u0027s,"},{"Start":"00:16.530 ","End":"00:18.090","Text":"how are we going to separate it?"},{"Start":"00:18.090 ","End":"00:22.035","Text":"Well, they actually gave us an exercise that we\u0027ll be able to factor."},{"Start":"00:22.035 ","End":"00:28.320","Text":"For example, if we notice here we have an xy and here we have minus 3x."},{"Start":"00:28.320 ","End":"00:32.940","Text":"We could take x out and be left with y minus 3."},{"Start":"00:32.940 ","End":"00:39.530","Text":"Similarly, if we look at this term and this term and we take 3 outside the bracket,"},{"Start":"00:39.530 ","End":"00:42.395","Text":"we\u0027ll also be left with y minus 3."},{"Start":"00:42.395 ","End":"00:48.350","Text":"What I\u0027m suggesting is that we write it this way where the green"},{"Start":"00:48.350 ","End":"00:55.219","Text":"corresponds to the green terms and the yellow corresponds to the yellow terms."},{"Start":"00:55.219 ","End":"00:59.000","Text":"Now we can take y minus 3 outside the brackets."},{"Start":"00:59.000 ","End":"01:02.810","Text":"Let me first erase this highlighting because I want to highlight it differently now."},{"Start":"01:02.810 ","End":"01:06.330","Text":"This is the y minus 3 and the y minus 3,"},{"Start":"01:06.330 ","End":"01:08.595","Text":"that\u0027s common to both."},{"Start":"01:08.595 ","End":"01:14.090","Text":"What we can do is take this same y minus 3 outside the brackets,"},{"Start":"01:14.090 ","End":"01:17.180","Text":"and what we\u0027re left with is the x plus 3 which is here."},{"Start":"01:17.180 ","End":"01:19.400","Text":"We\u0027re getting close to separating the variables."},{"Start":"01:19.400 ","End":"01:23.505","Text":"Let\u0027s just multiply by dx to get rid of fractions."},{"Start":"01:23.505 ","End":"01:29.040","Text":"Now, what we can do is take the y minus 3 to the left."},{"Start":"01:29.040 ","End":"01:36.445","Text":"What we\u0027re left with is 1 over y minus 3 dy is equal to x plus 3 dx."},{"Start":"01:36.445 ","End":"01:38.960","Text":"But because we divided by y minus 3,"},{"Start":"01:38.960 ","End":"01:40.610","Text":"we have to add this condition,"},{"Start":"01:40.610 ","End":"01:42.050","Text":"y not equal to 3,"},{"Start":"01:42.050 ","End":"01:44.360","Text":"and we\u0027ll return to that in due course."},{"Start":"01:44.360 ","End":"01:47.180","Text":"Just put an integral sign in front of each of them."},{"Start":"01:47.180 ","End":"01:51.814","Text":"These are relatively easy integrals, practically immediate."},{"Start":"01:51.814 ","End":"01:54.785","Text":"The integral of 1 over y minus 3,"},{"Start":"01:54.785 ","End":"01:56.180","Text":"it\u0027s like 1 over y."},{"Start":"01:56.180 ","End":"01:57.260","Text":"It\u0027s a natural log."},{"Start":"01:57.260 ","End":"01:59.870","Text":"Here we have a polynomial in short."},{"Start":"01:59.870 ","End":"02:03.650","Text":"This is a natural log of y minus 3, an absolute value."},{"Start":"02:03.650 ","End":"02:07.380","Text":"Here we have x squared over 2 plus 3x plus the constant."},{"Start":"02:07.380 ","End":"02:09.575","Text":"Now what are we going to do about the constant?"},{"Start":"02:09.575 ","End":"02:14.240","Text":"Well, remember that we had an initial condition that y of 1 was minus 1."},{"Start":"02:14.240 ","End":"02:17.300","Text":"We can substitute that when x is 1,"},{"Start":"02:17.300 ","End":"02:19.175","Text":"y is minus 1."},{"Start":"02:19.175 ","End":"02:20.720","Text":"If I put x equals 1,"},{"Start":"02:20.720 ","End":"02:22.670","Text":"y equals minus 1 here,"},{"Start":"02:22.670 ","End":"02:28.140","Text":"then y is minus 1 and x is 1 here and here."},{"Start":"02:28.140 ","End":"02:32.495","Text":"What this gives us is that the constant is equal to,"},{"Start":"02:32.495 ","End":"02:35.255","Text":"the natural log of this minus this,"},{"Start":"02:35.255 ","End":"02:37.660","Text":"an absolute value is natural log of 4,"},{"Start":"02:37.660 ","End":"02:39.980","Text":"and then we take these two over to the other side."},{"Start":"02:39.980 ","End":"02:41.900","Text":"This is 3 and this is a 1/2 together,"},{"Start":"02:41.900 ","End":"02:45.305","Text":"3.5, which goes over to the other side and then we switch sides."},{"Start":"02:45.305 ","End":"02:47.540","Text":"Anyway, this is what c is equal to,"},{"Start":"02:47.540 ","End":"02:49.235","Text":"and we can put c here,"},{"Start":"02:49.235 ","End":"02:52.730","Text":"and then we get that the natural log of y minus 3,"},{"Start":"02:52.730 ","End":"02:55.400","Text":"just copying from here with a 3.5 here."},{"Start":"02:55.400 ","End":"02:57.785","Text":"In effect, this is the answer,"},{"Start":"02:57.785 ","End":"03:04.265","Text":"but we have to go back and see what restriction was that y is not equal to 3."},{"Start":"03:04.265 ","End":"03:06.335","Text":"Let\u0027s just go back and visit it."},{"Start":"03:06.335 ","End":"03:10.250","Text":"You see here we assumed that y is not equal to"},{"Start":"03:10.250 ","End":"03:14.240","Text":"3 and we have to check possibly y equals 3 is a solution,"},{"Start":"03:14.240 ","End":"03:18.140","Text":"but only if enough to go back to the original equation because y equals 3,"},{"Start":"03:18.140 ","End":"03:22.655","Text":"it doesn\u0027t fit with the initial condition that y of 1 equals minus 1,"},{"Start":"03:22.655 ","End":"03:27.004","Text":"it would have to be 3 here in order for it possibly to succeed."},{"Start":"03:27.004 ","End":"03:30.830","Text":"This exclusion y not equal to 3 is acceptable,"},{"Start":"03:30.830 ","End":"03:34.470","Text":"and this is the solution. I\u0027m done."}],"Thumbnail":null,"ID":7639},{"Watched":false,"Name":"Exercise 6","Duration":"3m 3s","ChapterTopicVideoID":7567,"CourseChapterTopicPlaylistID":4218,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.100","Text":"Here we have a differential equation,"},{"Start":"00:02.100 ","End":"00:03.450","Text":"not in the usual form,"},{"Start":"00:03.450 ","End":"00:05.880","Text":"dx and dy appear separately."},{"Start":"00:05.880 ","End":"00:09.585","Text":"This looks like a case for separation of variables."},{"Start":"00:09.585 ","End":"00:12.225","Text":"Let\u0027s see if we can organize something here."},{"Start":"00:12.225 ","End":"00:15.930","Text":"I\u0027m going to combine this term with this term,"},{"Start":"00:15.930 ","End":"00:17.430","Text":"they have y in common."},{"Start":"00:17.430 ","End":"00:21.180","Text":"I\u0027m going to combine this term with this term,"},{"Start":"00:21.180 ","End":"00:23.205","Text":"they have y^2 in common."},{"Start":"00:23.205 ","End":"00:26.760","Text":"I\u0027m going to combine this and this,"},{"Start":"00:26.760 ","End":"00:28.140","Text":"they have the 2 in common,"},{"Start":"00:28.140 ","End":"00:32.150","Text":"and I\u0027m going to combine this with this which have the 4 in common."},{"Start":"00:32.150 ","End":"00:37.565","Text":"What we get is the y from here with x^2 minus 1,"},{"Start":"00:37.565 ","End":"00:40.140","Text":"the 2 here with x^2 minus 1,"},{"Start":"00:40.140 ","End":"00:44.435","Text":"these 2 yellow terms give me the y^2 bit,"},{"Start":"00:44.435 ","End":"00:46.520","Text":"and these 2 give me this."},{"Start":"00:46.520 ","End":"00:51.140","Text":"Now I see that here I have x^2 minus 1 in common to take"},{"Start":"00:51.140 ","End":"00:56.050","Text":"out and here I have x plus 1 in common to take out and then we get this."},{"Start":"00:56.050 ","End":"00:59.835","Text":"Now we are closer to doing separation of variables."},{"Start":"00:59.835 ","End":"01:02.195","Text":"But let\u0027s just factorize some more,"},{"Start":"01:02.195 ","End":"01:07.730","Text":"x^2 minus 1 and y^2 minus 4 can be factorized using the famous difference of squares."},{"Start":"01:07.730 ","End":"01:11.460","Text":"We get, this goes to y minus 2y plus 2,"},{"Start":"01:11.460 ","End":"01:14.290","Text":"and this goes to x minus 1x plus 1."},{"Start":"01:14.290 ","End":"01:16.865","Text":"Now notice that we have some things in common."},{"Start":"01:16.865 ","End":"01:21.830","Text":"For example, the x plus 1 here matches the x plus 1"},{"Start":"01:21.830 ","End":"01:27.245","Text":"here and the y plus 2 here matches the y plus 2 here."},{"Start":"01:27.245 ","End":"01:30.140","Text":"I would like to divide by these,"},{"Start":"01:30.140 ","End":"01:32.795","Text":"but they could be 0."},{"Start":"01:32.795 ","End":"01:36.950","Text":"Let\u0027s assume that x is not equal to minus 1 and y not equal to"},{"Start":"01:36.950 ","End":"01:41.215","Text":"minus 2 and I\u0027ll return to this later to allow for the eventuality."},{"Start":"01:41.215 ","End":"01:45.077","Text":"Then we can cancel the ones that I marked,"},{"Start":"01:45.077 ","End":"01:47.990","Text":"see what we get then, just this."},{"Start":"01:47.990 ","End":"01:51.890","Text":"Now it\u0027s pretty clear how to separate the variables."},{"Start":"01:51.890 ","End":"01:54.530","Text":"Just move the ys onto the right,"},{"Start":"01:54.530 ","End":"01:56.809","Text":"and once we have the variables separated,"},{"Start":"01:56.809 ","End":"01:58.280","Text":"we do an integration."},{"Start":"01:58.280 ","End":"02:01.340","Text":"The integrals are pretty straightforward."},{"Start":"02:01.340 ","End":"02:03.260","Text":"Essentially, this is the solution."},{"Start":"02:03.260 ","End":"02:10.160","Text":"It is not necessary to isolate y in terms of x or vice versa it\u0027s not always possible."},{"Start":"02:10.160 ","End":"02:11.630","Text":"We could leave it at that,"},{"Start":"02:11.630 ","End":"02:16.250","Text":"but there\u0027s still a debt I owe to explain about the restrictions here,"},{"Start":"02:16.250 ","End":"02:20.365","Text":"a restriction on x is not a restriction as such in terms of the solution,"},{"Start":"02:20.365 ","End":"02:23.273","Text":"it\u0027s just restricting the domain of definition."},{"Start":"02:23.273 ","End":"02:26.810","Text":"y = minus 2 is a possibility which we need to check out."},{"Start":"02:26.810 ","End":"02:29.780","Text":"Suppose we have the function y = 2, the constant function."},{"Start":"02:29.780 ","End":"02:31.265","Text":"If we plug it in here,"},{"Start":"02:31.265 ","End":"02:32.855","Text":"y = minus 2,"},{"Start":"02:32.855 ","End":"02:40.070","Text":"then this becomes 0 and this becomes 0 and that satisfies the differential equation."},{"Start":"02:40.070 ","End":"02:46.780","Text":"In actual fact, we get an extra solution and that extra solution is just y = 2."},{"Start":"02:46.780 ","End":"02:52.520","Text":"A solution that arises in this fashion is sometimes called a singular solution."},{"Start":"02:52.520 ","End":"02:54.215","Text":"It\u0027s not a term you have to remember."},{"Start":"02:54.215 ","End":"02:57.920","Text":"Many events we get 2 solutions, the normal,"},{"Start":"02:57.920 ","End":"03:04.050","Text":"so-to-speak solution, and a singular solution. We\u0027re done."}],"Thumbnail":null,"ID":7640},{"Watched":false,"Name":"Exercise 7","Duration":"2m 16s","ChapterTopicVideoID":7568,"CourseChapterTopicPlaylistID":4218,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.890","Text":"Here, we have a differential equation to solve and if it looks a bit different,"},{"Start":"00:04.890 ","End":"00:09.315","Text":"it\u0027s because the independent variable is t and not x,"},{"Start":"00:09.315 ","End":"00:10.485","Text":"that\u0027s no big deal."},{"Start":"00:10.485 ","End":"00:16.575","Text":"I want to do this with separation of variables and just want to emphasize here,"},{"Start":"00:16.575 ","End":"00:21.060","Text":"don\u0027t be confused by the t just means that y is a function of t instead of x and"},{"Start":"00:21.060 ","End":"00:25.635","Text":"separation of variables means we get something containing function of y,"},{"Start":"00:25.635 ","End":"00:28.890","Text":"dy on this side and another function of t, dt."},{"Start":"00:28.890 ","End":"00:32.160","Text":"It just means that all the y\u0027s go on the left and all the t\u0027s go on"},{"Start":"00:32.160 ","End":"00:35.615","Text":"the right and that\u0027s all there is to it, so let\u0027s begin."},{"Start":"00:35.615 ","End":"00:38.120","Text":"I\u0027ll just copy the exercise again,"},{"Start":"00:38.120 ","End":"00:42.080","Text":"and then we can bring the y\u0027s over to the left,"},{"Start":"00:42.080 ","End":"00:44.975","Text":"and then the t\u0027s are already on the right."},{"Start":"00:44.975 ","End":"00:48.695","Text":"There\u0027s no problem with dividing by y^2 plus 4 by the way,"},{"Start":"00:48.695 ","End":"00:52.535","Text":"because it\u0027s always positive and not zero."},{"Start":"00:52.535 ","End":"00:57.860","Text":"Just scroll up a bit and then we can put an integral sign in front of both sides."},{"Start":"00:57.860 ","End":"01:02.045","Text":"The integral here\u0027s given by the formula here."},{"Start":"01:02.045 ","End":"01:03.950","Text":"If we let a=2,"},{"Start":"01:03.950 ","End":"01:05.705","Text":"we\u0027ll get this here."},{"Start":"01:05.705 ","End":"01:07.746","Text":"So what we get is from the formula,"},{"Start":"01:07.746 ","End":"01:12.830","Text":"the left hand side is this and here the integral of 2t is just this."},{"Start":"01:12.830 ","End":"01:15.290","Text":"I could have said just t^2 right away,"},{"Start":"01:15.290 ","End":"01:17.105","Text":"but 2t^2 over 2, fine,"},{"Start":"01:17.105 ","End":"01:22.730","Text":"multiply both sides by 2 and then get rid of this 2 and this 2."},{"Start":"01:22.730 ","End":"01:26.660","Text":"2c is just as much any constant of c,"},{"Start":"01:26.660 ","End":"01:32.615","Text":"so just call it k instead of 2c and now we have an arctangent equation."},{"Start":"01:32.615 ","End":"01:35.960","Text":"Remember, arctangent is the inverse function of tangent."},{"Start":"01:35.960 ","End":"01:41.030","Text":"To say that arctangent of this angle is this is to say that the tangent of this is that."},{"Start":"01:41.030 ","End":"01:43.445","Text":"In other words, here it is a formula,"},{"Start":"01:43.445 ","End":"01:45.320","Text":"the arctangent of a is b,"},{"Start":"01:45.320 ","End":"01:47.120","Text":"then a is the arctangent of b,"},{"Start":"01:47.120 ","End":"01:48.320","Text":"or the other way round,"},{"Start":"01:48.320 ","End":"01:50.450","Text":"never mind, they\u0027re inverse functions."},{"Start":"01:50.450 ","End":"01:54.570","Text":"What we get is that y over 2 is the tangent of 2t^2 plus"},{"Start":"01:54.570 ","End":"01:59.315","Text":"k and if we multiply by 2, we\u0027ve isolated y."},{"Start":"01:59.315 ","End":"02:01.580","Text":"Of course, at this point already,"},{"Start":"02:01.580 ","End":"02:04.740","Text":"we solve the differential equation,"},{"Start":"02:05.500 ","End":"02:10.370","Text":"but the thing to do is often to isolate y in terms of x or t,"},{"Start":"02:10.370 ","End":"02:14.360","Text":"whatever the independent variable is not preferable if possible."},{"Start":"02:14.360 ","End":"02:16.620","Text":"Anyway, we\u0027re done."}],"Thumbnail":null,"ID":7641},{"Watched":false,"Name":"Exercise 8","Duration":"1m 58s","ChapterTopicVideoID":7569,"CourseChapterTopicPlaylistID":4218,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.745","Text":"Here we have to solve a differential equation."},{"Start":"00:02.745 ","End":"00:05.625","Text":"I\u0027m going to do it by separation of variables."},{"Start":"00:05.625 ","End":"00:09.015","Text":"Note that, x is a function of t,"},{"Start":"00:09.015 ","End":"00:11.100","Text":"usually it\u0027s y as a function of x here,"},{"Start":"00:11.100 ","End":"00:13.979","Text":"x is a function of t. To separate"},{"Start":"00:13.979 ","End":"00:17.730","Text":"the variables means we get some expression or function of x,"},{"Start":"00:17.730 ","End":"00:21.075","Text":"dx on the left and some function of tg of t,"},{"Start":"00:21.075 ","End":"00:22.845","Text":"dt on the right."},{"Start":"00:22.845 ","End":"00:25.530","Text":"Just copying the exercise first,"},{"Start":"00:25.530 ","End":"00:27.390","Text":"and the first thing we are doing,"},{"Start":"00:27.390 ","End":"00:32.700","Text":"in this case, is getting this in a more convenient form in anticipation of the integral."},{"Start":"00:32.700 ","End":"00:34.910","Text":"This is a process called completing the square."},{"Start":"00:34.910 ","End":"00:37.580","Text":"We\u0027ve covered this before and we get it in this form."},{"Start":"00:37.580 ","End":"00:39.350","Text":"If you multiply this out,"},{"Start":"00:39.350 ","End":"00:40.580","Text":"you\u0027ll see you get this,"},{"Start":"00:40.580 ","End":"00:43.040","Text":"so we won\u0027t go into more details on that."},{"Start":"00:43.040 ","End":"00:49.760","Text":"Then we multiply both sides by dt and divide both sides by this."},{"Start":"00:49.760 ","End":"00:51.470","Text":"Basically, the x is on the left,"},{"Start":"00:51.470 ","End":"00:53.480","Text":"the t is on the right function of x,"},{"Start":"00:53.480 ","End":"00:54.770","Text":"dx on something,"},{"Start":"00:54.770 ","End":"00:57.275","Text":"well 1dt, it\u0027s a function of t,"},{"Start":"00:57.275 ","End":"01:00.140","Text":"and then put an integral sign in front of both."},{"Start":"01:00.140 ","End":"01:03.060","Text":"Then this integral is not immediately obvious,"},{"Start":"01:03.060 ","End":"01:05.330","Text":"so we rip out the formula sheet,"},{"Start":"01:05.330 ","End":"01:10.055","Text":"find the suitable formula a is equal to 1 here,"},{"Start":"01:10.055 ","End":"01:12.770","Text":"and we choose the minus rather than the plus,"},{"Start":"01:12.770 ","End":"01:18.540","Text":"then what we will get is the arctangent of x-a is x-1."},{"Start":"01:18.540 ","End":"01:21.440","Text":"Then the integral of 1dt is just t,"},{"Start":"01:21.440 ","End":"01:24.320","Text":"and we add the constant to just ones on the right."},{"Start":"01:24.320 ","End":"01:27.769","Text":"That\u0027s pretty much the end of solving a differential equation,"},{"Start":"01:27.769 ","End":"01:29.150","Text":"but it\u0027s not usually enough."},{"Start":"01:29.150 ","End":"01:31.370","Text":"Usually, we can keep going to isolate"},{"Start":"01:31.370 ","End":"01:35.670","Text":"the dependent variable x in terms of the independent variable t, we keep going."},{"Start":"01:35.670 ","End":"01:37.910","Text":"The arctangent is the inverse of tangent,"},{"Start":"01:37.910 ","End":"01:40.100","Text":"and there\u0027s a nice formulas that says that,"},{"Start":"01:40.100 ","End":"01:42.230","Text":"if the arctangent of this is this,"},{"Start":"01:42.230 ","End":"01:45.710","Text":"then the tangent of this is this, inverse functions."},{"Start":"01:45.710 ","End":"01:50.090","Text":"What it means in our case is that x+1 is the tangent of t+c."},{"Start":"01:50.090 ","End":"01:54.530","Text":"We are just 1 step away to saying that x is 1 plus anyway,"},{"Start":"01:54.530 ","End":"01:58.800","Text":"we have x as a function of t and we are done."}],"Thumbnail":null,"ID":7642},{"Watched":false,"Name":"Exercise 9","Duration":"2m 27s","ChapterTopicVideoID":7570,"CourseChapterTopicPlaylistID":4218,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"Here we have a differential equation with"},{"Start":"00:02.550 ","End":"00:07.560","Text":"initial condition and we\u0027ll solve it using the separation of variables method."},{"Start":"00:07.560 ","End":"00:11.790","Text":"Let\u0027s just rewrite the y prime as dy over dx"},{"Start":"00:11.790 ","End":"00:16.395","Text":"and then we can bring the y^2 sine x to the other side."},{"Start":"00:16.395 ","End":"00:19.290","Text":"I want to divide now by y^2."},{"Start":"00:19.290 ","End":"00:22.140","Text":"I have to note that y should not be 0,"},{"Start":"00:22.140 ","End":"00:23.955","Text":"I\u0027ll return to this later."},{"Start":"00:23.955 ","End":"00:27.645","Text":"Then we can put the y^2 over here and the dx over here,"},{"Start":"00:27.645 ","End":"00:30.030","Text":"and our variables are separated, y\u0027s on the left,"},{"Start":"00:30.030 ","End":"00:31.110","Text":"x\u0027s on the right,"},{"Start":"00:31.110 ","End":"00:33.180","Text":"and now we can just integrate."},{"Start":"00:33.180 ","End":"00:36.045","Text":"These are both straightforward integrals."},{"Start":"00:36.045 ","End":"00:38.260","Text":"I\u0027ll write this is y^(-2),"},{"Start":"00:38.260 ","End":"00:42.620","Text":"and then we can see that this is just y^(-1) over minus 1,"},{"Start":"00:42.620 ","End":"00:45.124","Text":"and the integral of minus sine is cosine,"},{"Start":"00:45.124 ","End":"00:47.600","Text":"but must have a constant of integration."},{"Start":"00:47.600 ","End":"00:51.635","Text":"This is now the result of the differential equation,"},{"Start":"00:51.635 ","End":"00:55.654","Text":"but it\u0027s customary to isolate y if possible. Let\u0027s continue."},{"Start":"00:55.654 ","End":"00:58.915","Text":"Just rewrite this as minus 1 over y."},{"Start":"00:58.915 ","End":"01:02.030","Text":"Then we can do the inverse reciprocal."},{"Start":"01:02.030 ","End":"01:04.880","Text":"First of all, if you multiply it by y and divide by"},{"Start":"01:04.880 ","End":"01:08.105","Text":"cosine x plus c and change sides this is what you get."},{"Start":"01:08.105 ","End":"01:09.905","Text":"We still have a constant here,"},{"Start":"01:09.905 ","End":"01:14.255","Text":"but we do have an initial condition that when x is Pi,"},{"Start":"01:14.255 ","End":"01:15.935","Text":"y is equal to 1."},{"Start":"01:15.935 ","End":"01:18.890","Text":"If I put 1 here and Pi here,"},{"Start":"01:18.890 ","End":"01:20.180","Text":"so Pi instead of x,"},{"Start":"01:20.180 ","End":"01:23.120","Text":"1 instead of y, that should hold true and easy to"},{"Start":"01:23.120 ","End":"01:26.210","Text":"see that this gives us that c equals 0."},{"Start":"01:26.210 ","End":"01:32.660","Text":"Because cosine of Pi is minus 1 and you can see it was 1 over minus 1 and a minus is 1."},{"Start":"01:32.660 ","End":"01:39.065","Text":"Now we can put c in the answer and get that y is just minus 1 over cosine x."},{"Start":"01:39.065 ","End":"01:41.480","Text":"But before we finish,"},{"Start":"01:41.480 ","End":"01:47.615","Text":"we got to remember that there\u0027s a data that I said I\u0027d go back to the y not equal to 0."},{"Start":"01:47.615 ","End":"01:51.785","Text":"Now we have to check the possibility what happens if y equals 0."},{"Start":"01:51.785 ","End":"01:55.430","Text":"Well, we don\u0027t have to go all the way back to the original equation, though we could."},{"Start":"01:55.430 ","End":"01:58.145","Text":"In fact, if y equals 0,"},{"Start":"01:58.145 ","End":"02:00.240","Text":"then y prime is 0,"},{"Start":"02:00.240 ","End":"02:01.500","Text":"and y is 0, and it works."},{"Start":"02:01.500 ","End":"02:02.880","Text":"Or you could see it here,"},{"Start":"02:02.880 ","End":"02:04.890","Text":"y is 0, dy, dx is 0,"},{"Start":"02:04.890 ","End":"02:06.005","Text":"y is 0, it works."},{"Start":"02:06.005 ","End":"02:12.830","Text":"However, the solution y equals 0 would not fit the initial condition."},{"Start":"02:12.830 ","End":"02:15.290","Text":"We thought we might have had a singular solution,"},{"Start":"02:15.290 ","End":"02:16.685","Text":"but no, we don\u0027t."},{"Start":"02:16.685 ","End":"02:21.260","Text":"The only solution is therefore, here it is."},{"Start":"02:21.260 ","End":"02:23.765","Text":"This is the only solution."},{"Start":"02:23.765 ","End":"02:27.630","Text":"That is it. Highlighted, we are done."}],"Thumbnail":null,"ID":7643},{"Watched":false,"Name":"Exercise 10","Duration":"2m 12s","ChapterTopicVideoID":7571,"CourseChapterTopicPlaylistID":4218,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.870","Text":"Here we have a differential equation with an initial condition."},{"Start":"00:03.870 ","End":"00:07.185","Text":"I want to solve it using separation of variables."},{"Start":"00:07.185 ","End":"00:12.615","Text":"I\u0027m just going to rewrite this as some people don\u0027t know what secant is."},{"Start":"00:12.615 ","End":"00:14.610","Text":"Secant is 1/cos."},{"Start":"00:14.610 ","End":"00:17.135","Text":"I\u0027ll just write it in the form everyone knows,"},{"Start":"00:17.135 ","End":"00:21.390","Text":"1/cos^2x, and now I want to separate the variables."},{"Start":"00:21.390 ","End":"00:23.445","Text":"I want to bring the dx over here,"},{"Start":"00:23.445 ","End":"00:26.295","Text":"and then I want to put the y down there."},{"Start":"00:26.295 ","End":"00:29.190","Text":"That\u0027s right. We\u0027ll have the x\u0027s on the right and the y\u0027s on the left."},{"Start":"00:29.190 ","End":"00:30.720","Text":"Of course, if we divide by y,"},{"Start":"00:30.720 ","End":"00:33.450","Text":"we have to add a condition that y≠0,"},{"Start":"00:33.450 ","End":"00:36.120","Text":"and I\u0027ll return to this condition at the end."},{"Start":"00:36.120 ","End":"00:39.180","Text":"Meanwhile, we have already the separation so we can integrate,"},{"Start":"00:39.180 ","End":"00:42.825","Text":"and these are 2 fairly straightforward integrals."},{"Start":"00:42.825 ","End":"00:47.870","Text":"They\u0027re immediate, 1/cos^2 happens to be the tangent."},{"Start":"00:47.870 ","End":"00:50.681","Text":"You might not remember that\u0027s on the formula sheet. It\u0027s an immediate."},{"Start":"00:50.681 ","End":"00:53.840","Text":"The integral of 1/y is natural log of absolute value."},{"Start":"00:53.840 ","End":"00:56.315","Text":"At this point, plug in the original,"},{"Start":"00:56.315 ","End":"00:58.610","Text":"I mean the initial condition. Let\u0027s see what it was."},{"Start":"00:58.610 ","End":"01:02.960","Text":"It was the y(0)=5."},{"Start":"01:02.960 ","End":"01:04.205","Text":"That means if here,"},{"Start":"01:04.205 ","End":"01:07.520","Text":"I put x is 0, y is 5."},{"Start":"01:07.520 ","End":"01:15.520","Text":"We get the natural log of 5 is tangent of 0 + c. Now tangent of 0 is 0,"},{"Start":"01:15.520 ","End":"01:20.270","Text":"so that gives us that c=ln5,"},{"Start":"01:20.270 ","End":"01:22.475","Text":"and that means if we put the c back in here,"},{"Start":"01:22.475 ","End":"01:25.670","Text":"then we get that the answer is this."},{"Start":"01:25.670 ","End":"01:29.225","Text":"We could continue and isolate y."},{"Start":"01:29.225 ","End":"01:31.265","Text":"We can leave it at this also."},{"Start":"01:31.265 ","End":"01:38.015","Text":"Now, let\u0027s get back to the condition that we had that y≠0."},{"Start":"01:38.015 ","End":"01:41.360","Text":"Well, supposing that y=0."},{"Start":"01:41.360 ","End":"01:43.220","Text":"Let\u0027s say y is the 0 function."},{"Start":"01:43.220 ","End":"01:45.485","Text":"Let\u0027s go back up and see what would happen."},{"Start":"01:45.485 ","End":"01:49.635","Text":"If y is 0, then dy/dx is also 0."},{"Start":"01:49.635 ","End":"01:53.990","Text":"In actual fact, we do get a solution to the equation."},{"Start":"01:53.990 ","End":"01:57.770","Text":"However, the 0 solution does not fulfill"},{"Start":"01:57.770 ","End":"02:02.270","Text":"the initial condition because if y is the constant function 0, the y(0) is 0."},{"Start":"02:02.270 ","End":"02:03.470","Text":"So that doesn\u0027t work,"},{"Start":"02:03.470 ","End":"02:07.880","Text":"which means that the only solution we have is the one over here,"},{"Start":"02:07.880 ","End":"02:12.720","Text":"and this is the answer and we are done."}],"Thumbnail":null,"ID":7644},{"Watched":false,"Name":"Exercise 11","Duration":"2m 47s","ChapterTopicVideoID":7572,"CourseChapterTopicPlaylistID":4218,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"Here we have a differential equation with"},{"Start":"00:02.580 ","End":"00:07.155","Text":"an initial condition and we\u0027re going to solve it with separation of variables."},{"Start":"00:07.155 ","End":"00:10.465","Text":"Let me first multiply out to get rid of fractions."},{"Start":"00:10.465 ","End":"00:15.615","Text":"I get the dy times this equals this thing times dx."},{"Start":"00:15.615 ","End":"00:18.750","Text":"Then we can bring the xs onto the right."},{"Start":"00:18.750 ","End":"00:20.369","Text":"This thing goes in the denominator,"},{"Start":"00:20.369 ","End":"00:22.290","Text":"it\u0027s never 0, so we\u0027re okay,"},{"Start":"00:22.290 ","End":"00:24.480","Text":"y^3 over here to the denominator,"},{"Start":"00:24.480 ","End":"00:28.080","Text":"but it could be 0 so we have to warn the y is not equal to 0,"},{"Start":"00:28.080 ","End":"00:29.864","Text":"and I\u0027ll return to this later."},{"Start":"00:29.864 ","End":"00:32.085","Text":"Just put an integral sign in front now,"},{"Start":"00:32.085 ","End":"00:33.750","Text":"and we have 2 integrals."},{"Start":"00:33.750 ","End":"00:37.510","Text":"We need some formulas in case you\u0027ve forgotten, it was straightforward."},{"Start":"00:37.510 ","End":"00:40.020","Text":"I\u0027m going to use this formula for"},{"Start":"00:40.020 ","End":"00:44.400","Text":"the right-hand side and for the left-hand side, it\u0027s simpler."},{"Start":"00:44.400 ","End":"00:47.810","Text":"This says that if we have the square root of"},{"Start":"00:47.810 ","End":"00:51.139","Text":"a function on the denominator and its derivative in the numerator,"},{"Start":"00:51.139 ","End":"00:53.149","Text":"then this is the integral."},{"Start":"00:53.149 ","End":"00:56.105","Text":"The thing is if I take f is 1 plus x^2,"},{"Start":"00:56.105 ","End":"00:59.930","Text":"I would want to 2x on the numerator and not x."},{"Start":"00:59.930 ","End":"01:04.850","Text":"But that is easy enough to fix because we can just compensate,"},{"Start":"01:04.850 ","End":"01:07.775","Text":"put a 2 here and then compensate with a 2 here."},{"Start":"01:07.775 ","End":"01:10.280","Text":"Then we do have if 1 plus x^2 is f,"},{"Start":"01:10.280 ","End":"01:12.500","Text":"we do have f\u0027 in the numerator."},{"Start":"01:12.500 ","End":"01:13.699","Text":"At the same opportunity,"},{"Start":"01:13.699 ","End":"01:19.985","Text":"I wrote 1 over y^3 is y^-3 because when you use the usual exponent function formula."},{"Start":"01:19.985 ","End":"01:24.250","Text":"What we get is raise the power by 1 minus 2 and divide by it."},{"Start":"01:24.250 ","End":"01:28.865","Text":"Here we have from the formula twice the square root of f,"},{"Start":"01:28.865 ","End":"01:32.510","Text":"the 1/2 with the 2 cancels and we just get,"},{"Start":"01:32.510 ","End":"01:35.480","Text":"and we put the y^-2 into the denominator,"},{"Start":"01:35.480 ","End":"01:37.490","Text":"and this is what we get."},{"Start":"01:37.490 ","End":"01:44.540","Text":"Now there wasn\u0027t an initial condition that y(0)=1 which means that x is 0, y is 1."},{"Start":"01:44.540 ","End":"01:46.100","Text":"Sometimes people get it backwards."},{"Start":"01:46.100 ","End":"01:48.350","Text":"Put x as 0, y is 1,"},{"Start":"01:48.350 ","End":"01:50.105","Text":"and we get this,"},{"Start":"01:50.105 ","End":"01:54.470","Text":"and this gives us immediately that c is -1.5."},{"Start":"01:54.470 ","End":"01:56.180","Text":"You can do the arithmetic."},{"Start":"01:56.180 ","End":"02:00.450","Text":"That means if we put c back here as -1.5,"},{"Start":"02:00.450 ","End":"02:04.730","Text":"we get that this is the answer to the differential equation."},{"Start":"02:04.730 ","End":"02:07.520","Text":"We\u0027ll leave it at this and not try to isolate y."},{"Start":"02:07.520 ","End":"02:12.150","Text":"But there is a debt I owe you about the y=0 thing."},{"Start":"02:12.150 ","End":"02:14.540","Text":"Where it was it? Yes, here it is."},{"Start":"02:14.540 ","End":"02:16.565","Text":"What if we did take y=0?"},{"Start":"02:16.565 ","End":"02:18.725","Text":"The function y=0,"},{"Start":"02:18.725 ","End":"02:20.645","Text":"will it satisfy the equation?"},{"Start":"02:20.645 ","End":"02:25.565","Text":"Well, yes, because if y=0, then dy=0 and also y^3=0."},{"Start":"02:25.565 ","End":"02:31.415","Text":"However, the big thing is about y=0 is that it doesn\u0027t satisfy the initial condition."},{"Start":"02:31.415 ","End":"02:32.900","Text":"If we take the function 0,"},{"Start":"02:32.900 ","End":"02:35.720","Text":"then it\u0027s always 0 and it\u0027s not equal to 1 so that\u0027s ruled out."},{"Start":"02:35.720 ","End":"02:37.805","Text":"That would\u0027ve been a singular solution."},{"Start":"02:37.805 ","End":"02:40.400","Text":"The only solution that we have left is"},{"Start":"02:40.400 ","End":"02:43.715","Text":"this one that I wrote at the bottom and let me highlight it."},{"Start":"02:43.715 ","End":"02:47.820","Text":"This is our solution and we are done."}],"Thumbnail":null,"ID":7645},{"Watched":false,"Name":"Exercise 12","Duration":"4m 5s","ChapterTopicVideoID":7573,"CourseChapterTopicPlaylistID":4218,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"Here we have a word problem which will lead"},{"Start":"00:02.910 ","End":"00:05.790","Text":"us to a differential equation. Let\u0027s just read it."},{"Start":"00:05.790 ","End":"00:07.980","Text":"Let y of t denote the amount of"},{"Start":"00:07.980 ","End":"00:11.925","Text":"substance or population that is either growing or decaying."},{"Start":"00:11.925 ","End":"00:14.865","Text":"We assume that the time rate of change, this amount,"},{"Start":"00:14.865 ","End":"00:16.785","Text":"that means the derivative with respect to t,"},{"Start":"00:16.785 ","End":"00:20.850","Text":"this amount of substance is proportional to the amount of substance present."},{"Start":"00:20.850 ","End":"00:23.895","Text":"That implies that it grows, decay is exponentially."},{"Start":"00:23.895 ","End":"00:27.770","Text":"We\u0027re also given an initial condition that the start time t is 0,"},{"Start":"00:27.770 ","End":"00:30.725","Text":"the amount is given by y naught."},{"Start":"00:30.725 ","End":"00:33.455","Text":"We\u0027ve got to find the formula for the amount of any given time"},{"Start":"00:33.455 ","End":"00:36.500","Text":"t. The thing to focus on is first of all,"},{"Start":"00:36.500 ","End":"00:38.270","Text":"that we have a function y of t,"},{"Start":"00:38.270 ","End":"00:41.810","Text":"which is the amount as a function of time and the time rate of"},{"Start":"00:41.810 ","End":"00:45.555","Text":"change is dy/dt or y prime."},{"Start":"00:45.555 ","End":"00:49.460","Text":"That\u0027s one thing. We\u0027re given dy/dt and proportional"},{"Start":"00:49.460 ","End":"00:54.750","Text":"to means a constant times the amount of substance present is just y."},{"Start":"00:54.750 ","End":"00:58.365","Text":"Well, not writing, I\u0027m just exposing,"},{"Start":"00:58.365 ","End":"01:00.290","Text":"but dy/dt, as I say,"},{"Start":"01:00.290 ","End":"01:03.380","Text":"the time rate of change is proportional to y itself,"},{"Start":"01:03.380 ","End":"01:04.850","Text":"so it\u0027s k times y."},{"Start":"01:04.850 ","End":"01:07.265","Text":"The initial condition that when t is 0,"},{"Start":"01:07.265 ","End":"01:14.030","Text":"that means y when t is 0 is given by y naught and y is positive."},{"Start":"01:14.030 ","End":"01:17.420","Text":"If we find that we have a positive or negative solution,"},{"Start":"01:17.420 ","End":"01:18.740","Text":"we\u0027ll take the positive one."},{"Start":"01:18.740 ","End":"01:21.755","Text":"As I said, k is the constant of proportionality."},{"Start":"01:21.755 ","End":"01:26.270","Text":"I should have said,"},{"Start":"01:26.270 ","End":"01:28.565","Text":"I\u0027m going to do it by separation of variables."},{"Start":"01:28.565 ","End":"01:31.675","Text":"I want to get y\u0027s on the left and t\u0027s on the right."},{"Start":"01:31.675 ","End":"01:33.748","Text":"Y on the left is here,"},{"Start":"01:33.748 ","End":"01:36.860","Text":"and t on the right means I just have to bring the dt"},{"Start":"01:36.860 ","End":"01:41.030","Text":"over and it goes with the k. But since I divided by y,"},{"Start":"01:41.030 ","End":"01:44.764","Text":"I have to remember that y must not equal to 0,"},{"Start":"01:44.764 ","End":"01:48.440","Text":"but it\u0027s already says here that y has to be strictly positive,"},{"Start":"01:48.440 ","End":"01:49.730","Text":"so y is not 0."},{"Start":"01:49.730 ","End":"01:52.715","Text":"At the end, we need to check that this is indeed so."},{"Start":"01:52.715 ","End":"01:54.530","Text":"Once we have this separation,"},{"Start":"01:54.530 ","End":"01:58.340","Text":"then the next thing is just to put an integration sign in front of it."},{"Start":"01:58.340 ","End":"02:01.430","Text":"Now the integral of this is fairly straightforward."},{"Start":"02:01.430 ","End":"02:05.165","Text":"This is natural log and the integral of k as kt."},{"Start":"02:05.165 ","End":"02:12.140","Text":"We put the constant instead of as a regular constant as natural log of a constant."},{"Start":"02:12.140 ","End":"02:15.280","Text":"This is sometimes what we do when we have a natural log."},{"Start":"02:15.280 ","End":"02:17.150","Text":"You\u0027ll see it works out well,"},{"Start":"02:17.150 ","End":"02:24.595","Text":"because any number positive or negative could be the natural log of some positive number."},{"Start":"02:24.595 ","End":"02:28.115","Text":"I may be too early to uncover this spoiler,"},{"Start":"02:28.115 ","End":"02:29.353","Text":"just going to use a formula here."},{"Start":"02:29.353 ","End":"02:33.470","Text":"Because what I\u0027m going to do is take the natural log of C over to the other side."},{"Start":"02:33.470 ","End":"02:36.335","Text":"I\u0027ll get log of something minus log of something,"},{"Start":"02:36.335 ","End":"02:38.330","Text":"and that\u0027s the log of the quotient and that soul,"},{"Start":"02:38.330 ","End":"02:39.575","Text":"this goes to say."},{"Start":"02:39.575 ","End":"02:44.025","Text":"So we have the natural logarithm of y over C is kt."},{"Start":"02:44.025 ","End":"02:47.270","Text":"But then I want to get rid of the natural logarithm."},{"Start":"02:47.270 ","End":"02:49.145","Text":"I want to try and isolate y."},{"Start":"02:49.145 ","End":"02:51.110","Text":"There is a another formula,"},{"Start":"02:51.110 ","End":"02:52.340","Text":"it\u0027s almost the definition."},{"Start":"02:52.340 ","End":"02:58.340","Text":"The natural log of A is B means that the exponent of B is A. I can"},{"Start":"02:58.340 ","End":"03:01.580","Text":"rewrite this pretty much by definition that y over"},{"Start":"03:01.580 ","End":"03:05.135","Text":"C is the exponent of e to the power of,"},{"Start":"03:05.135 ","End":"03:09.680","Text":"then we just bring you over to the other side and we have the answer almost,"},{"Start":"03:09.680 ","End":"03:12.710","Text":"but we still have to find out what the constant C is."},{"Start":"03:12.710 ","End":"03:14.990","Text":"That\u0027s where the initial condition comes in,"},{"Start":"03:14.990 ","End":"03:16.490","Text":"that if we put t is 0,"},{"Start":"03:16.490 ","End":"03:18.410","Text":"then y is y is 0."},{"Start":"03:18.410 ","End":"03:21.740","Text":"That gives us that y_0 here,"},{"Start":"03:21.740 ","End":"03:24.455","Text":"t equals 0 here 0 times k is 0."},{"Start":"03:24.455 ","End":"03:25.670","Text":"This is what we get."},{"Start":"03:25.670 ","End":"03:27.260","Text":"Now, e^0 is 1,"},{"Start":"03:27.260 ","End":"03:29.840","Text":"so that y_0 is just C,"},{"Start":"03:29.840 ","End":"03:33.770","Text":"and then if I put that back in the formula here,"},{"Start":"03:33.770 ","End":"03:37.400","Text":"then I will get that y is y_0,"},{"Start":"03:37.400 ","End":"03:39.320","Text":"which is Ce^kt,"},{"Start":"03:39.320 ","End":"03:40.745","Text":"and that is the answer."},{"Start":"03:40.745 ","End":"03:44.030","Text":"We still just have to go back and check about the y equals"},{"Start":"03:44.030 ","End":"03:47.510","Text":"0 business or that y is positive,"},{"Start":"03:47.510 ","End":"03:50.630","Text":"we divide it by y over here,"},{"Start":"03:50.630 ","End":"03:53.840","Text":"and we said we have to check that y is positive."},{"Start":"03:53.840 ","End":"03:56.000","Text":"Let\u0027s see, let\u0027s go down here."},{"Start":"03:56.000 ","End":"04:00.890","Text":"Now y is 0 was given to be positive and e to the power of is always positive,"},{"Start":"04:00.890 ","End":"04:03.590","Text":"so y is indeed positive so everything\u0027s okay,"},{"Start":"04:03.590 ","End":"04:06.450","Text":"we\u0027ve got our answer and I\u0027m done."}],"Thumbnail":null,"ID":7646},{"Watched":false,"Name":"Exercise 13","Duration":"11m 3s","ChapterTopicVideoID":7577,"CourseChapterTopicPlaylistID":4218,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.875","Text":"In this exercise, we have an exponential growth problem involving population."},{"Start":"00:07.875 ","End":"00:14.864","Text":"We\u0027re given that it grows at a rate of 2 percent per year,"},{"Start":"00:14.864 ","End":"00:16.740","Text":"we assume it\u0027s constant,"},{"Start":"00:16.740 ","End":"00:19.500","Text":"of course every year it\u0027s different and it\u0027s changing."},{"Start":"00:19.500 ","End":"00:22.080","Text":"But, this is just a simplified model,"},{"Start":"00:22.080 ","End":"00:27.405","Text":"and we say 2 percent every year since 1980."},{"Start":"00:27.405 ","End":"00:31.090","Text":"At that time it was 4 billion."},{"Start":"00:31.130 ","End":"00:34.250","Text":"We have 3 questions."},{"Start":"00:34.250 ","End":"00:39.590","Text":"What will the population of the earth be in 2020?"},{"Start":"00:39.590 ","End":"00:43.270","Text":"Assuming that continues at this rate."},{"Start":"00:43.270 ","End":"00:47.450","Text":"Then going backwards before 1980,"},{"Start":"00:47.450 ","End":"00:51.950","Text":"we can estimate what was it in 1974, again,"},{"Start":"00:51.950 ","End":"00:58.360","Text":"assuming that it was still growing at 2 percent back in \u002774."},{"Start":"00:58.360 ","End":"01:05.740","Text":"The third question, doesn\u0027t give us a year it asks for a target population of 50 billion."},{"Start":"01:05.740 ","End":"01:08.645","Text":"We\u0027re asking, what year would that be reached?"},{"Start":"01:08.645 ","End":"01:12.530","Text":"Again, assuming that this growth rate continues."},{"Start":"01:12.530 ","End":"01:16.505","Text":"There are 1 or 2 depending how you look at it."},{"Start":"01:16.505 ","End":"01:20.850","Text":"This is the main formula that we\u0027re going to use."},{"Start":"01:21.730 ","End":"01:27.320","Text":"But, I\u0027ll explain there are 2 measures of growth."},{"Start":"01:27.320 ","End":"01:29.930","Text":"One of them is the percentage method in this case,"},{"Start":"01:29.930 ","End":"01:37.130","Text":"for example, we would say that P is 2 because it\u0027s the percent."},{"Start":"01:37.130 ","End":"01:42.350","Text":"But, it\u0027s more convenient to work with factors than with percentages."},{"Start":"01:42.350 ","End":"01:44.460","Text":"If I take a P is 2,"},{"Start":"01:44.460 ","End":"01:47.000","Text":"like if I had a 100 at the end of the year,"},{"Start":"01:47.000 ","End":"01:53.375","Text":"I\u0027d have 102, and 102/100 is 1.02."},{"Start":"01:53.375 ","End":"01:58.753","Text":"We get used to going back and forth between percentages,"},{"Start":"01:58.753 ","End":"02:04.195","Text":"and factors as it with the growth factor 1.02,"},{"Start":"02:04.195 ","End":"02:08.720","Text":"you multiply the population by each year to get the next year\u0027s population."},{"Start":"02:08.720 ","End":"02:12.095","Text":"It\u0027s this Q that we use in this formula."},{"Start":"02:12.095 ","End":"02:14.780","Text":"I\u0027m using letter M, though."},{"Start":"02:14.780 ","End":"02:18.890","Text":"Other letters are used if we\u0027re talking about radioactivity and stuff,"},{"Start":"02:18.890 ","End":"02:22.520","Text":"they often use the letter Q for quantity."},{"Start":"02:22.520 ","End":"02:23.990","Text":"M for amount."},{"Start":"02:23.990 ","End":"02:25.520","Text":"Yeah, amount begins with an a,"},{"Start":"02:25.520 ","End":"02:28.355","Text":"but M sounds like amount."},{"Start":"02:28.355 ","End":"02:34.775","Text":"Anyway, this is the amount after time t years."},{"Start":"02:34.775 ","End":"02:39.905","Text":"M naught is the initial amount."},{"Start":"02:39.905 ","End":"02:44.550","Text":"Naught here corresponds to 1980."},{"Start":"02:44.570 ","End":"02:48.770","Text":"If we\u0027re using 1980 as a baseline,"},{"Start":"02:48.770 ","End":"02:51.785","Text":"time is 0 in 1980,"},{"Start":"02:51.785 ","End":"02:58.695","Text":"then the year 2020 is the year where t is 40 years."},{"Start":"02:58.695 ","End":"03:02.850","Text":"Subtract this minus this, and 1974,"},{"Start":"03:02.850 ","End":"03:10.345","Text":"t was minus 6 years before the start time."},{"Start":"03:10.345 ","End":"03:14.460","Text":"When we solve C and we get a value of t,"},{"Start":"03:14.460 ","End":"03:23.300","Text":"we have to remember at the end to add that value of t to 1980 to get the actual year."},{"Start":"03:23.300 ","End":"03:26.254","Text":"We\u0027re 1980 based."},{"Start":"03:26.254 ","End":"03:28.040","Text":"That\u0027s a lot of introduction."},{"Start":"03:28.040 ","End":"03:32.045","Text":"Let\u0027s start."},{"Start":"03:32.045 ","End":"03:34.740","Text":"Let\u0027s say this is the formula."},{"Start":"03:35.350 ","End":"03:40.665","Text":"Time t is M naught times q^t."},{"Start":"03:40.665 ","End":"03:43.815","Text":"In our problem we\u0027ve already found Q."},{"Start":"03:43.815 ","End":"03:46.960","Text":"In part A,"},{"Start":"03:48.590 ","End":"03:56.190","Text":"we know that t=40."},{"Start":"03:56.190 ","End":"03:58.950","Text":"In all the parts,"},{"Start":"03:58.950 ","End":"04:01.331","Text":"Q is 1.02,"},{"Start":"04:01.331 ","End":"04:03.915","Text":"that doesn\u0027t change,"},{"Start":"04:03.915 ","End":"04:07.140","Text":"and in all the parts,"},{"Start":"04:07.140 ","End":"04:11.640","Text":"M naught 4 billion."},{"Start":"04:11.640 ","End":"04:14.865","Text":"I\u0027ll just write it in words."},{"Start":"04:14.865 ","End":"04:20.890","Text":"We could write it out as 4 with 9 zeros."},{"Start":"04:22.390 ","End":"04:26.120","Text":"We\u0027ve got everything that we need."},{"Start":"04:26.120 ","End":"04:29.490","Text":"We just need to find what is M,"},{"Start":"04:29.530 ","End":"04:36.425","Text":"the amount, population at time t by substituting in the formula."},{"Start":"04:36.425 ","End":"04:39.150","Text":"It\u0027s 4 billion,"},{"Start":"04:40.100 ","End":"04:43.680","Text":"I think I\u0027ll just write it out with 9 zeros,"},{"Start":"04:43.680 ","End":"04:49.020","Text":"1,2,3,4,5,6,7,8,9"},{"Start":"04:49.020 ","End":"04:51.280","Text":"times"},{"Start":"04:51.680 ","End":"05:02.415","Text":"1.02^40."},{"Start":"05:02.415 ","End":"05:05.395","Text":"See what that comes out to."},{"Start":"05:05.395 ","End":"05:09.080","Text":"Obviously, we need a calculator for this."},{"Start":"05:09.960 ","End":"05:13.735","Text":"This is what I get on my calculator,"},{"Start":"05:13.735 ","End":"05:23.330","Text":"and of course we don\u0027t put any decimals if we round off to a whole person."},{"Start":"05:24.220 ","End":"05:30.875","Text":"I\u0027ll just comment that it looks like it\u0027s more than doubled and 40 years."},{"Start":"05:30.875 ","End":"05:37.970","Text":"On to part b, very similar."},{"Start":"05:37.970 ","End":"05:45.035","Text":"The only difference is that this time t= minus 6,"},{"Start":"05:45.035 ","End":"05:49.655","Text":"and all the other numbers are the same."},{"Start":"05:49.655 ","End":"05:54.450","Text":"Basically we get that the amount,"},{"Start":"05:54.830 ","End":"05:59.400","Text":"actually I should have written this as not just a general"},{"Start":"05:59.400 ","End":"06:04.200","Text":"t. Should have written this as M40."},{"Start":"06:04.200 ","End":"06:09.695","Text":"Where they say t is time measured from baseline 1980."},{"Start":"06:09.695 ","End":"06:14.340","Text":"Now we want time minus 6,"},{"Start":"06:14.340 ","End":"06:18.010","Text":"which is what our 1974 is."},{"Start":"06:18.010 ","End":"06:24.870","Text":"Exactly the same formula except instead of 40 we put minus 6, so it\u0027s 4,000,000,000."},{"Start":"06:27.160 ","End":"06:37.115","Text":"Put some commas in times 1.02 to the power of minus 6."},{"Start":"06:37.115 ","End":"06:41.130","Text":"Once again, a calculator problem."},{"Start":"06:41.440 ","End":"06:49.380","Text":"Here I copied what my calculator said, 3.5 billion more."},{"Start":"06:50.570 ","End":"06:53.670","Text":"Just trust that this is the answer."},{"Start":"06:53.670 ","End":"06:55.400","Text":"Now we go on to Part C,"},{"Start":"06:55.400 ","End":"06:56.929","Text":"which is a bit different,"},{"Start":"06:56.929 ","End":"07:00.200","Text":"is where we don\u0027t know the year t,"},{"Start":"07:00.200 ","End":"07:04.195","Text":"but we know that the M of the year,"},{"Start":"07:04.195 ","End":"07:10.290","Text":"basically we know that m at time t is 50,000,000,000."},{"Start":"07:15.730 ","End":"07:17.960","Text":"On the other hand,"},{"Start":"07:17.960 ","End":"07:20.705","Text":"we know the formula for Mt,"},{"Start":"07:20.705 ","End":"07:23.720","Text":"which is what we have up here,"},{"Start":"07:23.720 ","End":"07:25.520","Text":"which is M0,"},{"Start":"07:25.520 ","End":"07:31.550","Text":"which is 4,000,000,000"},{"Start":"07:31.550 ","End":"07:39.340","Text":"times 1.02^t."},{"Start":"07:39.340 ","End":"07:41.400","Text":"Now I have an equation,"},{"Start":"07:41.400 ","End":"07:44.925","Text":"if I just forget the first part that this equals this,"},{"Start":"07:44.925 ","End":"07:47.480","Text":"again, an exponential equation."},{"Start":"07:47.480 ","End":"07:55.730","Text":"What I do is I can first of all divide both sides by 4,000,000,000."},{"Start":"07:55.730 ","End":"07:57.980","Text":"I\u0027ll just switch sides,"},{"Start":"07:57.980 ","End":"08:03.677","Text":"so I get 1.02^t"},{"Start":"08:03.677 ","End":"08:09.370","Text":"equals 50/4 is 12.5."},{"Start":"08:10.250 ","End":"08:13.363","Text":"The billion cancels with the billion,"},{"Start":"08:13.363 ","End":"08:19.385","Text":"and they just get 50/4, which is 12.5."},{"Start":"08:19.385 ","End":"08:25.340","Text":"The way we solve this is that we take the logarithm of both sides."},{"Start":"08:25.340 ","End":"08:29.335","Text":"It doesn\u0027t really matter what base we use."},{"Start":"08:29.335 ","End":"08:33.666","Text":"I\u0027ll go with base 10 logarithms."},{"Start":"08:33.666 ","End":"08:40.625","Text":"I\u0027ll take log of both sides or log to the base 10."},{"Start":"08:40.625 ","End":"08:46.070","Text":"Now the log of the exponent comes in front of,"},{"Start":"08:46.070 ","End":"08:57.490","Text":"so I get t log 1.02 equals log of 12.5."},{"Start":"08:57.490 ","End":"09:00.770","Text":"If you use another base if you use the log_e,"},{"Start":"09:00.770 ","End":"09:03.575","Text":"you should get the same answer in the end."},{"Start":"09:03.575 ","End":"09:05.240","Text":"Now we divide,"},{"Start":"09:05.240 ","End":"09:10.760","Text":"and we get that t is equal to log of"},{"Start":"09:10.760 ","End":"09:18.340","Text":"12.5 divided by log of 1.02."},{"Start":"09:19.700 ","End":"09:23.820","Text":"Once again, a calculator job."},{"Start":"09:23.820 ","End":"09:32.500","Text":"I make it a 127.545 and so on."},{"Start":"09:34.280 ","End":"09:37.750","Text":"I don\u0027t know when they said 1980,"},{"Start":"09:37.750 ","End":"09:41.180","Text":"if it\u0027s January 1st or the middle of the year."},{"Start":"09:42.720 ","End":"09:46.450","Text":"This is just my approach."},{"Start":"09:46.450 ","End":"09:49.728","Text":"I would say a 127 or 128,"},{"Start":"09:49.728 ","End":"09:52.750","Text":"and give the answer as this year or this year."},{"Start":"09:52.750 ","End":"09:57.845","Text":"Now remember we have to add this to 1980."},{"Start":"09:57.845 ","End":"10:05.400","Text":"The year is equal to"},{"Start":"10:05.400 ","End":"10:14.520","Text":"1980 plus 127, or maybe 128."},{"Start":"10:14.520 ","End":"10:18.240","Text":"If we took 127,"},{"Start":"10:18.240 ","End":"10:23.425","Text":"then we get the year 2107."},{"Start":"10:23.425 ","End":"10:29.010","Text":"Now let\u0027s say this or 2108 or somewhere in-between."},{"Start":"10:29.010 ","End":"10:33.140","Text":"We don\u0027t know exactly which month or what."},{"Start":"10:33.140 ","End":"10:36.010","Text":"You can leave it a bit vague like that."},{"Start":"10:36.010 ","End":"10:39.640","Text":"But just remember to add your answer to 1980,"},{"Start":"10:39.640 ","End":"10:42.805","Text":"you could decide to round up or round down also,"},{"Start":"10:42.805 ","End":"10:45.890","Text":"this is less important."},{"Start":"10:47.630 ","End":"10:57.950","Text":"That\u0027s the answer to Part C. I guess I forgot to indicate that this was Part C anyway,"},{"Start":"10:57.950 ","End":"10:59.630","Text":"this is that was Part B,"},{"Start":"10:59.630 ","End":"11:01.700","Text":"this is Part C. That\u0027s the last part,"},{"Start":"11:01.700 ","End":"11:03.870","Text":"and so we\u0027re done."}],"Thumbnail":null,"ID":7650}],"ID":4218},{"Name":"Homogeneous Equations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Homogeneous ODEs","Duration":"3m 28s","ChapterTopicVideoID":7588,"CourseChapterTopicPlaylistID":4219,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"We just finished talking about homogeneous functions,"},{"Start":"00:02.910 ","End":"00:06.765","Text":"and now we\u0027ll talk about homogeneous ordinary differential equations."},{"Start":"00:06.765 ","End":"00:08.580","Text":"Let\u0027s give a definition."},{"Start":"00:08.580 ","End":"00:11.580","Text":"A differential equation is called homogeneous if it"},{"Start":"00:11.580 ","End":"00:14.723","Text":"can be brought to the form some function of x and y dx,"},{"Start":"00:14.723 ","End":"00:17.745","Text":"another function of x and y dy=0."},{"Start":"00:17.745 ","End":"00:20.430","Text":"But there is an important condition attached to m and n,"},{"Start":"00:20.430 ","End":"00:24.060","Text":"and that is that both of them have to be homogeneous more than that,"},{"Start":"00:24.060 ","End":"00:27.390","Text":"what\u0027s important is that they have to have the same degree."},{"Start":"00:27.390 ","End":"00:29.220","Text":"This is homogeneous of Degree 3,"},{"Start":"00:29.220 ","End":"00:30.660","Text":"and this is homogeneous of Degree 4,"},{"Start":"00:30.660 ","End":"00:32.580","Text":"that\u0027s no good, same degree."},{"Start":"00:32.580 ","End":"00:34.530","Text":"Example; the following,"},{"Start":"00:34.530 ","End":"00:36.390","Text":"It\u0027s certainly in this form,"},{"Start":"00:36.390 ","End":"00:41.330","Text":"and I could take this as m and this as n,"},{"Start":"00:41.330 ","End":"00:44.810","Text":"and it\u0027s an easy exercise for you to check that"},{"Start":"00:44.810 ","End":"00:49.820","Text":"both m and n as functions of homogeneous both of Degree 2,"},{"Start":"00:49.820 ","End":"00:52.730","Text":"and so this is a homogeneous ODE."},{"Start":"00:52.730 ","End":"00:54.870","Text":"Next example, here it is,"},{"Start":"00:54.870 ","End":"00:56.300","Text":"I\u0027m getting to too much detail."},{"Start":"00:56.300 ","End":"00:59.300","Text":"This is m, this is n,"},{"Start":"00:59.300 ","End":"01:01.340","Text":"a little bit of work and you know how to do this."},{"Start":"01:01.340 ","End":"01:02.990","Text":"You can check that m and n,"},{"Start":"01:02.990 ","End":"01:08.210","Text":"each of them separately is homogeneous and both of them come out to be a Degree 3."},{"Start":"01:08.210 ","End":"01:10.730","Text":"The next example, it doesn\u0027t look like this,"},{"Start":"01:10.730 ","End":"01:14.045","Text":"but we\u0027re allowed to do a bit of work because it says here,"},{"Start":"01:14.045 ","End":"01:15.740","Text":"if it can be brought to the form,"},{"Start":"01:15.740 ","End":"01:17.650","Text":"so let\u0027s bring it to that form,"},{"Start":"01:17.650 ","End":"01:22.460","Text":"and what we can do is first of all use the notation of Leibniz,"},{"Start":"01:22.460 ","End":"01:23.975","Text":"the d notation,"},{"Start":"01:23.975 ","End":"01:27.860","Text":"and write it as dy by dx instead of y\u0027."},{"Start":"01:27.860 ","End":"01:31.880","Text":"Next thing you want to do is cross multiply and tidy up a bit."},{"Start":"01:31.880 ","End":"01:34.670","Text":"This is the cross multiplication y on the right,"},{"Start":"01:34.670 ","End":"01:35.810","Text":"x is on the left,"},{"Start":"01:35.810 ","End":"01:38.480","Text":"and then if we bring this to the other side,"},{"Start":"01:38.480 ","End":"01:40.090","Text":"we get this only,"},{"Start":"01:40.090 ","End":"01:42.170","Text":"the definition calls for a plus."},{"Start":"01:42.170 ","End":"01:46.385","Text":"So just reverse the order here and write a plus,"},{"Start":"01:46.385 ","End":"01:50.495","Text":"and now we can identify a function M of x, y,"},{"Start":"01:50.495 ","End":"01:51.710","Text":"and N of x,"},{"Start":"01:51.710 ","End":"01:55.235","Text":"y and a quick check will show you that each of these is homogeneous."},{"Start":"01:55.235 ","End":"01:56.485","Text":"This is of Degree 1."},{"Start":"01:56.485 ","End":"01:57.780","Text":"This is of Degree 1,"},{"Start":"01:57.780 ","End":"02:01.385","Text":"so we\u0027ve got an ordinary differential equation which is homogeneous,"},{"Start":"02:01.385 ","End":"02:03.470","Text":"and that\u0027s it for examples."},{"Start":"02:03.470 ","End":"02:04.595","Text":"The next thing is,"},{"Start":"02:04.595 ","End":"02:06.005","Text":"how do we solve them?"},{"Start":"02:06.005 ","End":"02:08.315","Text":"As you can see, it\u0027s a 3 step approach."},{"Start":"02:08.315 ","End":"02:09.650","Text":"In Step 1,"},{"Start":"02:09.650 ","End":"02:13.280","Text":"we naturally just verify that it is indeed homogeneous."},{"Start":"02:13.280 ","End":"02:15.230","Text":"I could have called this Step 0,"},{"Start":"02:15.230 ","End":"02:18.140","Text":"but you don\u0027t want to apply the technique to something that\u0027s not"},{"Start":"02:18.140 ","End":"02:22.615","Text":"a homogeneous equation because the technique is tailored to homogeneous."},{"Start":"02:22.615 ","End":"02:25.310","Text":"The main step starts with a substitution,"},{"Start":"02:25.310 ","End":"02:29.630","Text":"where we substitute y=v times x and v is a new variable,"},{"Start":"02:29.630 ","End":"02:31.550","Text":"so we get rid of y and have v.x,"},{"Start":"02:31.550 ","End":"02:33.740","Text":"but of course we also have to replace dy."},{"Start":"02:33.740 ","End":"02:38.630","Text":"This here just follows straight from the product rule that d of"},{"Start":"02:38.630 ","End":"02:44.080","Text":"a product is d of the first times the second and the second times the first,"},{"Start":"02:44.080 ","End":"02:45.380","Text":"and after you\u0027ve done that,"},{"Start":"02:45.380 ","End":"02:48.450","Text":"you\u0027ll get a equation in v and x."},{"Start":"02:48.450 ","End":"02:49.730","Text":"If you check it,"},{"Start":"02:49.730 ","End":"02:53.225","Text":"you\u0027ll see that it\u0027s solvable by separation of variables."},{"Start":"02:53.225 ","End":"02:55.310","Text":"Another technique we\u0027ve already learned just to get"},{"Start":"02:55.310 ","End":"02:58.320","Text":"a separable ODE is separation of variables,"},{"Start":"02:58.320 ","End":"03:00.095","Text":"and after we\u0027ve got the solution,"},{"Start":"03:00.095 ","End":"03:03.185","Text":"the final step which sometimes people forget to do is"},{"Start":"03:03.185 ","End":"03:06.995","Text":"we have a solution now in terms of V,"},{"Start":"03:06.995 ","End":"03:08.930","Text":"we have V and X,"},{"Start":"03:08.930 ","End":"03:10.280","Text":"and we don\u0027t want v,"},{"Start":"03:10.280 ","End":"03:11.780","Text":"We want to get back to y."},{"Start":"03:11.780 ","End":"03:13.325","Text":"Wherever we see v,"},{"Start":"03:13.325 ","End":"03:15.470","Text":"we replace it by y over x,"},{"Start":"03:15.470 ","End":"03:17.540","Text":"which is just what you would get here if you"},{"Start":"03:17.540 ","End":"03:20.300","Text":"extracted v. This is a bit theoretical I know,"},{"Start":"03:20.300 ","End":"03:23.690","Text":"but there are lots of examples following this,"},{"Start":"03:23.690 ","End":"03:26.795","Text":"and this was just an outline of the method used,"},{"Start":"03:26.795 ","End":"03:28.920","Text":"and so that\u0027s it."}],"Thumbnail":null,"ID":7651},{"Watched":false,"Name":"Homogeneous Functions","Duration":"4m 22s","ChapterTopicVideoID":7587,"CourseChapterTopicPlaylistID":4219,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.090","Text":"Here I\u0027m going to be talking about homogeneous functions and"},{"Start":"00:03.090 ","End":"00:06.960","Text":"preparation of token about homogeneous differential equations."},{"Start":"00:06.960 ","End":"00:09.519","Text":"Can\u0027t assume that you know what these are,"},{"Start":"00:09.519 ","End":"00:10.875","Text":"so here it goes."},{"Start":"00:10.875 ","End":"00:12.660","Text":"Function of 2 variables,"},{"Start":"00:12.660 ","End":"00:16.245","Text":"f(x,y) is called homogeneous of degree n,"},{"Start":"00:16.245 ","End":"00:20.490","Text":"where n is not necessarily positive, could be 0."},{"Start":"00:20.490 ","End":"00:22.785","Text":"Anyway, a whole number, an integer,"},{"Start":"00:22.785 ","End":"00:24.450","Text":"if for all Lambda,"},{"Start":"00:24.450 ","End":"00:28.050","Text":"Lambda is just a Greek letter for any number."},{"Start":"00:28.050 ","End":"00:29.130","Text":"For all Lambda,"},{"Start":"00:29.130 ","End":"00:30.377","Text":"for all numbers, f(Lambda x,"},{"Start":"00:30.377 ","End":"00:35.705","Text":"Lambda y), equals Lambda to the power of n, f(x) and y."},{"Start":"00:35.705 ","End":"00:37.475","Text":"That\u0027s why it doesn\u0027t make much sense."},{"Start":"00:37.475 ","End":"00:41.360","Text":"So I\u0027m going to bring you 4 examples and by then it should be clear."},{"Start":"00:41.360 ","End":"00:43.220","Text":"Just want to say that this can be generalized."},{"Start":"00:43.220 ","End":"00:46.130","Text":"It isn\u0027t just a function of 2 variables."},{"Start":"00:46.130 ","End":"00:48.950","Text":"It could be a function of 1, 2, 3, 4, 5,"},{"Start":"00:48.950 ","End":"00:50.300","Text":"any number of variables,"},{"Start":"00:50.300 ","End":"00:53.885","Text":"but we\u0027ll encounter it mostly with 2 variables."},{"Start":"00:53.885 ","End":"00:58.010","Text":"We start with the first example in 2 variables,"},{"Start":"00:58.010 ","End":"01:00.010","Text":"f(x,y) is as follows."},{"Start":"01:00.010 ","End":"01:03.260","Text":"I claim that it\u0027s homogeneous of degree 2."},{"Start":"01:03.260 ","End":"01:05.585","Text":"We don\u0027t just say homogeneous, we usually say,"},{"Start":"01:05.585 ","End":"01:07.204","Text":"homogeneous of degree,"},{"Start":"01:07.204 ","End":"01:08.840","Text":"whatever the n is."},{"Start":"01:08.840 ","End":"01:10.625","Text":"Using this definition,"},{"Start":"01:10.625 ","End":"01:13.885","Text":"and don\u0027t be alarmed by using Greek letters such as Lambda,"},{"Start":"01:13.885 ","End":"01:16.780","Text":"what I do is I substitute instead of x and y,"},{"Start":"01:16.780 ","End":"01:18.820","Text":"Lambda x Lambda y as here,"},{"Start":"01:18.820 ","End":"01:20.185","Text":"and see what that is."},{"Start":"01:20.185 ","End":"01:22.315","Text":"Instead of y, I have Lambda y,"},{"Start":"01:22.315 ","End":"01:24.070","Text":"instead of x, I get Lambda x,"},{"Start":"01:24.070 ","End":"01:25.850","Text":"instead of y, I have Lambda y."},{"Start":"01:25.850 ","End":"01:27.810","Text":"Notice that here I have Lambda squared,"},{"Start":"01:27.810 ","End":"01:29.730","Text":"and here Lambda with Lambda is Lambda squared."},{"Start":"01:29.730 ","End":"01:32.495","Text":"So I can pull the Lambda squared outside the brackets."},{"Start":"01:32.495 ","End":"01:36.350","Text":"What we\u0027re left with is y^2 plus 2xy."},{"Start":"01:36.350 ","End":"01:43.940","Text":"Now, all that remains is to notice that y^2 plus 2xy is our original function f(x,y)."},{"Start":"01:43.940 ","End":"01:46.475","Text":"We started from here and we got to here,"},{"Start":"01:46.475 ","End":"01:50.140","Text":"which is what we would expect from the definition where n is 2,"},{"Start":"01:50.140 ","End":"01:52.925","Text":"and so that\u0027s our first example."},{"Start":"01:52.925 ","End":"01:55.460","Text":"Continuing, second example,"},{"Start":"01:55.460 ","End":"01:59.340","Text":"this function f(x,y) is as follows,"},{"Start":"01:59.340 ","End":"02:02.060","Text":"claim is it is homogeneous of degree 1."},{"Start":"02:02.060 ","End":"02:05.225","Text":"Once again, we\u0027ll substitute Lambda x for x,"},{"Start":"02:05.225 ","End":"02:07.475","Text":"and Lambda y for y."},{"Start":"02:07.475 ","End":"02:09.620","Text":"Everywhere you see x, you put Lambda x,"},{"Start":"02:09.620 ","End":"02:12.335","Text":"and so where you see y, you put Lambda y, and so on."},{"Start":"02:12.335 ","End":"02:17.430","Text":"Now, we can take Lambda squared outside the brackets in each of these terms."},{"Start":"02:17.430 ","End":"02:20.955","Text":"What we end up with is Lambda squared,"},{"Start":"02:20.955 ","End":"02:22.650","Text":"4x^2 plus 3y^2,"},{"Start":"02:22.650 ","End":"02:24.120","Text":"and here just Lambda."},{"Start":"02:24.120 ","End":"02:25.260","Text":"This is top 2,"},{"Start":"02:25.260 ","End":"02:27.745","Text":"you take Lambda squared out of each and the bottom one,"},{"Start":"02:27.745 ","End":"02:29.680","Text":"just Lambda, and we\u0027re left with this."},{"Start":"02:29.680 ","End":"02:31.325","Text":"If you notice,"},{"Start":"02:31.325 ","End":"02:35.855","Text":"Lambda squared over Lambda is just Lambda to the 1,"},{"Start":"02:35.855 ","End":"02:37.610","Text":"it\u0027s even better to make it explicit,"},{"Start":"02:37.610 ","End":"02:38.650","Text":"if I just write Lambda,"},{"Start":"02:38.650 ","End":"02:40.465","Text":"you might not notice this degree 1."},{"Start":"02:40.465 ","End":"02:43.435","Text":"Lambda is Lambda to the 1, and this is the original function."},{"Start":"02:43.435 ","End":"02:46.570","Text":"It\u0027s ultimately Lambda to the 1, f(x,y)."},{"Start":"02:46.570 ","End":"02:47.995","Text":"If we get from here to here,"},{"Start":"02:47.995 ","End":"02:51.110","Text":"so this one fits the end in the definition of homogeneous,"},{"Start":"02:51.110 ","End":"02:53.495","Text":"and so it\u0027s homogeneous of degree 1."},{"Start":"02:53.495 ","End":"02:55.300","Text":"After example b,"},{"Start":"02:55.300 ","End":"02:57.675","Text":"we\u0027re going to do example c,"},{"Start":"02:57.675 ","End":"03:01.715","Text":"and claim is that this is homogeneous of degree 3."},{"Start":"03:01.715 ","End":"03:07.850","Text":"We\u0027ll proceed as usual by replacing x and y with Lambda x, Lambda y."},{"Start":"03:07.850 ","End":"03:11.360","Text":"Now, x doesn\u0027t explicitly appear here. So just ignore it."},{"Start":"03:11.360 ","End":"03:14.085","Text":"Just replace the y by Lambda y,"},{"Start":"03:14.085 ","End":"03:17.340","Text":"and then this is equal to Lambda cubed,"},{"Start":"03:17.340 ","End":"03:20.340","Text":"y^3, and y^3 is the original equation."},{"Start":"03:20.340 ","End":"03:23.310","Text":"We get that f of this equals this."},{"Start":"03:23.310 ","End":"03:27.560","Text":"It\u0027s the definition of homogeneous with n being replaced by 3."},{"Start":"03:27.560 ","End":"03:29.200","Text":"Let\u0027s give one more example."},{"Start":"03:29.200 ","End":"03:30.840","Text":"Here\u0027s our function."},{"Start":"03:30.840 ","End":"03:33.395","Text":"The claim is it\u0027s homogeneous of degree 0,"},{"Start":"03:33.395 ","End":"03:37.940","Text":"homogeneous, the n can be any whole number, positive or 0."},{"Start":"03:37.940 ","End":"03:40.010","Text":"I don\u0027t think it applies to negative,"},{"Start":"03:40.010 ","End":"03:43.040","Text":"I\u0027ve never seen it, but 0 could be. Let\u0027s check."},{"Start":"03:43.040 ","End":"03:44.760","Text":"Well, as usual we put Lambda x,"},{"Start":"03:44.760 ","End":"03:46.095","Text":"Lambda y here,"},{"Start":"03:46.095 ","End":"03:50.855","Text":"and then we just replace x and y respectively by Lambda x Lambda y everywhere."},{"Start":"03:50.855 ","End":"03:53.210","Text":"Notice I can take Lambda outside the brackets"},{"Start":"03:53.210 ","End":"03:55.910","Text":"both in the numerator and in the denominator,"},{"Start":"03:55.910 ","End":"03:57.575","Text":"and the Lambda cancels,"},{"Start":"03:57.575 ","End":"03:59.885","Text":"and then we\u0027re back to the original function."},{"Start":"03:59.885 ","End":"04:02.530","Text":"But instead of just writing it as the original function,"},{"Start":"04:02.530 ","End":"04:04.485","Text":"you can write it as Lambda to the note,"},{"Start":"04:04.485 ","End":"04:08.145","Text":"which is 1, and this is the original function."},{"Start":"04:08.145 ","End":"04:10.005","Text":"We start from here, we end up here."},{"Start":"04:10.005 ","End":"04:15.050","Text":"That\u0027s the definition of homogeneous of degree n if we look at m being 0."},{"Start":"04:15.050 ","End":"04:16.520","Text":"That\u0027s the last example,"},{"Start":"04:16.520 ","End":"04:18.530","Text":"and we are done."},{"Start":"04:18.530 ","End":"04:23.100","Text":"The next clip should talk about homogeneous differential equations."}],"Thumbnail":null,"ID":7652},{"Watched":false,"Name":"Exercise 1","Duration":"4m 35s","ChapterTopicVideoID":7578,"CourseChapterTopicPlaylistID":4219,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.865","Text":"Here I have a differential equation to solve,"},{"Start":"00:02.865 ","End":"00:06.585","Text":"and it looks like it could be a homogeneous equation."},{"Start":"00:06.585 ","End":"00:08.295","Text":"Let\u0027s try and verify that."},{"Start":"00:08.295 ","End":"00:10.200","Text":"With the homogeneous equation,"},{"Start":"00:10.200 ","End":"00:12.360","Text":"we take this function here,"},{"Start":"00:12.360 ","End":"00:13.590","Text":"and this function here,"},{"Start":"00:13.590 ","End":"00:17.460","Text":"and try and see if they are homogeneous of the same order."},{"Start":"00:17.460 ","End":"00:19.875","Text":"This one we\u0027ll call M and this one we\u0027ll call"},{"Start":"00:19.875 ","End":"00:23.715","Text":"N. Let\u0027s start with M. If we substitute lambda x,"},{"Start":"00:23.715 ","End":"00:25.520","Text":"lambda y instead of x and y,"},{"Start":"00:25.520 ","End":"00:27.510","Text":"and you can follow the development here,"},{"Start":"00:27.510 ","End":"00:30.650","Text":"we end up with lambda^3 of M(x, y)."},{"Start":"00:30.650 ","End":"00:33.605","Text":"It means that M is homogeneous of order 3."},{"Start":"00:33.605 ","End":"00:35.720","Text":"If we do the same thing with N,"},{"Start":"00:35.720 ","End":"00:37.970","Text":"we\u0027ll find that it\u0027s also of order 3."},{"Start":"00:37.970 ","End":"00:39.680","Text":"Since they both have the same order,"},{"Start":"00:39.680 ","End":"00:41.840","Text":"this is a homogeneous equation."},{"Start":"00:41.840 ","End":"00:46.790","Text":"The theory behind the homogeneous equation is that if we make a certain substitution,"},{"Start":"00:46.790 ","End":"00:48.620","Text":"that is that y=v x,"},{"Start":"00:48.620 ","End":"00:50.653","Text":"where v is another variable,"},{"Start":"00:50.653 ","End":"00:52.070","Text":"and dy as follows,"},{"Start":"00:52.070 ","End":"00:55.355","Text":"then we\u0027ll end up with one that can be separated."},{"Start":"00:55.355 ","End":"00:58.475","Text":"In other words, we can use a separation of variables method on it."},{"Start":"00:58.475 ","End":"01:00.065","Text":"That\u0027s the theory anyway."},{"Start":"01:00.065 ","End":"01:07.255","Text":"I meant to say that later when we want to go from v back to x and y that v is y/x."},{"Start":"01:07.255 ","End":"01:12.365","Text":"In homogeneous equations, we\u0027re always going to assume that x is not equal to 0."},{"Start":"01:12.365 ","End":"01:14.810","Text":"Otherwise, we won\u0027t be able to do that."},{"Start":"01:14.810 ","End":"01:17.180","Text":"Let\u0027s continue."},{"Start":"01:17.180 ","End":"01:20.060","Text":"The next step is to make the substitution."},{"Start":"01:20.060 ","End":"01:23.330","Text":"Wherever I see y in this equation,"},{"Start":"01:23.330 ","End":"01:25.280","Text":"I replace it by vx,"},{"Start":"01:25.280 ","End":"01:27.770","Text":"that would be here and here."},{"Start":"01:27.770 ","End":"01:33.320","Text":"The dy is going to be replaced by this expression here."},{"Start":"01:33.320 ","End":"01:35.360","Text":"This is all from this equation."},{"Start":"01:35.360 ","End":"01:38.345","Text":"Now, what we want to do is expand this."},{"Start":"01:38.345 ","End":"01:40.790","Text":"We will get the following as x^3 here,"},{"Start":"01:40.790 ","End":"01:43.955","Text":"x^3 here, x^4, and also x^3."},{"Start":"01:43.955 ","End":"01:46.220","Text":"X^3 can be taken out of all of them,"},{"Start":"01:46.220 ","End":"01:48.125","Text":"and after we take the x^3 out,"},{"Start":"01:48.125 ","End":"01:49.834","Text":"we will be left with this."},{"Start":"01:49.834 ","End":"01:52.970","Text":"Again we\u0027re reminded, x is not 0 in the domain."},{"Start":"01:52.970 ","End":"01:54.725","Text":"Now we want to separate the variables."},{"Start":"01:54.725 ","End":"01:59.210","Text":"Let\u0027s just group together the terms containing dx."},{"Start":"01:59.210 ","End":"02:01.573","Text":"Let\u0027s say this term is one dx,"},{"Start":"02:01.573 ","End":"02:02.930","Text":"and this is v^3 dx,"},{"Start":"02:02.930 ","End":"02:04.010","Text":"and take them all together,"},{"Start":"02:04.010 ","End":"02:05.975","Text":"we get v^3 plus v^3 plus 1,"},{"Start":"02:05.975 ","End":"02:09.470","Text":"that would make it 2v^3 plus one dx."},{"Start":"02:09.470 ","End":"02:10.940","Text":"The rest of it is,"},{"Start":"02:10.940 ","End":"02:14.930","Text":"from here, just changed the order v^2 xdv."},{"Start":"02:14.930 ","End":"02:17.630","Text":"Bringing it over to the other side,"},{"Start":"02:17.630 ","End":"02:21.125","Text":"the last bit, we\u0027re preparing for separation of variables."},{"Start":"02:21.125 ","End":"02:25.970","Text":"Now assuming that 2v^3 plus 1 is not 0,"},{"Start":"02:25.970 ","End":"02:29.330","Text":"I\u0027m bringing the minus x over to the left,"},{"Start":"02:29.330 ","End":"02:32.570","Text":"minus with x goes here, minus an x."},{"Start":"02:32.570 ","End":"02:35.090","Text":"The 2v^3 plus 1 is going here."},{"Start":"02:35.090 ","End":"02:39.530","Text":"That\u0027s adding the additional restriction that 2v^3 plus 1 is not 0."},{"Start":"02:39.530 ","End":"02:41.960","Text":"I will return to this later, it\u0027s important."},{"Start":"02:41.960 ","End":"02:46.385","Text":"But meanwhile, we\u0027ll continue and we just put the integral sign in front of each."},{"Start":"02:46.385 ","End":"02:48.658","Text":"Now the standard trick is to try,"},{"Start":"02:48.658 ","End":"02:53.990","Text":"and go for the formula where we have that the integral of f \u0027/f"},{"Start":"02:53.990 ","End":"03:00.920","Text":"is just equal to the natural log of the denominator plus a constant."},{"Start":"03:00.920 ","End":"03:02.105","Text":"This is just the idea."},{"Start":"03:02.105 ","End":"03:04.865","Text":"Now if f is 2v^3+1,"},{"Start":"03:04.865 ","End":"03:07.190","Text":"df would be 6v^2,"},{"Start":"03:07.190 ","End":"03:09.710","Text":"and we only had one v^2,"},{"Start":"03:09.710 ","End":"03:12.380","Text":"so the standard trick is to make it what we want and"},{"Start":"03:12.380 ","End":"03:15.200","Text":"then compensate so if we put a 6 in the numerator,"},{"Start":"03:15.200 ","End":"03:16.805","Text":"we put a 6 in the denominator."},{"Start":"03:16.805 ","End":"03:18.800","Text":"Now we can apply this formula,"},{"Start":"03:18.800 ","End":"03:24.650","Text":"and what we get is the following integral of 1/x is natural log of x,"},{"Start":"03:24.650 ","End":"03:28.355","Text":"integral of this is natural log of the denominator,"},{"Start":"03:28.355 ","End":"03:30.785","Text":"absolute values here and here of course."},{"Start":"03:30.785 ","End":"03:35.570","Text":"That\u0027s basically the solution to the differential equation,"},{"Start":"03:35.570 ","End":"03:41.375","Text":"but not quite because we still have v in here and we need to substitute v back."},{"Start":"03:41.375 ","End":"03:49.267","Text":"Remember that v was equal to y/x so we need to replace v here by y/x,"},{"Start":"03:49.267 ","End":"03:50.885","Text":"and that will be the solution."},{"Start":"03:50.885 ","End":"03:54.605","Text":"That\u0027s one solution and I\u0027m going to highlight it."},{"Start":"03:54.605 ","End":"04:00.965","Text":"But I made a promise that I\u0027d come back to that point earlier where,"},{"Start":"04:00.965 ","End":"04:05.645","Text":"here it is, we said that 2v^3 plus 1 is not equal to 0."},{"Start":"04:05.645 ","End":"04:09.155","Text":"Now we have to check the possibility that it is 0."},{"Start":"04:09.155 ","End":"04:11.090","Text":"Let\u0027s see what happens if it is 0."},{"Start":"04:11.090 ","End":"04:13.520","Text":"If 2v^3 plus 1 is 0,"},{"Start":"04:13.520 ","End":"04:18.410","Text":"then v is some number minus 1 over 2^3,"},{"Start":"04:18.410 ","End":"04:22.880","Text":"and y being xv is just minus x over this number."},{"Start":"04:22.880 ","End":"04:25.490","Text":"That will certainly also be a solution,"},{"Start":"04:25.490 ","End":"04:28.780","Text":"and it\u0027s called the singular solution."},{"Start":"04:28.780 ","End":"04:30.650","Text":"I\u0027ll highlight that too,"},{"Start":"04:30.650 ","End":"04:33.980","Text":"that y equals this."},{"Start":"04:33.980 ","End":"04:36.090","Text":"Anyway, I\u0027m done."}],"Thumbnail":null,"ID":7653},{"Watched":false,"Name":"Exercise 2","Duration":"9m 42s","ChapterTopicVideoID":7579,"CourseChapterTopicPlaylistID":4219,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"Here we have a differential equation and I\u0027m"},{"Start":"00:03.000 ","End":"00:06.000","Text":"going to show that this is a homogeneous differential equation."},{"Start":"00:06.000 ","End":"00:07.530","Text":"I just copied it over here,"},{"Start":"00:07.530 ","End":"00:12.600","Text":"we can rewrite it slightly as dy over dx instead of y\u0027."},{"Start":"00:12.600 ","End":"00:18.160","Text":"Now, we can cross multiply and get that dy"},{"Start":"00:18.160 ","End":"00:23.818","Text":"times 2x minus dy=dx times 4y minus 3x."},{"Start":"00:23.818 ","End":"00:27.030","Text":"I prefer to bring it all to one side=0."},{"Start":"00:27.030 ","End":"00:31.020","Text":"Basically, this minus this is 0. Whereas the minus?"},{"Start":"00:31.020 ","End":"00:34.530","Text":"Well, I left it as plus but I reverse the order so that\u0027s okay."},{"Start":"00:34.530 ","End":"00:36.105","Text":"This is what we get."},{"Start":"00:36.105 ","End":"00:39.150","Text":"The first function of x and y I\u0027ll call M, the second one,"},{"Start":"00:39.150 ","End":"00:44.780","Text":"I\u0027ll call N. Our first task is to show that they are homogeneous,"},{"Start":"00:44.780 ","End":"00:45.965","Text":"both M and N,"},{"Start":"00:45.965 ","End":"00:47.720","Text":"and so let\u0027s look at that."},{"Start":"00:47.720 ","End":"00:50.090","Text":"M of Lambda x, Lambda y equals,"},{"Start":"00:50.090 ","End":"00:53.750","Text":"you know how to substitute and take out the brackets up to development,"},{"Start":"00:53.750 ","End":"00:58.640","Text":"we get Lambda times the original function M. This is homogeneous of order 1."},{"Start":"00:58.640 ","End":"01:00.260","Text":"It\u0027s Lambda^1."},{"Start":"01:00.260 ","End":"01:01.865","Text":"Similarly the second one,"},{"Start":"01:01.865 ","End":"01:05.930","Text":"we substitute and develop it a bit we get Lambda times N,"},{"Start":"01:05.930 ","End":"01:07.550","Text":"if we put Lambda x, Lambda y,"},{"Start":"01:07.550 ","End":"01:10.715","Text":"this is also homogeneous of Degree 1, same degree."},{"Start":"01:10.715 ","End":"01:14.690","Text":"Therefore, we have a homogeneous differential equation and this means that we"},{"Start":"01:14.690 ","End":"01:18.990","Text":"can solve it using a technique where we substitute as follows,"},{"Start":"01:18.990 ","End":"01:20.895","Text":"y=v times x,"},{"Start":"01:20.895 ","End":"01:23.655","Text":"and dy is like this."},{"Start":"01:23.655 ","End":"01:24.915","Text":"Then if we do this,"},{"Start":"01:24.915 ","End":"01:28.489","Text":"we\u0027re guaranteed that we\u0027re going to get an equation that is separable,"},{"Start":"01:28.489 ","End":"01:34.565","Text":"meaning we can separate variables to have v on one side and x on the other or vice versa."},{"Start":"01:34.565 ","End":"01:39.995","Text":"What I did here basically is if I see y as I do here and here,"},{"Start":"01:39.995 ","End":"01:42.710","Text":"then I put v.x in their place."},{"Start":"01:42.710 ","End":"01:46.100","Text":"Look, v.x is here and v.x is here."},{"Start":"01:46.100 ","End":"01:49.355","Text":"Wherever I see dy, then I put,"},{"Start":"01:49.355 ","End":"01:53.840","Text":"instead of it, dv times x plus dx times v which I\u0027ve done here."},{"Start":"01:53.840 ","End":"01:55.730","Text":"So this is just a substitution."},{"Start":"01:55.730 ","End":"02:00.245","Text":"We get this, which if we just rearrange it,"},{"Start":"02:00.245 ","End":"02:05.825","Text":"here I can bring the x out front because there\u0027s an x here and an x here and over here,"},{"Start":"02:05.825 ","End":"02:10.730","Text":"I can take also x outside of here so we end up with this."},{"Start":"02:10.730 ","End":"02:12.335","Text":"Let me scroll up a bit."},{"Start":"02:12.335 ","End":"02:15.440","Text":"The x can cancel from both sides,"},{"Start":"02:15.440 ","End":"02:16.580","Text":"and of course,"},{"Start":"02:16.580 ","End":"02:18.830","Text":"we have to say that x is not equal to 0."},{"Start":"02:18.830 ","End":"02:22.565","Text":"But in any event, we\u0027re assuming that x is not equal to 0,"},{"Start":"02:22.565 ","End":"02:24.995","Text":"because when we substitute back at the end,"},{"Start":"02:24.995 ","End":"02:29.300","Text":"this y=v.x is the same as v=y over x,"},{"Start":"02:29.300 ","End":"02:31.940","Text":"in any event, you want to assume that x is not equal to 0,"},{"Start":"02:31.940 ","End":"02:34.775","Text":"it just eliminates x from the domain of definition."},{"Start":"02:34.775 ","End":"02:38.450","Text":"Let\u0027s expand, we have v minus 2 times dv"},{"Start":"02:38.450 ","End":"02:42.710","Text":"times x. I\u0027ll just put the x in front and leave the dv at the end,"},{"Start":"02:42.710 ","End":"02:45.500","Text":"and v minus 2 times dx.v,"},{"Start":"02:45.500 ","End":"02:48.470","Text":"which I\u0027d like to just put the dx at the end."},{"Start":"02:48.470 ","End":"02:51.048","Text":"Here I have dx and here I have dx."},{"Start":"02:51.048 ","End":"02:56.295","Text":"So I\u0027m gathering these two together under this dx and I get v minus 2,"},{"Start":"02:56.295 ","End":"02:59.355","Text":"we took first and then the 4v minus 3."},{"Start":"02:59.355 ","End":"03:02.795","Text":"Then there\u0027s also the dv bit which is here,"},{"Start":"03:02.795 ","End":"03:05.180","Text":"which is here and just copied it straight."},{"Start":"03:05.180 ","End":"03:07.310","Text":"That\u0027s where we\u0027re up to now."},{"Start":"03:07.310 ","End":"03:12.035","Text":"Then the next step is just to simplify the dx bit,"},{"Start":"03:12.035 ","End":"03:16.950","Text":"which is v^2 minus 2v plus 4v minus 3,"},{"Start":"03:16.950 ","End":"03:19.575","Text":"and here just as it is."},{"Start":"03:19.575 ","End":"03:23.790","Text":"Then combining minus 2v plus 4v is 2v,"},{"Start":"03:23.790 ","End":"03:26.130","Text":"and this remains the same."},{"Start":"03:26.130 ","End":"03:27.680","Text":"What do we do next?"},{"Start":"03:27.680 ","End":"03:29.690","Text":"We want to separate the variables."},{"Start":"03:29.690 ","End":"03:33.290","Text":"Let me just copy this over to the next page."},{"Start":"03:33.290 ","End":"03:34.880","Text":"Here we are on the new page,"},{"Start":"03:34.880 ","End":"03:37.100","Text":"the same line has been copied."},{"Start":"03:37.100 ","End":"03:42.875","Text":"Want to modify it a bit so what we get is instead of the minus,"},{"Start":"03:42.875 ","End":"03:46.430","Text":"I put it as a plus and reverse the order of the subtraction,"},{"Start":"03:46.430 ","End":"03:47.825","Text":"then that should be okay."},{"Start":"03:47.825 ","End":"03:49.395","Text":"Then we can finally separate."},{"Start":"03:49.395 ","End":"03:51.620","Text":"We already said that xis not equal to 0,"},{"Start":"03:51.620 ","End":"03:54.110","Text":"so x can go into the denominator here and"},{"Start":"03:54.110 ","End":"03:57.535","Text":"the v^2 plus 2v minus 3 goes into the denominator here."},{"Start":"03:57.535 ","End":"03:59.520","Text":"There\u0027s something written in brackets here."},{"Start":"03:59.520 ","End":"04:02.310","Text":"Why have I write v not equal to 1v not equal to 3?"},{"Start":"04:02.310 ","End":"04:05.745","Text":"Because these are the 2 roots of this polynomial."},{"Start":"04:05.745 ","End":"04:07.620","Text":"The quadratic polynomial actually,"},{"Start":"04:07.620 ","End":"04:08.660","Text":"it has 2 roots,"},{"Start":"04:08.660 ","End":"04:10.235","Text":"which I did at the side,"},{"Start":"04:10.235 ","End":"04:13.385","Text":"and it cannot be either one of these."},{"Start":"04:13.385 ","End":"04:17.855","Text":"What I\u0027m saying is that if you actually do the exercise of factorization,"},{"Start":"04:17.855 ","End":"04:21.650","Text":"this thing is equal to v minus 1,"},{"Start":"04:21.650 ","End":"04:23.390","Text":"v plus 3,"},{"Start":"04:23.390 ","End":"04:25.490","Text":"and so in order for this not to be 0,"},{"Start":"04:25.490 ","End":"04:28.130","Text":"we must have that v is not equal to 1 because of"},{"Start":"04:28.130 ","End":"04:31.160","Text":"this and v not equal to minus 3 because of this."},{"Start":"04:31.160 ","End":"04:35.240","Text":"Having said that, what we have to do is put integral signs in front of each,"},{"Start":"04:35.240 ","End":"04:38.840","Text":"which is here and the next step is to integrate this."},{"Start":"04:38.840 ","End":"04:40.220","Text":"Here is the integral."},{"Start":"04:40.220 ","End":"04:45.275","Text":"The integral of 1 over x is natural log x and the integral of this,"},{"Start":"04:45.275 ","End":"04:47.375","Text":"it\u0027s not clear how I got to this,"},{"Start":"04:47.375 ","End":"04:49.984","Text":"but I\u0027ll just give you the integral,"},{"Start":"04:49.984 ","End":"04:51.875","Text":"this minus this plus C,"},{"Start":"04:51.875 ","End":"04:54.170","Text":"and at the end, I\u0027ll show you how I got this."},{"Start":"04:54.170 ","End":"04:55.595","Text":"I don\u0027t want to break the flow,"},{"Start":"04:55.595 ","End":"04:58.010","Text":"so let\u0027s just take my word for it for now and"},{"Start":"04:58.010 ","End":"05:01.280","Text":"I\u0027ll owe you at the end to show you how I got to this."},{"Start":"05:01.280 ","End":"05:04.775","Text":"We begin the substituting back part."},{"Start":"05:04.775 ","End":"05:09.020","Text":"Remember I said that at the end we\u0027re going to let V equals as rewrite it."},{"Start":"05:09.020 ","End":"05:13.190","Text":"We\u0027re going to set V is equal to y over x,"},{"Start":"05:13.190 ","End":"05:15.425","Text":"that was because y=xv."},{"Start":"05:15.425 ","End":"05:21.680","Text":"Wherever I see v which is here and here that\u0027s where I put the y over x."},{"Start":"05:21.680 ","End":"05:24.005","Text":"We\u0027re back in the world of x and y,"},{"Start":"05:24.005 ","End":"05:27.170","Text":"we essentially have the solution to the equation"},{"Start":"05:27.170 ","End":"05:30.824","Text":"maybe not in the tidiest form that\u0027s as possible,"},{"Start":"05:30.824 ","End":"05:32.775","Text":"but this is the answer."},{"Start":"05:32.775 ","End":"05:38.090","Text":"Now, I want to get back to here because I said that we\u0027re going to return to that."},{"Start":"05:38.090 ","End":"05:42.455","Text":"We can\u0027t just assume v not equal to 1 and v not equal to 3,"},{"Start":"05:42.455 ","End":"05:44.825","Text":"we have to allow for the possibility that it is."},{"Start":"05:44.825 ","End":"05:49.535","Text":"In fact, if we let v=1 then we get,"},{"Start":"05:49.535 ","End":"05:51.350","Text":"since y is v.x,"},{"Start":"05:51.350 ","End":"05:56.160","Text":"that y=x and if we let v=3,"},{"Start":"05:56.160 ","End":"05:57.570","Text":"then y is v.x,"},{"Start":"05:57.570 ","End":"06:00.780","Text":"then we get that y=minus 3."},{"Start":"06:00.780 ","End":"06:04.880","Text":"These 2 equations actually do provide 2 alternative solutions."},{"Start":"06:04.880 ","End":"06:06.935","Text":"We have 3 solutions,"},{"Start":"06:06.935 ","End":"06:08.585","Text":"this main one,"},{"Start":"06:08.585 ","End":"06:10.700","Text":"and these two,"},{"Start":"06:10.700 ","End":"06:15.350","Text":"y=x and y=minus 3x, are singular solution."},{"Start":"06:15.350 ","End":"06:17.570","Text":"You should really check that these works."},{"Start":"06:17.570 ","End":"06:21.410","Text":"For example, if y=x,"},{"Start":"06:21.410 ","End":"06:24.005","Text":"let me look for the beginning of the exercise."},{"Start":"06:24.005 ","End":"06:25.610","Text":"Don\u0027t usually do this,"},{"Start":"06:25.610 ","End":"06:27.800","Text":"but let\u0027s check that, for example,"},{"Start":"06:27.800 ","End":"06:31.640","Text":"the y=x is indeed a solution."},{"Start":"06:31.640 ","End":"06:34.675","Text":"Y\u0027 is equal to 1."},{"Start":"06:34.675 ","End":"06:35.985","Text":"That\u0027s the left-hand side."},{"Start":"06:35.985 ","End":"06:41.985","Text":"The right-hand side 4y minus 3x over 2x minus y,"},{"Start":"06:41.985 ","End":"06:46.275","Text":"if y=x and everything is x here so 4x minus 3x is x."},{"Start":"06:46.275 ","End":"06:48.165","Text":"2x minus x is x,"},{"Start":"06:48.165 ","End":"06:50.595","Text":"and this equals 1, does 1=1?"},{"Start":"06:50.595 ","End":"06:51.980","Text":"The answer is yes."},{"Start":"06:51.980 ","End":"06:53.870","Text":"Let\u0027s quickly do the other one also,"},{"Start":"06:53.870 ","End":"06:56.645","Text":"what about y equals minus 3x?"},{"Start":"06:56.645 ","End":"06:59.060","Text":"Y\u0027 is minus 3."},{"Start":"06:59.060 ","End":"07:01.715","Text":"Let\u0027s see the right-hand side we have 4y,"},{"Start":"07:01.715 ","End":"07:08.670","Text":"which is minus 12x minus 3x over 2x minus y,"},{"Start":"07:08.670 ","End":"07:10.140","Text":"which is minus 3x,"},{"Start":"07:10.140 ","End":"07:11.760","Text":"which is plus 3x."},{"Start":"07:11.760 ","End":"07:15.645","Text":"What do we get? Minus 15x over 5x,"},{"Start":"07:15.645 ","End":"07:17.415","Text":"which is minus 3."},{"Start":"07:17.415 ","End":"07:19.290","Text":"Does minus 3 equal minus 3?"},{"Start":"07:19.290 ","End":"07:22.190","Text":"Yes, so the 2 singular solutions actually work."},{"Start":"07:22.190 ","End":"07:23.420","Text":"But I don\u0027t usually do this,"},{"Start":"07:23.420 ","End":"07:29.585","Text":"we just accepted that they will and we just highlight the 3 solutions that we have,"},{"Start":"07:29.585 ","End":"07:30.845","Text":"this, this, and this."},{"Start":"07:30.845 ","End":"07:33.140","Text":"However, we\u0027re not completely done."},{"Start":"07:33.140 ","End":"07:35.720","Text":"I still need to show you how I got this integral."},{"Start":"07:35.720 ","End":"07:37.775","Text":"I promised that I would do it later,"},{"Start":"07:37.775 ","End":"07:41.720","Text":"that the integral of this is what\u0027s written here."},{"Start":"07:41.720 ","End":"07:45.800","Text":"Here we are in a new page and I copied the thing I\u0027m going to integrate."},{"Start":"07:45.800 ","End":"07:46.940","Text":"I didn\u0027t copy the integral,"},{"Start":"07:46.940 ","End":"07:48.410","Text":"will put the integral in the end."},{"Start":"07:48.410 ","End":"07:52.175","Text":"What we have to do is partial fractions decomposition."},{"Start":"07:52.175 ","End":"07:55.250","Text":"If hopefully, you remember that if not, I\u0027ll remind you."},{"Start":"07:55.250 ","End":"07:59.195","Text":"We already talked about factoring the denominator,"},{"Start":"07:59.195 ","End":"08:03.095","Text":"and it comes out to be v minus 1, v plus 3."},{"Start":"08:03.095 ","End":"08:07.310","Text":"Since we have a quadratic here and something of lesser degree, linear, on the top,"},{"Start":"08:07.310 ","End":"08:10.460","Text":"we can break it up into partial fractions as some"},{"Start":"08:10.460 ","End":"08:14.615","Text":"constant over v minus 1 plus some constant over v plus 3."},{"Start":"08:14.615 ","End":"08:18.920","Text":"The next thing we do is we multiply both sides by"},{"Start":"08:18.920 ","End":"08:21.470","Text":"this denominator so we get that 2"},{"Start":"08:21.470 ","End":"08:24.845","Text":"minus v is equal to this times this plus this times this."},{"Start":"08:24.845 ","End":"08:27.070","Text":"This was A times v plus 3,"},{"Start":"08:27.070 ","End":"08:30.645","Text":"B times v minus 1 because this times, this is this."},{"Start":"08:30.645 ","End":"08:35.810","Text":"Next, we try and solve A and B by plugging in appropriate values."},{"Start":"08:35.810 ","End":"08:38.090","Text":"We can put v=1,"},{"Start":"08:38.090 ","End":"08:41.315","Text":"and then we can put v is minus 3. I\u0027ll do them both."},{"Start":"08:41.315 ","End":"08:44.380","Text":"If we put v=1 here,"},{"Start":"08:44.380 ","End":"08:48.360","Text":"then this becomes 0 and here I get 1 plus 3 is 4,"},{"Start":"08:48.360 ","End":"08:52.605","Text":"and that gives us that A is 1 over 4."},{"Start":"08:52.605 ","End":"08:55.710","Text":"If we put v is minus 3,"},{"Start":"08:55.710 ","End":"08:58.370","Text":"then this thing becomes 0 and here we get minus 3"},{"Start":"08:58.370 ","End":"09:00.980","Text":"minus 1 which is minus 4 and here we get 2 minus,"},{"Start":"09:00.980 ","End":"09:03.769","Text":"minus 3, which is 5 and we get 5/4."},{"Start":"09:03.769 ","End":"09:07.205","Text":"If I put A is 1/4 and B is 5/4,"},{"Start":"09:07.205 ","End":"09:11.705","Text":"then we put the integral back in."},{"Start":"09:11.705 ","End":"09:15.680","Text":"This thing here is equal to this thing here,"},{"Start":"09:15.680 ","End":"09:17.540","Text":"but with A and B replaced,"},{"Start":"09:17.540 ","End":"09:19.175","Text":"so it\u0027s 1/4,"},{"Start":"09:19.175 ","End":"09:21.695","Text":"which is A over v plus 1,"},{"Start":"09:21.695 ","End":"09:25.365","Text":"and B which is minus 5/4 over the v plus 3."},{"Start":"09:25.365 ","End":"09:26.760","Text":"Each of these is straightforward,"},{"Start":"09:26.760 ","End":"09:28.620","Text":"it\u0027s a natural logarithm."},{"Start":"09:28.620 ","End":"09:31.955","Text":"Here we have 1/4, here we have minus 5/4,"},{"Start":"09:31.955 ","End":"09:34.580","Text":"and this is what we claimed before."},{"Start":"09:34.580 ","End":"09:37.700","Text":"If you check back, you\u0027ll see that this is exactly what we"},{"Start":"09:37.700 ","End":"09:42.580","Text":"did and so I\u0027m paying my debt and we are done."}],"Thumbnail":null,"ID":7654},{"Watched":false,"Name":"Exercise 3","Duration":"5m 26s","ChapterTopicVideoID":7580,"CourseChapterTopicPlaylistID":4219,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.220","Text":"Here we have a differential equation to solve."},{"Start":"00:02.220 ","End":"00:05.100","Text":"It will turn out to be a homogeneous equation."},{"Start":"00:05.100 ","End":"00:07.335","Text":"I\u0027m going to just copy it over here."},{"Start":"00:07.335 ","End":"00:09.360","Text":"What I\u0027m going to do now is manipulate it,"},{"Start":"00:09.360 ","End":"00:11.910","Text":"bring this over to this side,"},{"Start":"00:11.910 ","End":"00:17.250","Text":"then I recall that y\u0027 is just dy/dx."},{"Start":"00:17.250 ","End":"00:18.690","Text":"I put that instead of y\u0027,"},{"Start":"00:18.690 ","End":"00:20.520","Text":"and multiply by dx,"},{"Start":"00:20.520 ","End":"00:22.245","Text":"then I\u0027ll get the following."},{"Start":"00:22.245 ","End":"00:24.674","Text":"This bit in front of the dx,"},{"Start":"00:24.674 ","End":"00:26.538","Text":"I\u0027ll call M, M(x,"},{"Start":"00:26.538 ","End":"00:28.395","Text":"y) and this in front of the dy,"},{"Start":"00:28.395 ","End":"00:30.690","Text":"I\u0027ll call N, which is N(x, y)."},{"Start":"00:30.690 ","End":"00:33.330","Text":"To show that this equation is homogeneous,"},{"Start":"00:33.330 ","End":"00:37.230","Text":"I have to show that M and N are both homogeneous functions of the same order."},{"Start":"00:37.230 ","End":"00:41.990","Text":"First of all, let\u0027s do M. I\u0027m not going to dwell on all the details,"},{"Start":"00:41.990 ","End":"00:43.804","Text":"but you can follow this easily."},{"Start":"00:43.804 ","End":"00:46.835","Text":"It turns out to be homogeneous of order 2."},{"Start":"00:46.835 ","End":"00:50.150","Text":"Similarly N, If you do the work and substitute Lambda x,"},{"Start":"00:50.150 ","End":"00:52.475","Text":"Lambda y, the Lambda^2 comes out."},{"Start":"00:52.475 ","End":"00:55.145","Text":"Because of the 2, this is also homogeneous of degree 2."},{"Start":"00:55.145 ","End":"00:59.480","Text":"They\u0027re both homogeneous of Degree 2 which means that we have a homogeneous equation."},{"Start":"00:59.480 ","End":"01:02.810","Text":"There\u0027s a standard technique for solving the homogeneous equations."},{"Start":"01:02.810 ","End":"01:06.320","Text":"We make this substitution that y is v times x."},{"Start":"01:06.320 ","End":"01:07.715","Text":"These are new variable,"},{"Start":"01:07.715 ","End":"01:10.010","Text":"and dy is given by this,"},{"Start":"01:10.010 ","End":"01:11.749","Text":"which is just the product rule of this,"},{"Start":"01:11.749 ","End":"01:14.180","Text":"and maybe I\u0027ll add that later on,"},{"Start":"01:14.180 ","End":"01:18.494","Text":"we\u0027ll need to substitute back from v back to x and y,"},{"Start":"01:18.494 ","End":"01:22.720","Text":"that v is equal to y/x."},{"Start":"01:22.720 ","End":"01:28.280","Text":"In general, we\u0027re going to assume that x≠0 in our domain."},{"Start":"01:28.280 ","End":"01:31.940","Text":"The next step is to just substitute."},{"Start":"01:31.940 ","End":"01:33.290","Text":"Now what have I done here?"},{"Start":"01:33.290 ","End":"01:41.075","Text":"What I\u0027ve done is to take the equation which I have here and wherever I see the y,"},{"Start":"01:41.075 ","End":"01:43.745","Text":"which is here and here,"},{"Start":"01:43.745 ","End":"01:49.265","Text":"I replace it by v times x and whenever I see dy,"},{"Start":"01:49.265 ","End":"01:51.865","Text":"I replace it by this."},{"Start":"01:51.865 ","End":"01:54.950","Text":"If you see that\u0027s just exactly what I\u0027ve done."},{"Start":"01:54.950 ","End":"02:00.230","Text":"I\u0027ve just replaced the y here by vx, here and here."},{"Start":"02:00.230 ","End":"02:02.690","Text":"I\u0027ve replaced the dy by this."},{"Start":"02:02.690 ","End":"02:05.395","Text":"Now, we\u0027ve got to simplify a bit."},{"Start":"02:05.395 ","End":"02:07.950","Text":"First of all we open up brackets,"},{"Start":"02:07.950 ","End":"02:09.390","Text":"so we get all this."},{"Start":"02:09.390 ","End":"02:10.890","Text":"If you follow,"},{"Start":"02:10.890 ","End":"02:12.650","Text":"then we\u0027re going to separate."},{"Start":"02:12.650 ","End":"02:15.950","Text":"Let\u0027s say, we\u0027ll collect together all the dx\u0027s,"},{"Start":"02:15.950 ","End":"02:19.655","Text":"so we have a dx here and here and here,"},{"Start":"02:19.655 ","End":"02:23.135","Text":"and we have the dv\u0027s here and here."},{"Start":"02:23.135 ","End":"02:25.880","Text":"Then just collecting terms like in algebra,"},{"Start":"02:25.880 ","End":"02:29.120","Text":"you get this, you take out the dx\u0027s separately,"},{"Start":"02:29.120 ","End":"02:32.855","Text":"the yellow ones and then the purple ones and so on,"},{"Start":"02:32.855 ","End":"02:35.520","Text":"then some things cancel for a start."},{"Start":"02:35.520 ","End":"02:41.900","Text":"For example, this v^2x^2 and this x^2v^2 cancel because this I changed the order."},{"Start":"02:41.900 ","End":"02:43.160","Text":"You can see it\u0027s the same."},{"Start":"02:43.160 ","End":"02:44.750","Text":"That\u0027s as far this term goes,"},{"Start":"02:44.750 ","End":"02:46.220","Text":"and as far as the other term goes,"},{"Start":"02:46.220 ","End":"02:50.790","Text":"x^3 comes out of the brackets and only left with 1-v here."},{"Start":"02:50.790 ","End":"02:52.935","Text":"That\u0027s a big simplification,"},{"Start":"02:52.935 ","End":"02:56.810","Text":"and that brings us to the point just copied it basically."},{"Start":"02:56.810 ","End":"03:03.190","Text":"Now, we can divide by x^2 because x is not 0 like here."},{"Start":"03:03.190 ","End":"03:05.240","Text":"If we divide both sides by x^2,"},{"Start":"03:05.240 ","End":"03:07.369","Text":"which is the smallest of these 2 powers,"},{"Start":"03:07.369 ","End":"03:11.060","Text":"then we end up getting just vdx because the x^2 is gone."},{"Start":"03:11.060 ","End":"03:13.250","Text":"But here, one of the x\u0027s remains,"},{"Start":"03:13.250 ","End":"03:14.900","Text":"so this is what we get."},{"Start":"03:14.900 ","End":"03:18.170","Text":"Next thing to do is to take a common denominator."},{"Start":"03:18.170 ","End":"03:20.450","Text":"I\u0027m going to divide by v and by x."},{"Start":"03:20.450 ","End":"03:21.647","Text":"This will be dx/x."},{"Start":"03:21.647 ","End":"03:23.623","Text":"This will be over v. But,"},{"Start":"03:23.623 ","End":"03:25.265","Text":"I\u0027m going to do 2 steps in 1."},{"Start":"03:25.265 ","End":"03:27.530","Text":"I\u0027m also going to transfer one of them to the other side"},{"Start":"03:27.530 ","End":"03:29.780","Text":"and make it minus and sure if you look at it,"},{"Start":"03:29.780 ","End":"03:32.900","Text":"you\u0027ll see that this is what we can get after a common denominator."},{"Start":"03:32.900 ","End":"03:36.314","Text":"I\u0027m throwing one of them to the other side and calling it minus."},{"Start":"03:36.314 ","End":"03:40.790","Text":"This point, we\u0027ve pretty much already separated the variables."},{"Start":"03:40.790 ","End":"03:44.000","Text":"What I\u0027m going to do is just put an integral sign in front of each."},{"Start":"03:44.000 ","End":"03:47.850","Text":"But, I\u0027m going to continue on the next page because I\u0027m running out of space here."},{"Start":"03:47.850 ","End":"03:49.730","Text":"Here\u0027s the same as we had before,"},{"Start":"03:49.730 ","End":"03:52.220","Text":"just with an integral sign stuck in front."},{"Start":"03:52.220 ","End":"03:55.115","Text":"We need to do a bit of simplification on this term."},{"Start":"03:55.115 ","End":"03:57.440","Text":"This is 1/v -v/v,"},{"Start":"03:57.440 ","End":"03:59.270","Text":"and v/v is 1."},{"Start":"03:59.270 ","End":"04:02.090","Text":"This is the integral that we get."},{"Start":"04:02.090 ","End":"04:05.000","Text":"Now, 1/x is just natural logs,"},{"Start":"04:05.000 ","End":"04:07.670","Text":"so that\u0027s minus and the minus is there."},{"Start":"04:07.670 ","End":"04:11.249","Text":"The 1/v is also natural log with the absolute value."},{"Start":"04:11.249 ","End":"04:16.075","Text":"The -1 gives us -v. Then there\u0027s the constant of integration."},{"Start":"04:16.075 ","End":"04:19.505","Text":"Basically, after we substitute back,"},{"Start":"04:19.505 ","End":"04:22.190","Text":"remember that I said that when we substitute back,"},{"Start":"04:22.190 ","End":"04:26.765","Text":"we substitute v is equal to y/x."},{"Start":"04:26.765 ","End":"04:28.258","Text":"This v here,"},{"Start":"04:28.258 ","End":"04:35.075","Text":"and this v here become y/x and y/x and that\u0027s the solution."},{"Start":"04:35.075 ","End":"04:38.000","Text":"But, there\u0027s something I forgot to emphasize before."},{"Start":"04:38.000 ","End":"04:39.350","Text":"When we got to this stage,"},{"Start":"04:39.350 ","End":"04:41.270","Text":"before we did the integration,"},{"Start":"04:41.270 ","End":"04:42.870","Text":"we divide it by v. Now,"},{"Start":"04:42.870 ","End":"04:44.450","Text":"we already said that x is not 0,"},{"Start":"04:44.450 ","End":"04:47.270","Text":"but this implies that v≠0."},{"Start":"04:47.270 ","End":"04:49.820","Text":"Now, if v=0,"},{"Start":"04:49.820 ","End":"04:54.890","Text":"this actually provides a solution to the differential equation at this level."},{"Start":"04:54.890 ","End":"04:58.175","Text":"If v is 0, then dv is also 0."},{"Start":"04:58.175 ","End":"05:02.300","Text":"That gives us an extra singular solution where v is 0."},{"Start":"05:02.300 ","End":"05:04.700","Text":"Let me get back to where I was before."},{"Start":"05:04.700 ","End":"05:11.570","Text":"Here I was. We also have the possibility that in fact v=0 for this singular solution."},{"Start":"05:11.570 ","End":"05:13.220","Text":"If v is 0, then y,"},{"Start":"05:13.220 ","End":"05:15.670","Text":"which is vx is also 0."},{"Start":"05:15.670 ","End":"05:20.360","Text":"Actually, get a second solution that y=0."},{"Start":"05:20.360 ","End":"05:22.790","Text":"This is the regular solution."},{"Start":"05:22.790 ","End":"05:24.605","Text":"This is the singular solution."},{"Start":"05:24.605 ","End":"05:27.120","Text":"Anyway, we are done."}],"Thumbnail":null,"ID":7655},{"Watched":false,"Name":"Exercise 4","Duration":"5m 36s","ChapterTopicVideoID":7581,"CourseChapterTopicPlaylistID":4219,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.415","Text":"Here we have a differential equation to solve,"},{"Start":"00:02.415 ","End":"00:06.930","Text":"and I\u0027m going to show you that it\u0027s actually a homogeneous differential equation."},{"Start":"00:06.930 ","End":"00:09.690","Text":"If we call this one M and this one N,"},{"Start":"00:09.690 ","End":"00:11.550","Text":"then I\u0027ll show you that M and N are both"},{"Start":"00:11.550 ","End":"00:15.540","Text":"homogeneous functions of x and y of the same order."},{"Start":"00:15.540 ","End":"00:20.820","Text":"Let\u0027s start with M and then we\u0027ll do N. M(Lambda x, Lambda y)."},{"Start":"00:20.820 ","End":"00:22.410","Text":"Well, I won\u0027t bore you with all the details."},{"Start":"00:22.410 ","End":"00:24.420","Text":"We do the computations and it comes out to"},{"Start":"00:24.420 ","End":"00:28.035","Text":"be Lambda^2 times M. This is homogeneous of order 2."},{"Start":"00:28.035 ","End":"00:30.510","Text":"Similarly N you can follow the details later,"},{"Start":"00:30.510 ","End":"00:33.510","Text":"turns out to be Lambda^2N."},{"Start":"00:33.510 ","End":"00:36.330","Text":"Also we have a homogeneous of order 2."},{"Start":"00:36.330 ","End":"00:37.560","Text":"Both are order 2,"},{"Start":"00:37.560 ","End":"00:39.495","Text":"so the original equation is homogeneous,"},{"Start":"00:39.495 ","End":"00:43.410","Text":"which means that we solve it by the usual technique of substituting"},{"Start":"00:43.410 ","End":"00:49.610","Text":"y=v times x)and dy comes out to be this from the product rule."},{"Start":"00:49.610 ","End":"00:54.000","Text":"Later on we\u0027ll probably have to substitute back from v to x and y."},{"Start":"00:54.000 ","End":"00:59.480","Text":"From here we get that v is equal to y over x."},{"Start":"00:59.480 ","End":"01:03.080","Text":"That implies also that we don\u0027t really want x to be 0,"},{"Start":"01:03.080 ","End":"01:04.460","Text":"restriction on the domain."},{"Start":"01:04.460 ","End":"01:06.125","Text":"In this equation here,"},{"Start":"01:06.125 ","End":"01:09.590","Text":"wherever I see y, which is here,"},{"Start":"01:09.590 ","End":"01:11.660","Text":"and here, and here,"},{"Start":"01:11.660 ","End":"01:15.755","Text":"I will replace it with v. x."},{"Start":"01:15.755 ","End":"01:17.720","Text":"Wherever I see dy,"},{"Start":"01:17.720 ","End":"01:20.045","Text":"which can only be in one place here,"},{"Start":"01:20.045 ","End":"01:21.635","Text":"I\u0027ll replace it with this."},{"Start":"01:21.635 ","End":"01:23.750","Text":"If I do these 2 replacements,"},{"Start":"01:23.750 ","End":"01:26.660","Text":"then what I\u0027ll get is this."},{"Start":"01:26.660 ","End":"01:28.100","Text":"You can see here\u0027s y,"},{"Start":"01:28.100 ","End":"01:29.360","Text":"here\u0027s v. x, here\u0027s y,"},{"Start":"01:29.360 ","End":"01:31.025","Text":"here\u0027s v. x, here\u0027s y,"},{"Start":"01:31.025 ","End":"01:33.110","Text":"here\u0027s v. x, here\u0027s dy,"},{"Start":"01:33.110 ","End":"01:37.385","Text":"Now let\u0027s expand the brackets."},{"Start":"01:37.385 ","End":"01:39.920","Text":"What we get is this equation."},{"Start":"01:39.920 ","End":"01:42.185","Text":"This is routine, I\u0027m not going to go over it bit by bit."},{"Start":"01:42.185 ","End":"01:46.040","Text":"But do notice that we have x^2s everywhere,"},{"Start":"01:46.040 ","End":"01:47.255","Text":"we even have x^3,"},{"Start":"01:47.255 ","End":"01:52.205","Text":"but that means we can divide everything by x^2 provided that x is not 0,"},{"Start":"01:52.205 ","End":"01:54.350","Text":"which we already said it\u0027s not."},{"Start":"01:54.350 ","End":"01:57.545","Text":"Now I want to collect like terms together."},{"Start":"01:57.545 ","End":"02:01.070","Text":"I want the terms that have the dx in them,"},{"Start":"02:01.070 ","End":"02:03.005","Text":"which is this and this,"},{"Start":"02:03.005 ","End":"02:05.000","Text":"and this, and this,"},{"Start":"02:05.000 ","End":"02:09.095","Text":"I\u0027ll collect those and the terms that have the dv in them,"},{"Start":"02:09.095 ","End":"02:11.750","Text":"which is this, and this collects separately,"},{"Start":"02:11.750 ","End":"02:13.925","Text":"just like we collect like terms in algebra."},{"Start":"02:13.925 ","End":"02:17.300","Text":"This is algebra. What we get is the following,"},{"Start":"02:17.300 ","End":"02:19.100","Text":"where the dx is 1,"},{"Start":"02:19.100 ","End":"02:20.660","Text":"2, 3, 4 things."},{"Start":"02:20.660 ","End":"02:27.545","Text":"But altogether we got 3v plus v is 4v and v^2 plus v^2 is 2v^2."},{"Start":"02:27.545 ","End":"02:30.785","Text":"For the dv, we got x plus xv."},{"Start":"02:30.785 ","End":"02:32.660","Text":"But the x from these 2,"},{"Start":"02:32.660 ","End":"02:34.085","Text":"I took outside the brackets."},{"Start":"02:34.085 ","End":"02:36.080","Text":"We get this equation."},{"Start":"02:36.080 ","End":"02:39.920","Text":"Now, what we\u0027d like to do is start separating the variables already."},{"Start":"02:39.920 ","End":"02:41.990","Text":"You can see that the v\u0027s are separate from the x."},{"Start":"02:41.990 ","End":"02:44.920","Text":"We just have to change sides for some of the things."},{"Start":"02:44.920 ","End":"02:47.360","Text":"Let\u0027s start by throwing this onto the other side,"},{"Start":"02:47.360 ","End":"02:50.510","Text":"you want to take the 4v plus 2v^2 to this side,"},{"Start":"02:50.510 ","End":"02:52.685","Text":"and we want to take the x to this side,"},{"Start":"02:52.685 ","End":"02:54.620","Text":"and this is what we can do."},{"Start":"02:54.620 ","End":"02:59.765","Text":"However, we have to add that x is not 0 as it wasn\u0027t before,"},{"Start":"02:59.765 ","End":"03:04.460","Text":"but there\u0027s 2 extra conditions on v. You see this denominator,"},{"Start":"03:04.460 ","End":"03:09.365","Text":"I could write it as 2v times v plus 2,"},{"Start":"03:09.365 ","End":"03:14.220","Text":"and this will be 0 if v is equal to 0 or v is equal to negative 2."},{"Start":"03:14.220 ","End":"03:17.450","Text":"I have to add these 2 extra conditions."},{"Start":"03:17.450 ","End":"03:19.745","Text":"In fact if v is equal to 0,"},{"Start":"03:19.745 ","End":"03:21.860","Text":"this actually satisfies the equation."},{"Start":"03:21.860 ","End":"03:24.290","Text":"If v is 0 and then dv is 0,"},{"Start":"03:24.290 ","End":"03:25.370","Text":"and this thing works,"},{"Start":"03:25.370 ","End":"03:27.020","Text":"which means the original thing is the solution,"},{"Start":"03:27.020 ","End":"03:28.805","Text":"but let\u0027s deal with those at the end."},{"Start":"03:28.805 ","End":"03:32.620","Text":"Meanwhile, what we have to do is we\u0027ve separated the variables here."},{"Start":"03:32.620 ","End":"03:36.455","Text":"We just have to put an integral sign in front of each of them."},{"Start":"03:36.455 ","End":"03:42.455","Text":"Here I\u0027d like to use the formula that the integral of f\u0027 over f,"},{"Start":"03:42.455 ","End":"03:44.375","Text":"doesn\u0027t matter anymore variable in this case,"},{"Start":"03:44.375 ","End":"03:50.375","Text":"v is equal to the natural log of absolute value of f plus constant."},{"Start":"03:50.375 ","End":"03:54.860","Text":"Here I want to take f as 2v^2 plus 4v."},{"Start":"03:54.860 ","End":"03:56.780","Text":"However, on the numerator,"},{"Start":"03:56.780 ","End":"03:58.880","Text":"I would then have to have f\u0027,"},{"Start":"03:58.880 ","End":"04:01.025","Text":"which is 4v plus 4."},{"Start":"04:01.025 ","End":"04:03.505","Text":"I don\u0027t have 4v plus 4,"},{"Start":"04:03.505 ","End":"04:07.910","Text":"I have v plus 1 or 1 plus v. A factor of 4 is missing,"},{"Start":"04:07.910 ","End":"04:12.680","Text":"and I can correct that in the usual method by throwing in a 4,"},{"Start":"04:12.680 ","End":"04:13.880","Text":"which is what we needed,"},{"Start":"04:13.880 ","End":"04:17.000","Text":"but then compensating because I can\u0027t just ruin it,"},{"Start":"04:17.000 ","End":"04:19.670","Text":"by putting a 4 in the denominator also."},{"Start":"04:19.670 ","End":"04:22.445","Text":"Now we\u0027re already to use this formula."},{"Start":"04:22.445 ","End":"04:29.540","Text":"What we get is that minus the natural log of x from here and here the quarter,"},{"Start":"04:29.540 ","End":"04:33.995","Text":"and then from the formula natural log of the denominator, plus the constant."},{"Start":"04:33.995 ","End":"04:37.400","Text":"Finally, we have to substitute back."},{"Start":"04:37.400 ","End":"04:42.890","Text":"Remember that v was equal to y over x."},{"Start":"04:42.890 ","End":"04:46.040","Text":"If we substitute that back in there,"},{"Start":"04:46.040 ","End":"04:48.005","Text":"then we get the solution,"},{"Start":"04:48.005 ","End":"04:51.610","Text":"which is, well, just replacing v by y over x,"},{"Start":"04:51.610 ","End":"04:53.195","Text":"here, this is what we get."},{"Start":"04:53.195 ","End":"04:57.725","Text":"Now this is the main solution and I\u0027ll highlight it,"},{"Start":"04:57.725 ","End":"05:02.030","Text":"but we also had v=0 and v=2."},{"Start":"05:02.030 ","End":"05:04.690","Text":"I mentioned that these could also be solutions."},{"Start":"05:04.690 ","End":"05:07.030","Text":"v=0 is a solution."},{"Start":"05:07.030 ","End":"05:09.115","Text":"If v is y x,"},{"Start":"05:09.115 ","End":"05:13.300","Text":"that gives us that y=0 as a solution."},{"Start":"05:13.300 ","End":"05:16.560","Text":"The other one was v equals minus 2."},{"Start":"05:16.560 ","End":"05:18.195","Text":"Again, y is v. x,"},{"Start":"05:18.195 ","End":"05:21.120","Text":"that gives us y equals minus 2x."},{"Start":"05:21.120 ","End":"05:24.140","Text":"Altogether 3 solutions, the one we had above,"},{"Start":"05:24.140 ","End":"05:26.660","Text":"and these 2 solutions that are obtained in"},{"Start":"05:26.660 ","End":"05:30.350","Text":"this way from restrictions on domain of definition,"},{"Start":"05:30.350 ","End":"05:32.345","Text":"they\u0027re called singular solutions."},{"Start":"05:32.345 ","End":"05:37.860","Text":"We have 9 solution and 2 singular solutions and we are done."}],"Thumbnail":null,"ID":7656},{"Watched":false,"Name":"Exercise 5","Duration":"2m 32s","ChapterTopicVideoID":7582,"CourseChapterTopicPlaylistID":4219,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.860","Text":"In this exercise, we have to solve"},{"Start":"00:01.860 ","End":"00:04.020","Text":"this differential equation and"},{"Start":"00:04.020 ","End":"00:06.510","Text":"this will turn out to be homogeneous and that I\u0027ll show you."},{"Start":"00:06.510 ","End":"00:08.670","Text":"If we call this M and this N,"},{"Start":"00:08.670 ","End":"00:13.130","Text":"then what I have to do is to show that M is homogeneous and so is N,"},{"Start":"00:13.130 ","End":"00:14.225","Text":"but of the same order."},{"Start":"00:14.225 ","End":"00:15.995","Text":"Let\u0027s take them one at a time,"},{"Start":"00:15.995 ","End":"00:17.360","Text":"M of Lambda x,"},{"Start":"00:17.360 ","End":"00:20.310","Text":"Lambda y equals. I\u0027ll leave you to follow."},{"Start":"00:20.310 ","End":"00:23.280","Text":"This is Lambda times M of x,"},{"Start":"00:23.280 ","End":"00:26.130","Text":"y, and that means it\u0027s homogeneous of order 1."},{"Start":"00:26.130 ","End":"00:29.070","Text":"Similarly, I\u0027ll let you follow the details later."},{"Start":"00:29.070 ","End":"00:32.160","Text":"N of Lambda x, Lambda y is Lambda times N of x,"},{"Start":"00:32.160 ","End":"00:35.775","Text":"y so this is also homogeneous of order 1 is Lambda to the 1,"},{"Start":"00:35.775 ","End":"00:38.040","Text":"and here it\u0027s Lambda to the 1,1 equals 1,"},{"Start":"00:38.040 ","End":"00:40.695","Text":"so this is indeed a homogeneous equation."},{"Start":"00:40.695 ","End":"00:42.965","Text":"We use our usual substitution,"},{"Start":"00:42.965 ","End":"00:48.079","Text":"which is this and hopefully this will turn into equation which is separable."},{"Start":"00:48.079 ","End":"00:49.400","Text":"That\u0027s the theory anyway,"},{"Start":"00:49.400 ","End":"00:54.480","Text":"I just wanted to remind you that the end usually we have to also substitute back so"},{"Start":"00:54.480 ","End":"01:00.050","Text":"let me write that v is equal to y over x and it implies x is not equal to 0,"},{"Start":"01:00.050 ","End":"01:02.240","Text":"but we\u0027ll get that later on anyway."},{"Start":"01:02.240 ","End":"01:06.305","Text":"As you can see y here becomes vx here,"},{"Start":"01:06.305 ","End":"01:08.305","Text":"y over x becomes v,"},{"Start":"01:08.305 ","End":"01:12.230","Text":"because we can use that one and this y here over x"},{"Start":"01:12.230 ","End":"01:16.385","Text":"also becomes v and the dy becomes what was here,"},{"Start":"01:16.385 ","End":"01:20.605","Text":"dv times x plus dx times v. That\u0027s after substitution."},{"Start":"01:20.605 ","End":"01:23.340","Text":"Next, we want to open the brackets."},{"Start":"01:23.340 ","End":"01:27.080","Text":"If we do that, we\u0027ll get this and I won\u0027t check every detail"},{"Start":"01:27.080 ","End":"01:31.445","Text":"but what we see already is that some things cancel."},{"Start":"01:31.445 ","End":"01:33.770","Text":"This term cancels with this term."},{"Start":"01:33.770 ","End":"01:39.715","Text":"We get simply after the cancellation xdx plus x squared cosine v,"},{"Start":"01:39.715 ","End":"01:43.400","Text":"dv is 0 and that looks very harmless."},{"Start":"01:43.400 ","End":"01:45.890","Text":"I can see that x appears in both of them,"},{"Start":"01:45.890 ","End":"01:49.370","Text":"so I can divide both by x."},{"Start":"01:49.370 ","End":"01:52.175","Text":"Of course, I had the condition x not equal to 0."},{"Start":"01:52.175 ","End":"01:56.750","Text":"I\u0027ll leave the 1dx here and move this to the other side just to make it easier."},{"Start":"01:56.750 ","End":"01:58.040","Text":"Now I can divide,"},{"Start":"01:58.040 ","End":"02:00.890","Text":"the x, can go over to this side."},{"Start":"02:00.890 ","End":"02:02.600","Text":"Now here I only have xs in here,"},{"Start":"02:02.600 ","End":"02:06.230","Text":"I only have vs so I can make it to an integral,"},{"Start":"02:06.230 ","End":"02:08.885","Text":"an integration sign on both sides."},{"Start":"02:08.885 ","End":"02:14.670","Text":"The result is that the integral of 1 over x is natural log of absolute value of x,"},{"Start":"02:14.670 ","End":"02:17.225","Text":"the integral of minus cosine is minus sine."},{"Start":"02:17.225 ","End":"02:18.890","Text":"We have to add a constant, of course."},{"Start":"02:18.890 ","End":"02:24.200","Text":"The last thing to do is to substitute back instead of v we put y over"},{"Start":"02:24.200 ","End":"02:30.080","Text":"x and we end up with this and this is our solution."},{"Start":"02:30.080 ","End":"02:32.940","Text":"That\u0027s the answer and we\u0027re done."}],"Thumbnail":null,"ID":7657},{"Watched":false,"Name":"Exercise 6","Duration":"6m 28s","ChapterTopicVideoID":7583,"CourseChapterTopicPlaylistID":4219,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.105","Text":"Here we have to solve this differential equation that y\u0027 equals this nightmare."},{"Start":"00:06.105 ","End":"00:10.425","Text":"This exercise is going to be slightly different from the previous ones."},{"Start":"00:10.425 ","End":"00:14.625","Text":"First of all, I\u0027d like to say that this is going to be homogeneous."},{"Start":"00:14.625 ","End":"00:17.040","Text":"I\u0027m not going to actually check it this time."},{"Start":"00:17.040 ","End":"00:22.770","Text":"Usually, we make it in the form of some function m dx plus some function n dy."},{"Start":"00:22.770 ","End":"00:24.870","Text":"You actually don\u0027t have to"},{"Start":"00:24.870 ","End":"00:27.930","Text":"check because the worst that will happen is it just won\u0027t work."},{"Start":"00:27.930 ","End":"00:30.825","Text":"You won\u0027t be able to convert it to 1 that\u0027s separable."},{"Start":"00:30.825 ","End":"00:33.190","Text":"But I know this is homogeneous on the proof"},{"Start":"00:33.190 ","End":"00:36.090","Text":"that we\u0027ll be able to work through it to the end."},{"Start":"00:36.090 ","End":"00:38.640","Text":"I\u0027m not going to check. That\u0027s one thing that makes this different,"},{"Start":"00:38.640 ","End":"00:39.850","Text":"but that\u0027s not the real thing."},{"Start":"00:39.850 ","End":"00:41.990","Text":"The thing is that when we\u0027re satisfied that it\u0027s"},{"Start":"00:41.990 ","End":"00:44.615","Text":"homogeneous or we take it as an act of faith,"},{"Start":"00:44.615 ","End":"00:52.155","Text":"then we make the substitution that y=vx and dy is this from the product rule."},{"Start":"00:52.155 ","End":"00:54.740","Text":"We also replace y over x by v,"},{"Start":"00:54.740 ","End":"00:56.870","Text":"especially at the end when we switch back."},{"Start":"00:56.870 ","End":"01:01.145","Text":"Now, all this presupposes that we\u0027re taking y as a function of x."},{"Start":"01:01.145 ","End":"01:03.680","Text":"But the situation in a sense is fairly symmetrical,"},{"Start":"01:03.680 ","End":"01:05.480","Text":"there\u0027s no reason to presume that it\u0027s y,"},{"Start":"01:05.480 ","End":"01:06.920","Text":"which is a function of x."},{"Start":"01:06.920 ","End":"01:10.355","Text":"Conceivably, it could be that x is a function of y."},{"Start":"01:10.355 ","End":"01:11.900","Text":"Now, why am I concerned with that?"},{"Start":"01:11.900 ","End":"01:13.475","Text":"Because here, I see"},{"Start":"01:13.475 ","End":"01:17.825","Text":"that the substitution v=y over x is good when I have a lot of y over x."},{"Start":"01:17.825 ","End":"01:20.960","Text":"But here, I have x over y, x over y,"},{"Start":"01:20.960 ","End":"01:26.075","Text":"x over y. I\u0027m thinking that maybe we should use the parallel formula,"},{"Start":"01:26.075 ","End":"01:27.355","Text":"which is this here."},{"Start":"01:27.355 ","End":"01:29.450","Text":"Basically, it\u0027s the same as this,"},{"Start":"01:29.450 ","End":"01:32.870","Text":"but just replace y by x everywhere and x by y."},{"Start":"01:32.870 ","End":"01:35.210","Text":"This formula works equally well,"},{"Start":"01:35.210 ","End":"01:39.440","Text":"but it has the emphasis that we think of x as a function of y. I\u0027m going to"},{"Start":"01:39.440 ","End":"01:43.895","Text":"ignore this one and I\u0027m going to use this one for this exercise."},{"Start":"01:43.895 ","End":"01:48.700","Text":"Let\u0027s see. Let\u0027s start making the substitution and we\u0027ll see what we get."},{"Start":"01:48.700 ","End":"01:51.590","Text":"First, just let me copy this except that instead of y\u0027,"},{"Start":"01:51.590 ","End":"01:54.050","Text":"I\u0027m going to write dx over dy."},{"Start":"01:54.050 ","End":"01:59.303","Text":"y\u0027 is dy over dx."},{"Start":"01:59.303 ","End":"02:01.040","Text":"Because it\u0027s dx over dy,"},{"Start":"02:01.040 ","End":"02:03.545","Text":"I invert the fraction here."},{"Start":"02:03.545 ","End":"02:07.280","Text":"What\u0027s on the top is on the bottom and what\u0027s on the bottom is on the top."},{"Start":"02:07.280 ","End":"02:09.275","Text":"Now it\u0027s dx over dy."},{"Start":"02:09.275 ","End":"02:11.690","Text":"Now I can make the substitution,"},{"Start":"02:11.690 ","End":"02:13.910","Text":"instead of dx here, I\u0027ll put this."},{"Start":"02:13.910 ","End":"02:16.520","Text":"dx is this here,"},{"Start":"02:16.520 ","End":"02:18.515","Text":"dy stays the same."},{"Start":"02:18.515 ","End":"02:22.490","Text":"Wherever I see an x like x on its own here,"},{"Start":"02:22.490 ","End":"02:25.865","Text":"I put vy and x here is vy."},{"Start":"02:25.865 ","End":"02:27.545","Text":"If I see x over y,"},{"Start":"02:27.545 ","End":"02:31.475","Text":"I\u0027ll put it as v. Here I get a v^2 and here I get a v^2."},{"Start":"02:31.475 ","End":"02:34.550","Text":"Basically, the substitution gives me this."},{"Start":"02:34.550 ","End":"02:35.840","Text":"Now how do we proceed?"},{"Start":"02:35.840 ","End":"02:38.585","Text":"First, I\u0027ll give myself some more space here."},{"Start":"02:38.585 ","End":"02:42.590","Text":"What I\u0027m going to do on the left side is split it up into 2 fractions."},{"Start":"02:42.590 ","End":"02:51.275","Text":"What I\u0027m going to do on the right is cancel by y^2 because I have y^2 here,"},{"Start":"02:51.275 ","End":"02:54.380","Text":"y^2 here, and y^2 here,"},{"Start":"02:54.380 ","End":"02:55.445","Text":"there\u0027s a 2 out here,"},{"Start":"02:55.445 ","End":"02:57.815","Text":"and here, I have y times y."},{"Start":"02:57.815 ","End":"02:59.555","Text":"On the left, I\u0027m going to split it up,"},{"Start":"02:59.555 ","End":"03:01.820","Text":"on the right, I\u0027m taking y^2 out."},{"Start":"03:01.820 ","End":"03:05.520","Text":"What I end up with is dy times v over dy,"},{"Start":"03:05.520 ","End":"03:08.125","Text":"dy times v. Here,"},{"Start":"03:08.125 ","End":"03:09.890","Text":"I get it somewhat simpler."},{"Start":"03:09.890 ","End":"03:12.020","Text":"Next thing I notice is that this,"},{"Start":"03:12.020 ","End":"03:17.480","Text":"the dy over the dy cancels and this is just v. Let me throw the v to the other side."},{"Start":"03:17.480 ","End":"03:20.960","Text":"I get this bit equals this bit minus"},{"Start":"03:20.960 ","End":"03:24.920","Text":"v. Then what I want to do is leave the left side alone,"},{"Start":"03:24.920 ","End":"03:26.810","Text":"but put a common denominator here."},{"Start":"03:26.810 ","End":"03:28.895","Text":"I\u0027m going to show you what I end up with."},{"Start":"03:28.895 ","End":"03:34.370","Text":"How I get this is if I put everything over to (2ve)^v^2."},{"Start":"03:34.370 ","End":"03:37.415","Text":"Here I have to multiply by (2ve)^v^2."},{"Start":"03:37.415 ","End":"03:40.205","Text":"I get minus 2v^2 (e)v^2,"},{"Start":"03:40.205 ","End":"03:45.140","Text":"which just exactly cancels with this after I made a common denominator."},{"Start":"03:45.140 ","End":"03:47.600","Text":"This just boils down to this."},{"Start":"03:47.600 ","End":"03:49.490","Text":"Now it\u0027s looking a lot simpler."},{"Start":"03:49.490 ","End":"03:54.005","Text":"At this point, what I want to do is to separate the variables."},{"Start":"03:54.005 ","End":"03:55.670","Text":"Here, I have a dv,"},{"Start":"03:55.670 ","End":"03:58.835","Text":"the dy I\u0027m going to multiply by is going to be over here."},{"Start":"03:58.835 ","End":"04:02.525","Text":"Everything with v like this numerator is going to go down here."},{"Start":"04:02.525 ","End":"04:05.210","Text":"This 2v is going to go into the numerator here."},{"Start":"04:05.210 ","End":"04:07.759","Text":"This y is going to go into the denominator here."},{"Start":"04:07.759 ","End":"04:10.700","Text":"In short, we\u0027ll end up with this expression."},{"Start":"04:10.700 ","End":"04:11.864","Text":"You can check it."},{"Start":"04:11.864 ","End":"04:15.785","Text":"Here, all the v\u0027s are on the left with the dv and the y\u0027s are on the right with the dy."},{"Start":"04:15.785 ","End":"04:19.505","Text":"But of course, we now have to write a condition that y is not"},{"Start":"04:19.505 ","End":"04:23.590","Text":"equal to 0 in order for this right side to make sense."},{"Start":"04:23.590 ","End":"04:26.105","Text":"We\u0027ll return to this at the end."},{"Start":"04:26.105 ","End":"04:28.160","Text":"Now here, I have an integration to do,"},{"Start":"04:28.160 ","End":"04:32.450","Text":"and I want to put the integral sign in front of each of them."},{"Start":"04:32.450 ","End":"04:36.080","Text":"Both of these are fairly easy integrals because if I produce"},{"Start":"04:36.080 ","End":"04:40.865","Text":"the formula that the integral of the derivative of a function over a function,"},{"Start":"04:40.865 ","End":"04:42.950","Text":"then it doesn\u0027t have to be dx."},{"Start":"04:42.950 ","End":"04:46.700","Text":"This could be a function of v. This is just how it\u0027s usually phrased,"},{"Start":"04:46.700 ","End":"04:49.295","Text":"but the function could be in any variable."},{"Start":"04:49.295 ","End":"04:52.925","Text":"Using this formula and noticing that the derivative of 1 plus"},{"Start":"04:52.925 ","End":"04:57.245","Text":"(e)^v^2 is exactly 2v times (e)v^2."},{"Start":"04:57.245 ","End":"04:58.655","Text":"In fact, even here, it\u0027s true,"},{"Start":"04:58.655 ","End":"05:02.390","Text":"derivative of y is 1 if we\u0027re talking about functions of y."},{"Start":"05:02.390 ","End":"05:06.253","Text":"After the integration, what I\u0027ll get is this."},{"Start":"05:06.253 ","End":"05:11.330","Text":"Natural log of this denominator equals natural log of this denominator,"},{"Start":"05:11.330 ","End":"05:13.325","Text":"with the constant of integration."},{"Start":"05:13.325 ","End":"05:15.260","Text":"The last thing to do, don\u0027t forget,"},{"Start":"05:15.260 ","End":"05:18.800","Text":"is to go back from v to x and y."},{"Start":"05:18.800 ","End":"05:23.660","Text":"Remember that this time that we had that v was not y over x."},{"Start":"05:23.660 ","End":"05:26.450","Text":"This time, v was equal to x over y."},{"Start":"05:26.450 ","End":"05:28.745","Text":"If we substitute that,"},{"Start":"05:28.745 ","End":"05:31.369","Text":"then we get that this is our answer,"},{"Start":"05:31.369 ","End":"05:37.280","Text":"but not completely because remember we also had this y not"},{"Start":"05:37.280 ","End":"05:43.460","Text":"equal to 0 and we have to check possibly y=0 is a solution."},{"Start":"05:43.460 ","End":"05:47.135","Text":"Let\u0027s go right back all the way to the top if we want,"},{"Start":"05:47.135 ","End":"05:48.430","Text":"and let\u0027s look at it."},{"Start":"05:48.430 ","End":"05:51.140","Text":"In fact, I\u0027ll even go to the original equation."},{"Start":"05:51.140 ","End":"05:55.640","Text":"The original equation, if y is equal to the function zero,"},{"Start":"05:55.640 ","End":"05:57.920","Text":"then this numerator is 0,"},{"Start":"05:57.920 ","End":"05:59.480","Text":"and the denominator here,"},{"Start":"05:59.480 ","End":"06:00.995","Text":"I have 0 because of y."},{"Start":"06:00.995 ","End":"06:02.315","Text":"Here, I have 0,"},{"Start":"06:02.315 ","End":"06:06.775","Text":"but this is not 0 because I\u0027ll have e to the power of,"},{"Start":"06:06.775 ","End":"06:10.110","Text":"and y\u0027 is also 0."},{"Start":"06:10.110 ","End":"06:12.285","Text":"We get 0=0."},{"Start":"06:12.285 ","End":"06:15.530","Text":"That means that y=0 is also a solution."},{"Start":"06:15.530 ","End":"06:17.990","Text":"The solution that we get this way is called,"},{"Start":"06:17.990 ","End":"06:19.910","Text":"if you remember, a singular solution."},{"Start":"06:19.910 ","End":"06:28.440","Text":"The last thing we have is also the singular solution y=0. We\u0027re done."}],"Thumbnail":null,"ID":7658},{"Watched":false,"Name":"Exercise 7","Duration":"5m 13s","ChapterTopicVideoID":7584,"CourseChapterTopicPlaylistID":4219,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"Here we have to solve this differential equation,"},{"Start":"00:02.700 ","End":"00:05.010","Text":"which is given with an initial condition."},{"Start":"00:05.010 ","End":"00:08.625","Text":"I\u0027m going to show that this equation is actually homogeneous"},{"Start":"00:08.625 ","End":"00:12.585","Text":"if we label this bit M and this bit N,"},{"Start":"00:12.585 ","End":"00:17.865","Text":"our task is now to show that M is a homogeneous function and so is N,"},{"Start":"00:17.865 ","End":"00:19.695","Text":"and they both have the same order."},{"Start":"00:19.695 ","End":"00:24.750","Text":"Let\u0027s start with M and I\u0027m not going to go into this in great detail,"},{"Start":"00:24.750 ","End":"00:29.985","Text":"but we have to show this is equal to Lambda to some power times M. Well,"},{"Start":"00:29.985 ","End":"00:31.920","Text":"I\u0027m going to let you follow the development."},{"Start":"00:31.920 ","End":"00:34.170","Text":"Basically, we take Lambda squared out here,"},{"Start":"00:34.170 ","End":"00:36.344","Text":"then we bring Lambda out in front,"},{"Start":"00:36.344 ","End":"00:41.459","Text":"and we can see that this is equal to Lambda times the original function."},{"Start":"00:41.459 ","End":"00:47.100","Text":"This is homogeneous of order 1 because it\u0027s Lambda to the power of 1."},{"Start":"00:47.100 ","End":"00:48.840","Text":"The same thing with N,"},{"Start":"00:48.840 ","End":"00:53.735","Text":"if we just substitute set of x and y. Lambda x and Lambda y,"},{"Start":"00:53.735 ","End":"00:57.320","Text":"then we just get Lambda times N. Both of these"},{"Start":"00:57.320 ","End":"01:01.640","Text":"show that M and N are homogeneous of order 1,"},{"Start":"01:01.640 ","End":"01:04.565","Text":"so the equation is indeed homogeneous."},{"Start":"01:04.565 ","End":"01:10.880","Text":"If so then we make this substitution that y is v times x and therefore,"},{"Start":"01:10.880 ","End":"01:14.570","Text":"dy is this, which is from the product rule from here."},{"Start":"01:14.570 ","End":"01:23.030","Text":"Also, useful to have is that when we substitute back v=y over x."},{"Start":"01:23.030 ","End":"01:25.895","Text":"Continuing, this is what I get,"},{"Start":"01:25.895 ","End":"01:28.580","Text":"if you look at the previous,"},{"Start":"01:28.580 ","End":"01:30.500","Text":"then this one here was y,"},{"Start":"01:30.500 ","End":"01:31.990","Text":"and then instead of it I put v.x,"},{"Start":"01:31.990 ","End":"01:33.545","Text":"here we had a y,"},{"Start":"01:33.545 ","End":"01:35.015","Text":"and I put v.x,"},{"Start":"01:35.015 ","End":"01:36.810","Text":"and here I had dy,"},{"Start":"01:36.810 ","End":"01:39.105","Text":"and I put this instead of it."},{"Start":"01:39.105 ","End":"01:41.555","Text":"Now, let\u0027s get to work on simplifying this a bit."},{"Start":"01:41.555 ","End":"01:43.910","Text":"If I take this x^2 here,"},{"Start":"01:43.910 ","End":"01:48.055","Text":"and there\u0027s actually x^2 here also and I want to take this outside the brackets,"},{"Start":"01:48.055 ","End":"01:51.950","Text":"and so this is what I get and I\u0027ve left the rest as is."},{"Start":"01:51.950 ","End":"01:57.380","Text":"Then I can take x^2 outside the square root sign,"},{"Start":"01:57.380 ","End":"02:02.255","Text":"but there\u0027s a small subtle point here is that actually"},{"Start":"02:02.255 ","End":"02:07.735","Text":"the square root of x^2 is actually not equal to x."},{"Start":"02:07.735 ","End":"02:11.150","Text":"It\u0027s equal to x only if x is positive,"},{"Start":"02:11.150 ","End":"02:16.060","Text":"but it\u0027s actually minus x if x is negative."},{"Start":"02:16.060 ","End":"02:18.725","Text":"There\u0027s going to be 2 whole branches here,"},{"Start":"02:18.725 ","End":"02:21.960","Text":"but the solutions will be very similar for positive or negative."},{"Start":"02:21.960 ","End":"02:24.730","Text":"Let\u0027s just assume that x is bigger than 0."},{"Start":"02:24.730 ","End":"02:26.105","Text":"If we want to,"},{"Start":"02:26.105 ","End":"02:29.855","Text":"we could do it again with x less than 0 remarkably similar."},{"Start":"02:29.855 ","End":"02:32.195","Text":"Let\u0027s just restrict x to be positive."},{"Start":"02:32.195 ","End":"02:35.920","Text":"Now, the idea is to get this to be a separable equation."},{"Start":"02:35.920 ","End":"02:37.820","Text":"It means separation of variables."},{"Start":"02:37.820 ","End":"02:42.545","Text":"I want to put the dxs together and the dvs together."},{"Start":"02:42.545 ","End":"02:44.796","Text":"Here, I have dx and here I have dx."},{"Start":"02:44.796 ","End":"02:46.715","Text":"I\u0027ve combined this with this,"},{"Start":"02:46.715 ","End":"02:49.505","Text":"and that\u0027s what I have here and this is over here."},{"Start":"02:49.505 ","End":"02:53.670","Text":"If I look at it, v.x here and minus v.x here cancels,"},{"Start":"02:53.670 ","End":"02:55.725","Text":"that makes it still simpler."},{"Start":"02:55.725 ","End":"02:59.600","Text":"What we\u0027re left with is this minus this equals 0,"},{"Start":"02:59.600 ","End":"03:03.170","Text":"and now we can easily separate the variables."},{"Start":"03:03.170 ","End":"03:09.150","Text":"First of all, I can cancel by x because we said that x is positive, so it\u0027s not 0."},{"Start":"03:09.150 ","End":"03:12.410","Text":"Then we can move this to the other side and now we can do"},{"Start":"03:12.410 ","End":"03:17.420","Text":"the divisions where we can put the vs on the right and the xs on the left."},{"Start":"03:17.420 ","End":"03:19.475","Text":"This is what we get."},{"Start":"03:19.475 ","End":"03:21.560","Text":"The square root goes in the denominator here,"},{"Start":"03:21.560 ","End":"03:23.405","Text":"this x goes into the denominator here."},{"Start":"03:23.405 ","End":"03:24.785","Text":"X is not 0,"},{"Start":"03:24.785 ","End":"03:27.380","Text":"1 plus v^2 can never be 0 either."},{"Start":"03:27.380 ","End":"03:28.817","Text":"It\u0027s always positive."},{"Start":"03:28.817 ","End":"03:33.185","Text":"We can just put an integral sign in front of each of these."},{"Start":"03:33.185 ","End":"03:36.965","Text":"We use the formula sheet to do the integral on the right,"},{"Start":"03:36.965 ","End":"03:39.915","Text":"this integral is just natural logarithm."},{"Start":"03:39.915 ","End":"03:42.605","Text":"We get that the integral of this is this."},{"Start":"03:42.605 ","End":"03:46.565","Text":"Now, this is sometimes called the hyperbolic arcsinh."},{"Start":"03:46.565 ","End":"03:49.100","Text":"This function here, if you haven\u0027t seen it before,"},{"Start":"03:49.100 ","End":"03:50.435","Text":"and with the minus 1."},{"Start":"03:50.435 ","End":"03:54.200","Text":"This is arcsinh(v) and,"},{"Start":"03:54.200 ","End":"03:59.885","Text":"and arcsinh hyperbolic is the inverse function of sinh hyperbolic."},{"Start":"03:59.885 ","End":"04:03.415","Text":"You can look it up on the Internet if you\u0027re not familiar with it."},{"Start":"04:03.415 ","End":"04:06.650","Text":"All we have to do now is substitute back."},{"Start":"04:06.650 ","End":"04:11.209","Text":"Instead of v, we put y over x and this is the answer,"},{"Start":"04:11.209 ","End":"04:13.130","Text":"but not quite, because if you remember,"},{"Start":"04:13.130 ","End":"04:19.130","Text":"we had an initial condition and the initial condition said that y(1) is 0,"},{"Start":"04:19.130 ","End":"04:22.605","Text":"which means that when x is 1, y is 0."},{"Start":"04:22.605 ","End":"04:24.590","Text":"If we plug that into here,"},{"Start":"04:24.590 ","End":"04:31.550","Text":"then we get the natural log of 1 because x is 1 is sinh hyperbolic of y is 0,"},{"Start":"04:31.550 ","End":"04:33.335","Text":"0 over 1 is 0."},{"Start":"04:33.335 ","End":"04:37.400","Text":"Inverse sinh hyperbolic or arcsinh hyperbolic of 0"},{"Start":"04:37.400 ","End":"04:41.870","Text":"happens to be 0 and the natural logarithm of 1 is also 0."},{"Start":"04:41.870 ","End":"04:45.020","Text":"We get 0=0 plus c,"},{"Start":"04:45.020 ","End":"04:47.495","Text":"which makes c to be 0,"},{"Start":"04:47.495 ","End":"04:50.733","Text":"and now we can put the 0 back here."},{"Start":"04:50.733 ","End":"04:55.730","Text":"We get that this is really the answer with the initial condition."},{"Start":"04:55.730 ","End":"05:00.723","Text":"But I\u0027d like to remind you that we did assume that x is bigger than 0."},{"Start":"05:00.723 ","End":"05:05.150","Text":"There may be another solution for x less than 0."},{"Start":"05:05.150 ","End":"05:06.590","Text":"It might come out the same,"},{"Start":"05:06.590 ","End":"05:09.440","Text":"except with an absolute value or a minus x,"},{"Start":"05:09.440 ","End":"05:14.310","Text":"but I\u0027m going to leave it for x bigger than 0 and we\u0027re done."}],"Thumbnail":null,"ID":7659},{"Watched":false,"Name":"Exercise 8","Duration":"9m 50s","ChapterTopicVideoID":7585,"CourseChapterTopicPlaylistID":4219,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.960","Text":"In this exercise, we have a differential equation to solve,"},{"Start":"00:03.960 ","End":"00:06.945","Text":"but it\u0027s a little bit different than what we usually have,"},{"Start":"00:06.945 ","End":"00:09.030","Text":"because we usually have x,"},{"Start":"00:09.030 ","End":"00:10.935","Text":"and y as the variables."},{"Start":"00:10.935 ","End":"00:12.308","Text":"But here we have t,"},{"Start":"00:12.308 ","End":"00:14.100","Text":"and x as the variable."},{"Start":"00:14.100 ","End":"00:16.710","Text":"I\u0027m going to show this as homogeneous."},{"Start":"00:16.710 ","End":"00:18.990","Text":"But, our usual substitution,"},{"Start":"00:18.990 ","End":"00:22.230","Text":"which is that when y is a function of x,"},{"Start":"00:22.230 ","End":"00:27.705","Text":"we take the substitution y equals v times x and dy is this, and so on."},{"Start":"00:27.705 ","End":"00:31.485","Text":"But here we\u0027re going to take not y as a function of x,"},{"Start":"00:31.485 ","End":"00:33.850","Text":"but x is a function of t. It\u0027s often,"},{"Start":"00:33.850 ","End":"00:35.040","Text":"physics this is the case,"},{"Start":"00:35.040 ","End":"00:36.510","Text":"t is the time variable,"},{"Start":"00:36.510 ","End":"00:37.995","Text":"and x is the displacement,"},{"Start":"00:37.995 ","End":"00:40.909","Text":"in which case we use the corresponding formula."},{"Start":"00:40.909 ","End":"00:43.280","Text":"You can get from here to here quite mechanically,"},{"Start":"00:43.280 ","End":"00:45.980","Text":"if wherever you see y you put x,"},{"Start":"00:45.980 ","End":"00:48.590","Text":"and wherever you see x, you put t,"},{"Start":"00:48.590 ","End":"00:51.080","Text":"then this is the formula you will get,"},{"Start":"00:51.080 ","End":"00:55.730","Text":"so we\u0027ll be using the lower one because it\u0027s x,"},{"Start":"00:55.730 ","End":"00:58.085","Text":"and t. I\u0027m just going to highlight it."},{"Start":"00:58.085 ","End":"01:05.360","Text":"In this case, we use to substitute at the end that v is equal to y over x."},{"Start":"01:05.360 ","End":"01:09.620","Text":"In our case, we\u0027re going to have the v equals instead of y over x,"},{"Start":"01:09.620 ","End":"01:12.650","Text":"x over t. When we substitute back,"},{"Start":"01:12.650 ","End":"01:14.435","Text":"I said it was homogeneous,"},{"Start":"01:14.435 ","End":"01:16.325","Text":"but I haven\u0027t shown it yet."},{"Start":"01:16.325 ","End":"01:19.475","Text":"Let\u0027s just get to work on showing that it\u0027s homogeneous."},{"Start":"01:19.475 ","End":"01:23.905","Text":"I let the bit in front of dt be m,"},{"Start":"01:23.905 ","End":"01:25.400","Text":"and the bit in front of dx,"},{"Start":"01:25.400 ","End":"01:29.300","Text":"we\u0027ll call that n. We have to show that each of the functions m,"},{"Start":"01:29.300 ","End":"01:33.965","Text":"and n are homogeneous functions of a certain order and the orders have to be equal."},{"Start":"01:33.965 ","End":"01:36.905","Text":"As per m, I\u0027m not going to go into detail."},{"Start":"01:36.905 ","End":"01:38.480","Text":"You can just follow this line."},{"Start":"01:38.480 ","End":"01:41.900","Text":"Basically we get that m is lambda t is lambda x so at the end is"},{"Start":"01:41.900 ","End":"01:45.775","Text":"lambda cube times m. This is homogeneous of order 3."},{"Start":"01:45.775 ","End":"01:50.110","Text":"Similarly, n is also homogeneous of order 3."},{"Start":"01:50.110 ","End":"01:51.530","Text":"We get from lambda t,"},{"Start":"01:51.530 ","End":"01:53.735","Text":"lambda x, lambda comes out cubed."},{"Start":"01:53.735 ","End":"01:59.255","Text":"This is homogeneous, which means that we can use the substitution,"},{"Start":"01:59.255 ","End":"02:00.410","Text":"like we said before,"},{"Start":"02:00.410 ","End":"02:02.360","Text":"the one that\u0027s highlighted here."},{"Start":"02:02.360 ","End":"02:10.565","Text":"What I\u0027m going to do is substitute x equals vt for x here, here, here,"},{"Start":"02:10.565 ","End":"02:12.785","Text":"here, and here,"},{"Start":"02:12.785 ","End":"02:19.550","Text":"and I\u0027m going to substitute dx is this in just one place that would be here."},{"Start":"02:19.550 ","End":"02:21.410","Text":"If I do all that,"},{"Start":"02:21.410 ","End":"02:24.890","Text":"then what we\u0027ll get is this mess here,"},{"Start":"02:24.890 ","End":"02:27.410","Text":"which isn\u0027t so bad as it looks."},{"Start":"02:27.410 ","End":"02:29.355","Text":"Basically, you see here x,"},{"Start":"02:29.355 ","End":"02:30.990","Text":"here vt, here x,"},{"Start":"02:30.990 ","End":"02:32.460","Text":"here vt and so on."},{"Start":"02:32.460 ","End":"02:36.280","Text":"These vt\u0027s are instead of the x\u0027s and instead of dx, we have this."},{"Start":"02:36.280 ","End":"02:39.335","Text":"Now we have to try, and simplify this a bit."},{"Start":"02:39.335 ","End":"02:44.690","Text":"We can expand, open up the brackets here and we\u0027ll get to v^2,"},{"Start":"02:44.690 ","End":"02:48.530","Text":"t cubed and so on and so on and so on."},{"Start":"02:48.530 ","End":"02:56.750","Text":"Just opening up all these powers of vt. Then we can collect together all the dt terms,"},{"Start":"02:56.750 ","End":"03:02.255","Text":"here and here, and dv is here."},{"Start":"03:02.255 ","End":"03:06.770","Text":"This bit of the dt I just copied over here."},{"Start":"03:06.770 ","End":"03:09.746","Text":"This dt has the V next to it."},{"Start":"03:09.746 ","End":"03:12.890","Text":"If I multiply all of this by v,"},{"Start":"03:12.890 ","End":"03:14.060","Text":"so it\u0027s just like this,"},{"Start":"03:14.060 ","End":"03:16.250","Text":"but with the powers of v raised by 1."},{"Start":"03:16.250 ","End":"03:18.868","Text":"It\u0027s here v to the fourth here, v cubed,"},{"Start":"03:18.868 ","End":"03:20.250","Text":"here, v^2,"},{"Start":"03:20.250 ","End":"03:22.275","Text":"and that\u0027s the dt stuff."},{"Start":"03:22.275 ","End":"03:29.000","Text":"Now the dv stuff is all of this multiplied by t. So here\u0027s the T and all of this."},{"Start":"03:29.000 ","End":"03:30.470","Text":"I just took the 2 outside,"},{"Start":"03:30.470 ","End":"03:32.330","Text":"so I\u0027ve got the coefficients 2, 3,"},{"Start":"03:32.330 ","End":"03:34.730","Text":"and 1 instead of what it was."},{"Start":"03:34.730 ","End":"03:36.950","Text":"We can take stuff outside the brackets."},{"Start":"03:36.950 ","End":"03:41.817","Text":"For example, in all of the dt terms we have v^2 t cubed at least."},{"Start":"03:41.817 ","End":"03:43.055","Text":"If I take this out,"},{"Start":"03:43.055 ","End":"03:45.580","Text":"then what I\u0027m left with is,"},{"Start":"03:45.580 ","End":"03:49.235","Text":"this 4 becomes because I have a 2 here,"},{"Start":"03:49.235 ","End":"03:51.215","Text":"and a 2 here for this term,"},{"Start":"03:51.215 ","End":"03:52.730","Text":"2 and 2 is 4,"},{"Start":"03:52.730 ","End":"03:56.780","Text":"this 8v comes from this and this 2 and 6 is 8."},{"Start":"03:56.780 ","End":"03:59.150","Text":"Well it\u0027s minus 2, and minus 6 is minus 8."},{"Start":"03:59.150 ","End":"04:01.205","Text":"The last one as this is 4,"},{"Start":"04:01.205 ","End":"04:02.975","Text":"so that gives us the dt bit."},{"Start":"04:02.975 ","End":"04:05.040","Text":"Now from the dv,"},{"Start":"04:05.040 ","End":"04:11.405","Text":"I have 2 with the t. I can take out another vt cubed,"},{"Start":"04:11.405 ","End":"04:12.935","Text":"which is in all of these,"},{"Start":"04:12.935 ","End":"04:16.040","Text":"and what I\u0027m left with after I take it out is this."},{"Start":"04:16.040 ","End":"04:19.820","Text":"We\u0027re getting close now to the separation of the variables because look,"},{"Start":"04:19.820 ","End":"04:21.110","Text":"this only contains v,"},{"Start":"04:21.110 ","End":"04:27.140","Text":"this only contains v. Just take a 4 out of here first to make the numbers smaller."},{"Start":"04:27.140 ","End":"04:29.675","Text":"Then we notice something else,"},{"Start":"04:29.675 ","End":"04:34.325","Text":"that 1 minus 2v plus v^2 is one minus v^2."},{"Start":"04:34.325 ","End":"04:41.340","Text":"Similarly, this term in parentheses is equal to this bit here."},{"Start":"04:41.340 ","End":"04:46.190","Text":"Because in general, if we have a quadratic polynomial in v,"},{"Start":"04:46.190 ","End":"04:49.490","Text":"av^2 plus bv plus c,"},{"Start":"04:49.490 ","End":"04:58.245","Text":"this factorizes into a times v minus v1 times v minus v2,"},{"Start":"04:58.245 ","End":"05:04.070","Text":"where v1 and v2 are the solutions to the equation."},{"Start":"05:04.070 ","End":"05:07.310","Text":"This equals 0 and we know how to solve a quadratic equation."},{"Start":"05:07.310 ","End":"05:09.470","Text":"If we solve this quadratic equation,"},{"Start":"05:09.470 ","End":"05:16.245","Text":"we\u0027ll get that v equals 1 or v equals 1.5."},{"Start":"05:16.245 ","End":"05:18.140","Text":"That\u0027s if we let this equal 0,"},{"Start":"05:18.140 ","End":"05:19.400","Text":"these are the solutions."},{"Start":"05:19.400 ","End":"05:21.485","Text":"The half is 0.5."},{"Start":"05:21.485 ","End":"05:22.880","Text":"So according to this formula,"},{"Start":"05:22.880 ","End":"05:24.940","Text":"we take the a, which is the 2,"},{"Start":"05:24.940 ","End":"05:29.060","Text":"and then v minus the first root and v minus the second root."},{"Start":"05:29.060 ","End":"05:31.505","Text":"So that\u0027s how we get to this stage."},{"Start":"05:31.505 ","End":"05:33.980","Text":"Now we\u0027re going to simplify a lot more."},{"Start":"05:33.980 ","End":"05:38.215","Text":"I\u0027d like to cancel 1 minus v with v minus 1."},{"Start":"05:38.215 ","End":"05:42.160","Text":"But first, I have to write this as one minus v and I\u0027ll put a minus in front."},{"Start":"05:42.160 ","End":"05:45.940","Text":"Also want to combine the 2 with the 2 to make it 4."},{"Start":"05:45.940 ","End":"05:48.780","Text":"So here we get the 4 from the 2 times 2."},{"Start":"05:48.780 ","End":"05:50.450","Text":"Here I have 1 minus V,"},{"Start":"05:50.450 ","End":"05:52.055","Text":"but I put a minus here."},{"Start":"05:52.055 ","End":"05:54.385","Text":"Now, I can cancel things."},{"Start":"05:54.385 ","End":"05:58.350","Text":"For example, this 4 can go with this 4,"},{"Start":"05:58.350 ","End":"06:00.710","Text":"and 0 over 4 is still 0."},{"Start":"06:00.710 ","End":"06:05.570","Text":"I can cancel 1 minus v with one of the 1 minus v\u0027s here,"},{"Start":"06:05.570 ","End":"06:09.365","Text":"so we\u0027re just left with 1 minus v. It\u0027s as if I\u0027ve canceled this 2 here."},{"Start":"06:09.365 ","End":"06:12.620","Text":"This v can cancel with one of the v\u0027s in the v^2,"},{"Start":"06:12.620 ","End":"06:15.170","Text":"so I just cancel this 2."},{"Start":"06:15.170 ","End":"06:20.180","Text":"Then, another thing I can cancel t cubed and t to the fourth,"},{"Start":"06:20.180 ","End":"06:25.174","Text":"so that just leaves me t. Now we\u0027ve really got something quite simple."},{"Start":"06:25.174 ","End":"06:28.545","Text":"V times 1 minus v dt."},{"Start":"06:28.545 ","End":"06:31.890","Text":"What\u0027s left is the t from here,"},{"Start":"06:31.890 ","End":"06:36.430","Text":"the v minus a half from here, dv equals 0."},{"Start":"06:36.430 ","End":"06:38.510","Text":"Now we counseled by t cubed,"},{"Start":"06:38.510 ","End":"06:40.400","Text":"so t is not 0."},{"Start":"06:40.400 ","End":"06:42.440","Text":"We canceled by 1 minus v,"},{"Start":"06:42.440 ","End":"06:44.240","Text":"so v is not equal to 1."},{"Start":"06:44.240 ","End":"06:46.100","Text":"We also divided by v,"},{"Start":"06:46.100 ","End":"06:47.630","Text":"so v is not equal to 0."},{"Start":"06:47.630 ","End":"06:49.460","Text":"So we have all these conditions,"},{"Start":"06:49.460 ","End":"06:52.445","Text":"but we will return to those if necessary."},{"Start":"06:52.445 ","End":"06:54.305","Text":"Meanwhile, let\u0027s continue here."},{"Start":"06:54.305 ","End":"06:58.175","Text":"Now it\u0027s easy to separate the variables."},{"Start":"06:58.175 ","End":"07:01.430","Text":"I copied what we had before."},{"Start":"07:01.430 ","End":"07:05.345","Text":"Now what we can get is we bring the t to this side,"},{"Start":"07:05.345 ","End":"07:08.540","Text":"and bring the V times 1 minus v to the other side."},{"Start":"07:08.540 ","End":"07:10.255","Text":"This is what we get."},{"Start":"07:10.255 ","End":"07:16.340","Text":"We have complete separation stuff with t times dt and stuff with v times dv."},{"Start":"07:16.340 ","End":"07:19.790","Text":"I\u0027ve copied the restrictions that we had before,"},{"Start":"07:19.790 ","End":"07:22.160","Text":"which definitely is still hold."},{"Start":"07:22.160 ","End":"07:25.175","Text":"Now, just put an integral sign in front of each."},{"Start":"07:25.175 ","End":"07:28.370","Text":"I want to expand this denominator to get this,"},{"Start":"07:28.370 ","End":"07:32.705","Text":"I\u0027m going to multiply top and bottom by minus 2 here."},{"Start":"07:32.705 ","End":"07:34.205","Text":"Why would I do that?"},{"Start":"07:34.205 ","End":"07:37.100","Text":"Well, I would like to have the derivative of"},{"Start":"07:37.100 ","End":"07:40.700","Text":"the denominator in the numerator because then I have a formula."},{"Start":"07:40.700 ","End":"07:42.905","Text":"I\u0027m reverse engineering it."},{"Start":"07:42.905 ","End":"07:46.895","Text":"I mean, I planned it so that the derivative v minus v^2,"},{"Start":"07:46.895 ","End":"07:50.390","Text":"which is 1 minus 2v, would be here."},{"Start":"07:50.390 ","End":"07:54.500","Text":"I noticed that in order to get from here to here,"},{"Start":"07:54.500 ","End":"07:56.330","Text":"if I multiplied it by minus 2,"},{"Start":"07:56.330 ","End":"07:59.330","Text":"this is what I would get because minus 2 times v is minus"},{"Start":"07:59.330 ","End":"08:03.260","Text":"2v and minus 2 times minus 0.5 is 1."},{"Start":"08:03.260 ","End":"08:04.730","Text":"So the minus 2,"},{"Start":"08:04.730 ","End":"08:07.265","Text":"I was working backwards in order to get this."},{"Start":"08:07.265 ","End":"08:08.930","Text":"Now once I have this,"},{"Start":"08:08.930 ","End":"08:11.600","Text":"and since this is the derivative of this,"},{"Start":"08:11.600 ","End":"08:13.370","Text":"I can get the integral."},{"Start":"08:13.370 ","End":"08:15.635","Text":"Here it\u0027s natural log of t,"},{"Start":"08:15.635 ","End":"08:20.270","Text":"and here it\u0027s natural log of v minus v^2."},{"Start":"08:20.270 ","End":"08:23.735","Text":"I have forgotten to write the 1 over minus 2 here,"},{"Start":"08:23.735 ","End":"08:25.100","Text":"there, well, there wasn\u0027t much room,"},{"Start":"08:25.100 ","End":"08:26.435","Text":"so I just indicated,"},{"Start":"08:26.435 ","End":"08:30.785","Text":"and that gives us the minus a half besides the natural logarithm."},{"Start":"08:30.785 ","End":"08:37.235","Text":"Now all that remains is to substitute back from v to x and t. If you remember,"},{"Start":"08:37.235 ","End":"08:41.600","Text":"v was equal to knock y over x this time,"},{"Start":"08:41.600 ","End":"08:45.875","Text":"but x over t. If we make that substitution,"},{"Start":"08:45.875 ","End":"08:49.325","Text":"then we will get v is x over t,"},{"Start":"08:49.325 ","End":"08:54.110","Text":"v^2 is x over t^2 and this is the solution."},{"Start":"08:54.110 ","End":"08:58.160","Text":"You have to remember that we also had some restrictions."},{"Start":"08:58.160 ","End":"09:02.120","Text":"Well, the t not equal to 0 is just a restriction on the domain."},{"Start":"09:02.120 ","End":"09:03.590","Text":"That makes sense anyway,"},{"Start":"09:03.590 ","End":"09:06.125","Text":"because we can\u0027t have the natural log of 0."},{"Start":"09:06.125 ","End":"09:07.775","Text":"But what about these two?"},{"Start":"09:07.775 ","End":"09:11.750","Text":"What would happen if v was equal to 1 or v was equal to 0?"},{"Start":"09:11.750 ","End":"09:16.220","Text":"Actually, they give solutions because it fits the differential equation."},{"Start":"09:16.220 ","End":"09:18.710","Text":"When we have the equals,"},{"Start":"09:18.710 ","End":"09:21.020","Text":"we\u0027re going to get two extra solutions."},{"Start":"09:21.020 ","End":"09:25.130","Text":"If v is 0, then we get x, which is vt,"},{"Start":"09:25.130 ","End":"09:30.030","Text":"so x is 0 as a function of t. The other one,"},{"Start":"09:30.030 ","End":"09:31.330","Text":"when v is 1,"},{"Start":"09:31.330 ","End":"09:36.680","Text":"will give us that x of t equals t. These are the two singular solutions."},{"Start":"09:36.680 ","End":"09:42.635","Text":"So basically we got x=0 is one singular solution,"},{"Start":"09:42.635 ","End":"09:47.030","Text":"and x=t is the other singular solution together with this solution."},{"Start":"09:47.030 ","End":"09:51.600","Text":"We actually have three solutions. I\u0027m done."}],"Thumbnail":null,"ID":7660},{"Watched":false,"Name":"Exercise 9","Duration":"3m 27s","ChapterTopicVideoID":7586,"CourseChapterTopicPlaylistID":4219,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.280","Text":"This exercise has 2 parts."},{"Start":"00:02.280 ","End":"00:07.050","Text":"The first part, we\u0027re given a differential equation but with an unknown n,"},{"Start":"00:07.050 ","End":"00:10.260","Text":"and we have to find n to make this homogeneous."},{"Start":"00:10.260 ","End":"00:11.415","Text":"In Part 2,"},{"Start":"00:11.415 ","End":"00:16.710","Text":"we use this value of n that we found and solve the differential equation."},{"Start":"00:16.710 ","End":"00:19.320","Text":"Let\u0027s get to Part 1 first,"},{"Start":"00:19.320 ","End":"00:26.430","Text":"and what we\u0027re going to do is to let this bit be m and this bit be n as we usually do,"},{"Start":"00:26.430 ","End":"00:28.620","Text":"and let\u0027s see if m is homogeneous."},{"Start":"00:28.620 ","End":"00:30.150","Text":"Turns out that m is homogeneous,"},{"Start":"00:30.150 ","End":"00:31.680","Text":"we\u0027re not going to go into all the details,"},{"Start":"00:31.680 ","End":"00:36.315","Text":"but it turns out to be homogeneous of order 2 and if we take n,"},{"Start":"00:36.315 ","End":"00:39.990","Text":"we find that it\u0027s also homogeneous of order 1 plus"},{"Start":"00:39.990 ","End":"00:45.370","Text":"n. This naturally leads to the equation 2=1 plus n,"},{"Start":"00:45.370 ","End":"00:48.410","Text":"because in order for the equation to be homogeneous,"},{"Start":"00:48.410 ","End":"00:51.935","Text":"the functions have to be homogeneous of the same order."},{"Start":"00:51.935 ","End":"00:54.785","Text":"Of course this gives us that n=1."},{"Start":"00:54.785 ","End":"00:56.860","Text":"Now, onto Part 2."},{"Start":"00:56.860 ","End":"00:58.645","Text":"Now, if we put n equals 1,"},{"Start":"00:58.645 ","End":"00:59.870","Text":"n was right here."},{"Start":"00:59.870 ","End":"01:02.560","Text":"We had y ^n and we put y^1,"},{"Start":"01:02.560 ","End":"01:04.910","Text":"so this is the equation we get and we know it\u0027s"},{"Start":"01:04.910 ","End":"01:08.510","Text":"homogeneous because each of these is homogeneous of order 2."},{"Start":"01:08.510 ","End":"01:09.905","Text":"The usual technique,"},{"Start":"01:09.905 ","End":"01:13.000","Text":"we substitute y equals v. x,"},{"Start":"01:13.000 ","End":"01:15.500","Text":"and then dy comes out to be this,"},{"Start":"01:15.500 ","End":"01:18.035","Text":"which is this with the product rule."},{"Start":"01:18.035 ","End":"01:24.575","Text":"Also we should remember at the end to replace v by y over x."},{"Start":"01:24.575 ","End":"01:30.380","Text":"Continuing, what I did here was wherever I had y here,"},{"Start":"01:30.380 ","End":"01:31.985","Text":"like here and here,"},{"Start":"01:31.985 ","End":"01:34.370","Text":"I replaced it with the y here,"},{"Start":"01:34.370 ","End":"01:37.663","Text":"which is v. x, so that\u0027s where I get v. x and v. x."},{"Start":"01:37.663 ","End":"01:42.590","Text":"The other thing I did was to replace the dy with whatever dy equals to,"},{"Start":"01:42.590 ","End":"01:44.650","Text":"which is this, another p is here."},{"Start":"01:44.650 ","End":"01:47.070","Text":"This is what we get after the substitution,"},{"Start":"01:47.070 ","End":"01:49.365","Text":"and if we expand,"},{"Start":"01:49.365 ","End":"01:51.045","Text":"we will get this,"},{"Start":"01:51.045 ","End":"01:52.935","Text":"and I would like to simplify this,"},{"Start":"01:52.935 ","End":"01:54.510","Text":"and I see that there\u0027s x^2,"},{"Start":"01:54.510 ","End":"01:56.700","Text":"x^2, x^2, x^3."},{"Start":"01:56.700 ","End":"02:00.334","Text":"What I can do is take x^2 out of here,"},{"Start":"02:00.334 ","End":"02:03.620","Text":"and then assuming that x is not equal to 0,"},{"Start":"02:03.620 ","End":"02:05.015","Text":"I\u0027ll make that a condition,"},{"Start":"02:05.015 ","End":"02:09.950","Text":"then we can divide the x^2 here and we can divide x squared into x^3,"},{"Start":"02:09.950 ","End":"02:11.240","Text":"that just leaves x,"},{"Start":"02:11.240 ","End":"02:13.250","Text":"if I cancel the 3."},{"Start":"02:13.250 ","End":"02:17.407","Text":"Now we get the 1 plus 2v^2(dx),"},{"Start":"02:17.407 ","End":"02:19.505","Text":"and bring this to the other side,"},{"Start":"02:19.505 ","End":"02:21.500","Text":"we get minus xv."},{"Start":"02:21.500 ","End":"02:24.650","Text":"dv. We\u0027ve already close to separating the variables."},{"Start":"02:24.650 ","End":"02:26.495","Text":"I divide by x,"},{"Start":"02:26.495 ","End":"02:28.595","Text":"which already we said is not 0,"},{"Start":"02:28.595 ","End":"02:31.400","Text":"and we divide here by 1 plus 2v^2,"},{"Start":"02:31.400 ","End":"02:32.840","Text":"which is always positive,"},{"Start":"02:32.840 ","End":"02:35.255","Text":"so no extra conditions needed."},{"Start":"02:35.255 ","End":"02:36.620","Text":"Once we have this,"},{"Start":"02:36.620 ","End":"02:41.780","Text":"we can put the integral sign in front of each and what I want to do is"},{"Start":"02:41.780 ","End":"02:47.210","Text":"my usual tricks of trying to get the derivative of the denominator in the numerator."},{"Start":"02:47.210 ","End":"02:50.330","Text":"The trick this time is to put a 4 here and here,"},{"Start":"02:50.330 ","End":"02:55.190","Text":"and then the derivative of 1 plus 2v^2 is 4v,"},{"Start":"02:55.190 ","End":"02:59.095","Text":"the integral of dx over x is natural log of absolute value of x,"},{"Start":"02:59.095 ","End":"03:01.320","Text":"and here the minus 1/4 stays."},{"Start":"03:01.320 ","End":"03:04.250","Text":"We get natural log of the denominator because it\u0027s derivative"},{"Start":"03:04.250 ","End":"03:07.460","Text":"is already on the numerator and plus the constant of integration."},{"Start":"03:07.460 ","End":"03:10.910","Text":"The last thing is not to forget to replace,"},{"Start":"03:10.910 ","End":"03:14.630","Text":"remember that v=y over x,"},{"Start":"03:14.630 ","End":"03:16.760","Text":"and if we replace that,"},{"Start":"03:16.760 ","End":"03:23.430","Text":"then we end up with this solution,"},{"Start":"03:23.430 ","End":"03:24.960","Text":"and I\u0027ll highlight that,"},{"Start":"03:24.960 ","End":"03:27.880","Text":"and we are done."}],"Thumbnail":null,"ID":7661}],"ID":4219},{"Name":"Homogeneous after Substitution","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Homogeneous after Substitution","Duration":"3m 31s","ChapterTopicVideoID":7594,"CourseChapterTopicPlaylistID":4220,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.585","Text":"This is really a tutorial disguised as an exercise."},{"Start":"00:03.585 ","End":"00:08.010","Text":"We\u0027re going to learn about how to solve a particular differential equation,"},{"Start":"00:08.010 ","End":"00:09.285","Text":"one of this form,"},{"Start":"00:09.285 ","End":"00:11.850","Text":"where we have something linear in x and y,"},{"Start":"00:11.850 ","End":"00:14.565","Text":"dx and something linear in x and y, dy."},{"Start":"00:14.565 ","End":"00:19.560","Text":"These a, b and c are constants and they vary from problem to problem."},{"Start":"00:19.560 ","End":"00:21.780","Text":"It turns out that there are 2 main cases and"},{"Start":"00:21.780 ","End":"00:24.000","Text":"each 1 of them has a totally different approach."},{"Start":"00:24.000 ","End":"00:28.965","Text":"It all depends on whether a_1 times b_2 is"},{"Start":"00:28.965 ","End":"00:34.830","Text":"equal to b_1 times a_2 and the second case will be where it\u0027s not equal."},{"Start":"00:34.830 ","End":"00:39.300","Text":"In Case 1, we can reduce this to a separable equation,"},{"Start":"00:39.300 ","End":"00:41.940","Text":"meaning using separation of variables."},{"Start":"00:41.940 ","End":"00:48.165","Text":"It turns out that Case 2 will be a homogeneous equation, so totally different."},{"Start":"00:48.165 ","End":"00:50.600","Text":"Now, I\u0027m just going to outline the steps because"},{"Start":"00:50.600 ","End":"00:53.060","Text":"really it won\u0027t make sense until you do the exercises."},{"Start":"00:53.060 ","End":"00:54.845","Text":"On the other hand, we can\u0027t do the exercises."},{"Start":"00:54.845 ","End":"00:57.185","Text":"We\u0027ve at least to explain the general approach."},{"Start":"00:57.185 ","End":"01:01.610","Text":"You might want to go back and forth to in the tutorial and say the first test."},{"Start":"01:01.610 ","End":"01:03.275","Text":"The main steps are as follows."},{"Start":"01:03.275 ","End":"01:08.105","Text":"We make a substitution and we declare z to be what\u0027s written here,"},{"Start":"01:08.105 ","End":"01:10.385","Text":"a_1x plus b_1y plus c_1."},{"Start":"01:10.385 ","End":"01:13.340","Text":"The next step is differentiation."},{"Start":"01:13.340 ","End":"01:16.910","Text":"We find out what is the derivative of z with respect to x."},{"Start":"01:16.910 ","End":"01:20.720","Text":"But we going to use the form dz by dx, dy by dx,"},{"Start":"01:20.720 ","End":"01:26.070","Text":"the Leibniz\u0027s notation and then we extract dy is just a bit of algebra,"},{"Start":"01:26.070 ","End":"01:27.920","Text":"multiply both sides by dx,"},{"Start":"01:27.920 ","End":"01:30.200","Text":"bring this to this side, divide by b_1,"},{"Start":"01:30.200 ","End":"01:36.650","Text":"and this is what we get and then if we put dy in here and we put z here,"},{"Start":"01:36.650 ","End":"01:38.570","Text":"turns out, as you\u0027ll see in the example,"},{"Start":"01:38.570 ","End":"01:43.595","Text":"that we can get a separable equation and it\u0027s going to be an x and z,"},{"Start":"01:43.595 ","End":"01:49.190","Text":"it turns out it become a function of z and we\u0027ll solve it for x and z and then not"},{"Start":"01:49.190 ","End":"01:55.060","Text":"to forget at the end to substitute z back to what it was so we can get it in x and y."},{"Start":"01:55.060 ","End":"01:57.150","Text":"Let\u0027s move on to Case 2,"},{"Start":"01:57.150 ","End":"01:58.755","Text":"where this is not equal."},{"Start":"01:58.755 ","End":"02:00.230","Text":"Here we are in Case 2,"},{"Start":"02:00.230 ","End":"02:02.550","Text":"I copied the original equation again."},{"Start":"02:02.550 ","End":"02:07.855","Text":"This time, this is going to be an inequality as opposed to the equality of k is 1."},{"Start":"02:07.855 ","End":"02:11.705","Text":"As I mentioned before, we\u0027re going to reduce it to a homogeneous equation,"},{"Start":"02:11.705 ","End":"02:14.495","Text":"whereas previously we reduce it to a separable equation."},{"Start":"02:14.495 ","End":"02:16.820","Text":"The first step is going to be to solve a system of"},{"Start":"02:16.820 ","End":"02:20.465","Text":"linear equation and this is the system we just take this part here,"},{"Start":"02:20.465 ","End":"02:23.255","Text":"set it to 0, the second part here to 0."},{"Start":"02:23.255 ","End":"02:25.080","Text":"A system of linear equations,"},{"Start":"02:25.080 ","End":"02:26.765","Text":"2 equations in 2 unknowns."},{"Start":"02:26.765 ","End":"02:31.730","Text":"It turns out that this condition guarantees that we\u0027re going to get a unique solution,"},{"Start":"02:31.730 ","End":"02:33.995","Text":"2 values, one for x and one for y."},{"Start":"02:33.995 ","End":"02:35.230","Text":"This guarantees it,"},{"Start":"02:35.230 ","End":"02:36.950","Text":"that\u0027s what we have to separate the cases."},{"Start":"02:36.950 ","End":"02:40.310","Text":"We have this unique solution and now we make a substitution."},{"Start":"02:40.310 ","End":"02:44.330","Text":"We substitute little x original x to be say another x,"},{"Start":"02:44.330 ","End":"02:46.820","Text":"a big X plus h. That\u0027s a solution,"},{"Start":"02:46.820 ","End":"02:50.420","Text":"and y equals y plus k from this solution and the idea"},{"Start":"02:50.420 ","End":"02:54.170","Text":"is to get rid of these 2 constants because these 2 constants,"},{"Start":"02:54.170 ","End":"02:56.435","Text":"so what stopped this from being homogeneous?"},{"Start":"02:56.435 ","End":"02:58.205","Text":"The substitution will do the trick."},{"Start":"02:58.205 ","End":"03:01.940","Text":"After we tidy it up and get a homogeneous, we solve it."},{"Start":"03:01.940 ","End":"03:06.905","Text":"But then not to forget at the end to back substitute that once we have X,"},{"Start":"03:06.905 ","End":"03:08.885","Text":"we can get back to little x."},{"Start":"03:08.885 ","End":"03:15.950","Text":"Well, I really should have said that we substitute X is equal to little x minus h and"},{"Start":"03:15.950 ","End":"03:24.005","Text":"Y will be little y minus and then we substitute that in our solution for big X and big Y."},{"Start":"03:24.005 ","End":"03:26.660","Text":"As I said, best look at an example."},{"Start":"03:26.660 ","End":"03:29.690","Text":"You might want to revisit this after you have done the first example,"},{"Start":"03:29.690 ","End":"03:32.070","Text":"and that\u0027s all for now."}],"Thumbnail":null,"ID":7662},{"Watched":false,"Name":"Exercise 1","Duration":"5m 35s","ChapterTopicVideoID":7589,"CourseChapterTopicPlaylistID":4220,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"In this exercise, we have this differential equation to solve,"},{"Start":"00:02.970 ","End":"00:05.250","Text":"which at first appears unfamiliar."},{"Start":"00:05.250 ","End":"00:11.280","Text":"We recognize that it\u0027s very close to this special differential equation we just studied."},{"Start":"00:11.280 ","End":"00:18.300","Text":"We\u0027re going to bring it to this form and then the big question is whether a_1b_2=a_2b_1."},{"Start":"00:18.300 ","End":"00:19.650","Text":"Then we split from there."},{"Start":"00:19.650 ","End":"00:21.390","Text":"If the answer is yes, we do it one way."},{"Start":"00:21.390 ","End":"00:22.770","Text":"The answer is no, the other way."},{"Start":"00:22.770 ","End":"00:26.330","Text":"Let\u0027s go back a bit in here and see how we get it into this form."},{"Start":"00:26.330 ","End":"00:29.000","Text":"Cross-multiplying gets me up to here."},{"Start":"00:29.000 ","End":"00:32.715","Text":"Next, we brought this over to the other side equal to 0."},{"Start":"00:32.715 ","End":"00:34.100","Text":"Now, we\u0027re very close to this."},{"Start":"00:34.100 ","End":"00:36.050","Text":"On here, there\u0027s a plus and here there\u0027s a minus."},{"Start":"00:36.050 ","End":"00:37.145","Text":"No problem."},{"Start":"00:37.145 ","End":"00:41.120","Text":"Write a plus here and just reverse the signs of what\u0027s in here."},{"Start":"00:41.120 ","End":"00:43.550","Text":"Now, we can identify the 6 constants,"},{"Start":"00:43.550 ","End":"00:46.250","Text":"a_1, b_1, c_1 and so on, just from here."},{"Start":"00:46.250 ","End":"00:49.040","Text":"I don\u0027t actually care about the c at the moment because I"},{"Start":"00:49.040 ","End":"00:51.970","Text":"need the a\u0027s and b\u0027s in order to answer this question."},{"Start":"00:51.970 ","End":"00:54.095","Text":"I need a bit more space here."},{"Start":"00:54.095 ","End":"00:57.170","Text":"Well, a_1 times b_2 is this times this,"},{"Start":"00:57.170 ","End":"00:58.385","Text":"that\u0027s minus 1,"},{"Start":"00:58.385 ","End":"01:03.110","Text":"and a_2 b_1 is product of these 2 also minus 1."},{"Start":"01:03.110 ","End":"01:07.400","Text":"The answer is yes they are equal and we in what we call it Case"},{"Start":"01:07.400 ","End":"01:11.675","Text":"1 in the tutorial and we are going to reduce it into a separable equation."},{"Start":"01:11.675 ","End":"01:15.560","Text":"What we do is we take this bit and we call it z,"},{"Start":"01:15.560 ","End":"01:18.460","Text":"and we\u0027re going to differentiate this in a moment but"},{"Start":"01:18.460 ","End":"01:22.235","Text":"before that we can also get this to be a function of z."},{"Start":"01:22.235 ","End":"01:23.750","Text":"Just do a bit of manipulation on this."},{"Start":"01:23.750 ","End":"01:25.790","Text":"For example, multiply it by minus."},{"Start":"01:25.790 ","End":"01:30.500","Text":"I\u0027ve got minus z equals minus x minus y minus 1."},{"Start":"01:30.500 ","End":"01:31.700","Text":"Very close."},{"Start":"01:31.700 ","End":"01:34.085","Text":"Now we\u0027ll have to do is add one to both sides."},{"Start":"01:34.085 ","End":"01:36.320","Text":"I\u0027ve got minus z, subtract 1,"},{"Start":"01:36.320 ","End":"01:39.180","Text":"and then it becomes minus x minus y minus 2."},{"Start":"01:39.180 ","End":"01:41.280","Text":"All this is just minus z,"},{"Start":"01:41.280 ","End":"01:43.845","Text":"minus 1. That\u0027s that part."},{"Start":"01:43.845 ","End":"01:49.505","Text":"If we record this fact here and now differentiate this."},{"Start":"01:49.505 ","End":"01:52.595","Text":"We\u0027ve got the Z by d x equals,"},{"Start":"01:52.595 ","End":"01:53.855","Text":"this gives me one,"},{"Start":"01:53.855 ","End":"01:58.495","Text":"y gives me dy by dx and the one gives me 0,"},{"Start":"01:58.495 ","End":"02:00.390","Text":"multiply both sides by dx,"},{"Start":"02:00.390 ","End":"02:07.820","Text":"we\u0027ve got DZ equals dx plus dy and we\u0027ve got dy equals dz minus dx."},{"Start":"02:07.820 ","End":"02:10.760","Text":"Lost. Go back up a bit. You can just about see it."},{"Start":"02:10.760 ","End":"02:12.530","Text":"What we\u0027re going to get is,"},{"Start":"02:12.530 ","End":"02:15.480","Text":"we\u0027re going to get z dx and then here,"},{"Start":"02:15.480 ","End":"02:17.400","Text":"minus z minus 1."},{"Start":"02:17.400 ","End":"02:22.310","Text":"Instead of dy, I\u0027m going to put what we got. This is what we get."},{"Start":"02:22.310 ","End":"02:24.065","Text":"Remember there was z for the first bit,"},{"Start":"02:24.065 ","End":"02:27.560","Text":"minus z minus 1 for the second bit and the dy we took from here."},{"Start":"02:27.560 ","End":"02:32.900","Text":"After this, we expand the brackets and put z just 0 on one side,"},{"Start":"02:32.900 ","End":"02:35.030","Text":"collect the dx\u0027s together,"},{"Start":"02:35.030 ","End":"02:36.950","Text":"collect the dz\u0027s together,"},{"Start":"02:36.950 ","End":"02:41.795","Text":"bring this to the other side and now clearly we have a separation of variables."},{"Start":"02:41.795 ","End":"02:45.920","Text":"What I need to do is to take this and bring it over to the other side."},{"Start":"02:45.920 ","End":"02:49.910","Text":"I\u0027ll just have 1dx here and this over this, dz."},{"Start":"02:49.910 ","End":"02:52.020","Text":"I want to move to a new page."},{"Start":"02:52.020 ","End":"02:55.055","Text":"Here we are with the 2z plus 1 on the denominator."},{"Start":"02:55.055 ","End":"02:56.360","Text":"Of course, to do this,"},{"Start":"02:56.360 ","End":"03:01.595","Text":"we have to ensure that z not be equal to minus 0.5."},{"Start":"03:01.595 ","End":"03:05.390","Text":"I\u0027ll put it just an asterisk here to remember when I come back and take"},{"Start":"03:05.390 ","End":"03:09.875","Text":"care of the case of what happens if z is equal to minus 0.5,"},{"Start":"03:09.875 ","End":"03:13.130","Text":"which take an integral sign in front of each of the 2 sides."},{"Start":"03:13.130 ","End":"03:16.070","Text":"Integral the left side is x,"},{"Start":"03:16.070 ","End":"03:18.050","Text":"the integral of the right-hand side,"},{"Start":"03:18.050 ","End":"03:22.880","Text":"I\u0027m going to write it and at the end I\u0027ll show you how I got from here to here."},{"Start":"03:22.880 ","End":"03:24.245","Text":"This I owe you,"},{"Start":"03:24.245 ","End":"03:25.490","Text":"we\u0027ll leave that to the end."},{"Start":"03:25.490 ","End":"03:27.935","Text":"At this point we have a solution for x and z,"},{"Start":"03:27.935 ","End":"03:30.515","Text":"that\u0027s why we have to substitute back for z."},{"Start":"03:30.515 ","End":"03:34.580","Text":"If we go back up, you\u0027ll see that z was x plus y plus 1."},{"Start":"03:34.580 ","End":"03:36.530","Text":"I see z, I put x plus y plus 1."},{"Start":"03:36.530 ","End":"03:39.410","Text":"I see z, I put x plus y plus 1."},{"Start":"03:39.410 ","End":"03:41.320","Text":"Otherwise it\u0027s just the same."},{"Start":"03:41.320 ","End":"03:44.825","Text":"That\u0027s the answer except that I owe you the asterisk, what happens?"},{"Start":"03:44.825 ","End":"03:46.940","Text":"Z is equal to minus 0.5."},{"Start":"03:46.940 ","End":"03:51.455","Text":"We get x plus y plus 1 is that because that\u0027s what z is and we get the solution,"},{"Start":"03:51.455 ","End":"03:54.425","Text":"y equals minus x minus 1.5,"},{"Start":"03:54.425 ","End":"03:57.470","Text":"and this is another solution to the differential equation."},{"Start":"03:57.470 ","End":"04:02.330","Text":"We have this family of solutions as c varies and we have one special solution,"},{"Start":"04:02.330 ","End":"04:06.620","Text":"but we\u0027re not quite done because I still have a debt of how I got from here to here."},{"Start":"04:06.620 ","End":"04:08.705","Text":"We\u0027re going to do this by substitution."},{"Start":"04:08.705 ","End":"04:12.695","Text":"We\u0027re going to let t equal the 2z plus 1."},{"Start":"04:12.695 ","End":"04:15.930","Text":"On differentiating this, we get the 2dz,"},{"Start":"04:15.930 ","End":"04:19.150","Text":"the one just drops out, 2dz equals dt."},{"Start":"04:19.150 ","End":"04:23.375","Text":"I\u0027m also going to need z in terms of t. From the first line,"},{"Start":"04:23.375 ","End":"04:30.605","Text":"I\u0027ll get that z is equal to t minus 1 over 2,"},{"Start":"04:30.605 ","End":"04:33.290","Text":"and that will give me the z plus 1."},{"Start":"04:33.290 ","End":"04:37.654","Text":"The numerator is t plus 1 over 2."},{"Start":"04:37.654 ","End":"04:40.670","Text":"Now I make the substitution and we\u0027ve got the numerator,"},{"Start":"04:40.670 ","End":"04:42.289","Text":"which is t plus 1 over 2."},{"Start":"04:42.289 ","End":"04:44.405","Text":"This doesn\u0027t look very good. This is a big,"},{"Start":"04:44.405 ","End":"04:46.070","Text":"should be a thicker dividing line."},{"Start":"04:46.070 ","End":"04:47.580","Text":"Smaller font here should have been,"},{"Start":"04:47.580 ","End":"04:50.960","Text":"and the denominator here is t and then dz,"},{"Start":"04:50.960 ","End":"04:53.375","Text":"we said 2dz is dt,"},{"Start":"04:53.375 ","End":"04:58.505","Text":"so we\u0027ve also got that dz is dt over 2."},{"Start":"04:58.505 ","End":"05:01.670","Text":"That\u0027s it goes here. Now this 1/2 and this 1/2,"},{"Start":"05:01.670 ","End":"05:03.965","Text":"I can pull out front and make it a quarter."},{"Start":"05:03.965 ","End":"05:06.440","Text":"I\u0027ve got t plus 1 over t,"},{"Start":"05:06.440 ","End":"05:08.450","Text":"dt, a bit of algebra."},{"Start":"05:08.450 ","End":"05:11.585","Text":"Just divide out t over t plus 1 over t,"},{"Start":"05:11.585 ","End":"05:14.645","Text":"and this is just 1 plus 1 over t here."},{"Start":"05:14.645 ","End":"05:19.160","Text":"Continuing, the integral of one is t. The integral"},{"Start":"05:19.160 ","End":"05:23.840","Text":"of one over t is natural log of t or absolute value of t. Finally,"},{"Start":"05:23.840 ","End":"05:28.835","Text":"we substitute back the t was 2z plus 1 here and here,"},{"Start":"05:28.835 ","End":"05:31.955","Text":"I should have put a plus c here but we do that once."},{"Start":"05:31.955 ","End":"05:36.540","Text":"That\u0027s the data I owed you and now we\u0027re done."}],"Thumbnail":null,"ID":7663},{"Watched":false,"Name":"Exercise 2","Duration":"3m 47s","ChapterTopicVideoID":7590,"CourseChapterTopicPlaylistID":4220,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.610","Text":"We have here a differential equation to solve and we"},{"Start":"00:02.610 ","End":"00:05.805","Text":"recognize that is one of those special cases where we have"},{"Start":"00:05.805 ","End":"00:12.000","Text":"a linear expression in x and y(dx) and another linear expression in x and y(dy),"},{"Start":"00:12.000 ","End":"00:13.935","Text":"and just copied it over here."},{"Start":"00:13.935 ","End":"00:18.015","Text":"The coefficients are given names a_1 b_1 c_1,"},{"Start":"00:18.015 ","End":"00:19.140","Text":"a_2, b_2,"},{"Start":"00:19.140 ","End":"00:24.570","Text":"c_2 and what interest me at the moment are the as and the bs 1,"},{"Start":"00:24.570 ","End":"00:26.610","Text":"2, 2, 4,"},{"Start":"00:26.610 ","End":"00:29.370","Text":"because if you remember the theory, there are 2 cases."},{"Start":"00:29.370 ","End":"00:33.150","Text":"If a_1 b_2 equals a_2 b_1 or otherwise,"},{"Start":"00:33.150 ","End":"00:37.965","Text":"and this case we do have equality because this times this is 4 and this times this is 4."},{"Start":"00:37.965 ","End":"00:40.910","Text":"When this happens, there\u0027s a way we can reduce"},{"Start":"00:40.910 ","End":"00:44.110","Text":"this to a separable differential equation."},{"Start":"00:44.110 ","End":"00:47.065","Text":"If you recall, the first step is a substitution,"},{"Start":"00:47.065 ","End":"00:49.280","Text":"we take say this part here,"},{"Start":"00:49.280 ","End":"00:51.720","Text":"and this one we call z,"},{"Start":"00:51.720 ","End":"00:53.520","Text":"so that\u0027s what this is."},{"Start":"00:53.520 ","End":"00:59.975","Text":"One of the things we do first is to try and express the other part also in terms of z,"},{"Start":"00:59.975 ","End":"01:01.490","Text":"this is guaranteed you see."},{"Start":"01:01.490 ","End":"01:04.565","Text":"The fact that this times this equals this times this"},{"Start":"01:04.565 ","End":"01:08.030","Text":"ensures that x and y are in the same proportion here and here,"},{"Start":"01:08.030 ","End":"01:09.590","Text":"1 to 2, 1 to 2,"},{"Start":"01:09.590 ","End":"01:13.870","Text":"so that I can manipulate this other bit and I\u0027m just going work on this bit."},{"Start":"01:13.870 ","End":"01:16.030","Text":"The first thing is to take the 2 out,"},{"Start":"01:16.030 ","End":"01:20.440","Text":"and that way you see I get the same x and y as here and here."},{"Start":"01:20.440 ","End":"01:22.665","Text":"But there\u0027s still a minus 1."},{"Start":"01:22.665 ","End":"01:27.425","Text":"Now, x plus 2y from this equation or from here,"},{"Start":"01:27.425 ","End":"01:29.585","Text":"is just z minus 3,"},{"Start":"01:29.585 ","End":"01:33.320","Text":"and so this comes out to be 2z minus 7."},{"Start":"01:33.320 ","End":"01:34.730","Text":"Well, it\u0027s just scrolled off,"},{"Start":"01:34.730 ","End":"01:39.075","Text":"but the other bit here was 2z minus 7."},{"Start":"01:39.075 ","End":"01:43.310","Text":"The problem is that if I do the substitution on your scope back up,"},{"Start":"01:43.310 ","End":"01:49.235","Text":"I\u0027ll still have a dy and that\u0027s where we do the second stage, which is differentiation."},{"Start":"01:49.235 ","End":"01:51.070","Text":"Let me go back down again,"},{"Start":"01:51.070 ","End":"01:53.565","Text":"but I want us to keep this in sight."},{"Start":"01:53.565 ","End":"01:56.210","Text":"What we get during the right-hand side first"},{"Start":"01:56.210 ","End":"01:58.760","Text":"is the derivative of z with respect to x is,"},{"Start":"01:58.760 ","End":"02:00.230","Text":"from here I get a 1,"},{"Start":"02:00.230 ","End":"02:04.040","Text":"from here I get 2dy over dx and the constant vanishes."},{"Start":"02:04.040 ","End":"02:06.560","Text":"But what I want to do is extract dy."},{"Start":"02:06.560 ","End":"02:10.910","Text":"I brought the one over to the other side and divide it by 2."},{"Start":"02:10.910 ","End":"02:16.670","Text":"If I multiply by dx and switch sides, you see that dy,"},{"Start":"02:16.670 ","End":"02:19.340","Text":"I\u0027ve got it in terms of dz and dx,"},{"Start":"02:19.340 ","End":"02:23.750","Text":"and now I can substitute it back in that expression above onto a new page."},{"Start":"02:23.750 ","End":"02:27.770","Text":"I recorded that we got the second expression was dy was"},{"Start":"02:27.770 ","End":"02:32.630","Text":"this and that dy was equal to that and I copied the original equation again,"},{"Start":"02:32.630 ","End":"02:35.375","Text":"and remember this is what was z."},{"Start":"02:35.375 ","End":"02:38.780","Text":"We now have something in z and x."},{"Start":"02:38.780 ","End":"02:41.150","Text":"This is z, this is dx,"},{"Start":"02:41.150 ","End":"02:48.850","Text":"this part from here 2z minus 7 and the dy from here substituting gives me this."},{"Start":"02:48.850 ","End":"02:52.460","Text":"Multiply by 2, expand the brackets,"},{"Start":"02:52.460 ","End":"02:54.965","Text":"collect like terms for dx and dz"},{"Start":"02:54.965 ","End":"02:59.510","Text":"and separate the variables x is on the left, z is on the right."},{"Start":"02:59.510 ","End":"03:01.160","Text":"Notice that instead of putting a minus,"},{"Start":"03:01.160 ","End":"03:03.170","Text":"I just switched the order of the subtraction,"},{"Start":"03:03.170 ","End":"03:05.405","Text":"put an integral sign in front of each."},{"Start":"03:05.405 ","End":"03:08.540","Text":"Do the integration straightforward enough."},{"Start":"03:08.540 ","End":"03:10.765","Text":"Don\u0027t forget to add the plus c,"},{"Start":"03:10.765 ","End":"03:15.380","Text":"and of course we have to back substitute because we have to replace z with what it was."},{"Start":"03:15.380 ","End":"03:19.040","Text":"We had x and y, so this was z and the same here,"},{"Start":"03:19.040 ","End":"03:22.639","Text":"just replacing z here and here with this expression,"},{"Start":"03:22.639 ","End":"03:25.850","Text":"open the brackets, subtract the 7x,"},{"Start":"03:25.850 ","End":"03:28.310","Text":"which appears both here and here."},{"Start":"03:28.310 ","End":"03:34.189","Text":"The other thing I did here is that this 21 with the c is just a constant."},{"Start":"03:34.189 ","End":"03:38.255","Text":"I mean, any constant can be written as 21 plus another constant."},{"Start":"03:38.255 ","End":"03:39.860","Text":"There\u0027s no need for the 21,"},{"Start":"03:39.860 ","End":"03:42.065","Text":"it gets swallowed up in the c, so to speak,"},{"Start":"03:42.065 ","End":"03:47.910","Text":"so I\u0027ll write it as a different letter call it k and that\u0027s the answer, so we\u0027re done."}],"Thumbnail":null,"ID":7664},{"Watched":false,"Name":"Exercise 3","Duration":"6m 39s","ChapterTopicVideoID":7591,"CourseChapterTopicPlaylistID":4220,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.745","Text":"We have this differential equation to solve."},{"Start":"00:02.745 ","End":"00:04.995","Text":"And if we rearrange it,"},{"Start":"00:04.995 ","End":"00:06.750","Text":"then it looks more familiar."},{"Start":"00:06.750 ","End":"00:16.215","Text":"Remember we had this a_1x plus b_1y plus c_1 dx plus a_2, b_2,"},{"Start":"00:16.215 ","End":"00:18.210","Text":"c_2 with dy, and so on,"},{"Start":"00:18.210 ","End":"00:24.810","Text":"and we studied this and there were 2 cases just have to write down the a_1, b_1,"},{"Start":"00:24.810 ","End":"00:27.450","Text":"a_2, b_2, these are the important ones,"},{"Start":"00:27.450 ","End":"00:30.660","Text":"and the 2 cases are whether this product a_1,"},{"Start":"00:30.660 ","End":"00:33.550","Text":"b_2 is equal or not equal to a_2, b_1."},{"Start":"00:33.550 ","End":"00:36.460","Text":"In this case, we have the not equals,"},{"Start":"00:36.460 ","End":"00:39.770","Text":"and the theory said that we do one thing if it\u0027s equal,"},{"Start":"00:39.770 ","End":"00:41.285","Text":"something else if it\u0027s not equal."},{"Start":"00:41.285 ","End":"00:45.410","Text":"In this case, in fact, we can reduce it to a homogeneous equation."},{"Start":"00:45.410 ","End":"00:49.370","Text":"As a trick that we use for getting rid of the constants,"},{"Start":"00:49.370 ","End":"00:52.250","Text":"the 5 and the 4 are what stop it from being homogeneous."},{"Start":"00:52.250 ","End":"00:57.890","Text":"What we do is we solve the system of linear equations in two unknowns,"},{"Start":"00:57.890 ","End":"01:01.085","Text":"we take this equals 0 and this equals 0."},{"Start":"01:01.085 ","End":"01:03.290","Text":"I\u0027m not going to go through the steps of the solution."},{"Start":"01:03.290 ","End":"01:04.865","Text":"I\u0027ll just tell you what it is."},{"Start":"01:04.865 ","End":"01:07.415","Text":"That\u0027s x_1 and that\u0027s y_1,"},{"Start":"01:07.415 ","End":"01:14.795","Text":"and then we make a substitution that our x is going to be x plus 1, the 1 from here,"},{"Start":"01:14.795 ","End":"01:16.160","Text":"and similarly for y,"},{"Start":"01:16.160 ","End":"01:19.220","Text":"y minus 2 dx and comes out the"},{"Start":"01:19.220 ","End":"01:23.030","Text":"same as with the capital or with the x and the same for dy,"},{"Start":"01:23.030 ","End":"01:28.070","Text":"and if we substitute all of these in this equation,"},{"Start":"01:28.070 ","End":"01:29.382","Text":"x, y, x,"},{"Start":"01:29.382 ","End":"01:32.045","Text":"and y, we get this,"},{"Start":"01:32.045 ","End":"01:34.035","Text":"and after simplification,"},{"Start":"01:34.035 ","End":"01:36.590","Text":"we get a homogeneous equation."},{"Start":"01:36.590 ","End":"01:38.176","Text":"We get this dx,"},{"Start":"01:38.176 ","End":"01:40.670","Text":"this dY equals 0,"},{"Start":"01:40.670 ","End":"01:41.930","Text":"that\u0027s our function M,"},{"Start":"01:41.930 ","End":"01:46.400","Text":"and that\u0027s our function N. M is homogeneous of degree 1,"},{"Start":"01:46.400 ","End":"01:50.580","Text":"because if I put Lambda x and Lambda y instead of x and Y,"},{"Start":"01:50.580 ","End":"01:53.280","Text":"I can take the Lambda out front and similarly for"},{"Start":"01:53.280 ","End":"01:56.570","Text":"N. Both of these are homogeneous of degree 1."},{"Start":"01:56.570 ","End":"01:58.025","Text":"It\u0027s Lambda to the 1,"},{"Start":"01:58.025 ","End":"01:59.960","Text":"and they\u0027re homogeneous of the same degree."},{"Start":"01:59.960 ","End":"02:02.995","Text":"That\u0027s important, that makes this a homogeneous equation."},{"Start":"02:02.995 ","End":"02:05.990","Text":"I hope you remember how to solve homogeneous equations."},{"Start":"02:05.990 ","End":"02:12.080","Text":"We make this substitution where y equals some new letter v times x."},{"Start":"02:12.080 ","End":"02:14.450","Text":"Sometimes it\u0027s more convenient the other way round,"},{"Start":"02:14.450 ","End":"02:15.740","Text":"the x equals vY,"},{"Start":"02:15.740 ","End":"02:17.535","Text":"but in this case let\u0027s do it this way,"},{"Start":"02:17.535 ","End":"02:20.375","Text":"and then we also need to substitute for dY."},{"Start":"02:20.375 ","End":"02:25.160","Text":"This is what we get when we substitute y here and here,"},{"Start":"02:25.160 ","End":"02:29.315","Text":"and also the dY here, multiply out."},{"Start":"02:29.315 ","End":"02:31.710","Text":"Notice that that canceled with that,"},{"Start":"02:31.710 ","End":"02:33.570","Text":"so we can simplify a bit,"},{"Start":"02:33.570 ","End":"02:40.045","Text":"collect together the dXs and the dvs is divided by x,"},{"Start":"02:40.045 ","End":"02:42.555","Text":"provided x is not equal to 0."},{"Start":"02:42.555 ","End":"02:44.690","Text":"Going to move on to the next page now,"},{"Start":"02:44.690 ","End":"02:47.000","Text":"but I copied the last line there."},{"Start":"02:47.000 ","End":"02:51.620","Text":"We move this to the other side and divide by v^2 minus 1."},{"Start":"02:51.620 ","End":"02:54.350","Text":"But the denominator here can\u0027t be 0."},{"Start":"02:54.350 ","End":"02:55.550","Text":"If this is 0,"},{"Start":"02:55.550 ","End":"02:57.485","Text":"it means that v is plus or minus 1,"},{"Start":"02:57.485 ","End":"02:58.870","Text":"so we exclude this."},{"Start":"02:58.870 ","End":"03:03.240","Text":"At the end we\u0027ll see what happens if v is plus or minus 1."},{"Start":"03:03.240 ","End":"03:05.495","Text":"We get what are called singular solutions."},{"Start":"03:05.495 ","End":"03:07.640","Text":"This condition is a bit different."},{"Start":"03:07.640 ","End":"03:11.135","Text":"It just means since x is x plus 1,"},{"Start":"03:11.135 ","End":"03:14.705","Text":"this implies that x is not equal to 1."},{"Start":"03:14.705 ","End":"03:18.709","Text":"That just a restriction on the function which is a solution."},{"Start":"03:18.709 ","End":"03:21.080","Text":"It won\u0027t apply to x equals 1,"},{"Start":"03:21.080 ","End":"03:23.615","Text":"but these will be treated differently."},{"Start":"03:23.615 ","End":"03:24.710","Text":"I\u0027ll do this at the end,"},{"Start":"03:24.710 ","End":"03:25.790","Text":"the plus 1 case,"},{"Start":"03:25.790 ","End":"03:28.190","Text":"I\u0027ll call asterisk on the minus 1 case,"},{"Start":"03:28.190 ","End":"03:30.260","Text":"double asterisk till the end."},{"Start":"03:30.260 ","End":"03:31.730","Text":"Meanwhile, let\u0027s continue,"},{"Start":"03:31.730 ","End":"03:34.325","Text":"take an integral sign in front of each."},{"Start":"03:34.325 ","End":"03:36.935","Text":"The integral of the left-hand side is this."},{"Start":"03:36.935 ","End":"03:38.705","Text":"As for the right-hand side,"},{"Start":"03:38.705 ","End":"03:40.820","Text":"I want to leave this to the end,"},{"Start":"03:40.820 ","End":"03:43.010","Text":"use this partial fractions."},{"Start":"03:43.010 ","End":"03:46.610","Text":"For those interested, I\u0027ll show you at the end how I got this integral."},{"Start":"03:46.610 ","End":"03:49.505","Text":"Meanwhile, I didn\u0027t want to stop the flow. Let\u0027s continue."},{"Start":"03:49.505 ","End":"03:55.295","Text":"We want to do some back substituting because we want to get back to x and y."},{"Start":"03:55.295 ","End":"03:59.200","Text":"Now, v was y over x, which is this."},{"Start":"03:59.200 ","End":"04:03.440","Text":"What do we get? X is x minus 1, that\u0027s on the left."},{"Start":"04:03.440 ","End":"04:08.030","Text":"Now here, we just replace v by this over this,"},{"Start":"04:08.030 ","End":"04:09.380","Text":"so that\u0027s here,"},{"Start":"04:09.380 ","End":"04:13.070","Text":"and here it\u0027s here and we\u0027re okay with this"},{"Start":"04:13.070 ","End":"04:16.850","Text":"because we already said that x is not equal to 1,"},{"Start":"04:16.850 ","End":"04:20.405","Text":"the solution will just skip x equals 1 from the domain."},{"Start":"04:20.405 ","End":"04:24.230","Text":"Now, remember we put aside 2 special cases with an asterisk."},{"Start":"04:24.230 ","End":"04:26.390","Text":"We had v is 1 or minus 1."},{"Start":"04:26.390 ","End":"04:29.270","Text":"The v equals 1 actually gives rise to a,"},{"Start":"04:29.270 ","End":"04:30.765","Text":"it\u0027s called a singular solution."},{"Start":"04:30.765 ","End":"04:32.510","Text":"Y over X is 1,"},{"Start":"04:32.510 ","End":"04:38.285","Text":"so Y is Y and replace y and x by what they are and we end up with this."},{"Start":"04:38.285 ","End":"04:40.430","Text":"This is actually also a solution."},{"Start":"04:40.430 ","End":"04:43.640","Text":"If you plug it into the original differential equation,"},{"Start":"04:43.640 ","End":"04:45.410","Text":"you\u0027ll see that it works,"},{"Start":"04:45.410 ","End":"04:49.100","Text":"and similarly, if v is minus 1 and we substitute it,"},{"Start":"04:49.100 ","End":"04:51.860","Text":"we get another singular solution."},{"Start":"04:51.860 ","End":"04:54.800","Text":"So there\u0027s actually the main solution."},{"Start":"04:54.800 ","End":"04:57.860","Text":"It\u0027s a family of solutions because it depends on the constant."},{"Start":"04:57.860 ","End":"05:00.573","Text":"Just omit x equals 1 from the domain,"},{"Start":"05:00.573 ","End":"05:02.705","Text":"and then we have the 2 singular solutions,"},{"Start":"05:02.705 ","End":"05:04.640","Text":"this one and this one."},{"Start":"05:04.640 ","End":"05:09.580","Text":"The only thing left is that I did an integral earlier involving partial fractions."},{"Start":"05:09.580 ","End":"05:12.770","Text":"For those interested, then please stay."},{"Start":"05:12.770 ","End":"05:14.990","Text":"If you\u0027re not, then we\u0027re done."},{"Start":"05:14.990 ","End":"05:17.765","Text":"I\u0027ll just settle that debt."},{"Start":"05:17.765 ","End":"05:19.543","Text":"This was the integral,"},{"Start":"05:19.543 ","End":"05:22.130","Text":"and we\u0027re going to use partial fractions."},{"Start":"05:22.130 ","End":"05:26.240","Text":"I want to write this since the denominator factors v"},{"Start":"05:26.240 ","End":"05:31.210","Text":"squared minus 1 into v minus 1, v plus 1."},{"Start":"05:31.210 ","End":"05:35.030","Text":"We look for this as a partial fraction decomposition of this form,"},{"Start":"05:35.030 ","End":"05:37.230","Text":"go and review partial fractions,"},{"Start":"05:37.230 ","End":"05:39.590","Text":"if we don\u0027t remember, ignore the integrals."},{"Start":"05:39.590 ","End":"05:43.280","Text":"What we\u0027re doing here is algebras letting this equal this,"},{"Start":"05:43.280 ","End":"05:45.095","Text":"should have put it as an extra step,"},{"Start":"05:45.095 ","End":"05:46.715","Text":"just strip the integral away,"},{"Start":"05:46.715 ","End":"05:48.710","Text":"and multiplying out,"},{"Start":"05:48.710 ","End":"05:52.850","Text":"cross-multiplying or putting a common denominator of v minus 1, v plus 1,"},{"Start":"05:52.850 ","End":"05:58.700","Text":"we get that 2 minus v is this times this plus this times this."},{"Start":"05:58.700 ","End":"06:03.350","Text":"A and B are the unknowns not v. This holds for all v supposedly,"},{"Start":"06:03.350 ","End":"06:05.765","Text":"so we can substitute what we like."},{"Start":"06:05.765 ","End":"06:10.130","Text":"A convenient value to substitute would be 1 because then this becomes 0,"},{"Start":"06:10.130 ","End":"06:13.340","Text":"and if you do that, you\u0027ll get that A equals 1.5."},{"Start":"06:13.340 ","End":"06:15.500","Text":"If we let v equals minus 1,"},{"Start":"06:15.500 ","End":"06:17.495","Text":"that makes this term 0,"},{"Start":"06:17.495 ","End":"06:20.540","Text":"so substitute and we get the value of B,"},{"Start":"06:20.540 ","End":"06:22.200","Text":"plug those here and here,"},{"Start":"06:22.200 ","End":"06:24.770","Text":"and so with these substituted values,"},{"Start":"06:24.770 ","End":"06:26.525","Text":"this is the integral we get,"},{"Start":"06:26.525 ","End":"06:28.910","Text":"and then we can do it separately for each piece."},{"Start":"06:28.910 ","End":"06:30.170","Text":"This gives us this,"},{"Start":"06:30.170 ","End":"06:31.220","Text":"this gives us this,"},{"Start":"06:31.220 ","End":"06:33.530","Text":"and we have the constant of integration,"},{"Start":"06:33.530 ","End":"06:35.810","Text":"and that\u0027s what we got above,"},{"Start":"06:35.810 ","End":"06:39.930","Text":"so everything\u0027s settled down and we\u0027re done."}],"Thumbnail":null,"ID":7665},{"Watched":false,"Name":"Exercise 4","Duration":"7m 37s","ChapterTopicVideoID":7592,"CourseChapterTopicPlaylistID":4220,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.770","Text":"In this exercise, we have to solve"},{"Start":"00:01.770 ","End":"00:06.720","Text":"this differential equation and I just slightly rewrote it right order x,"},{"Start":"00:06.720 ","End":"00:08.145","Text":"y and then constants."},{"Start":"00:08.145 ","End":"00:11.550","Text":"What I\u0027m aiming for is to cross-multiply and I want to bring it"},{"Start":"00:11.550 ","End":"00:16.060","Text":"to the form where we have ax plus by plus c times dx"},{"Start":"00:17.660 ","End":"00:23.340","Text":"and similarly for dy only I better put subscripts and a similar thing"},{"Start":"00:23.340 ","End":"00:30.030","Text":"for dy, (a_2x+b_2y+c_2)dy."},{"Start":"00:30.030 ","End":"00:31.400","Text":"If I bring it to this form,"},{"Start":"00:31.400 ","End":"00:34.040","Text":"then we have a technique to solve it and basically we"},{"Start":"00:34.040 ","End":"00:38.550","Text":"branch out as to whether a_1b_2=a_2b_1."},{"Start":"00:38.710 ","End":"00:43.080","Text":"Then we do one approach and convert it into a separable."},{"Start":"00:43.080 ","End":"00:48.290","Text":"If we happen to have a_1b_2 not equal to a_2b_1,"},{"Start":"00:48.290 ","End":"00:52.385","Text":"then we convert it into a homogeneous equation."},{"Start":"00:52.385 ","End":"00:55.670","Text":"First of all, I brought it into this form by"},{"Start":"00:55.670 ","End":"01:00.350","Text":"cross-multiplying and then bringing the right-hand side over to the left-hand side."},{"Start":"01:00.350 ","End":"01:03.250","Text":"That\u0027s why these like this, but with minuses."},{"Start":"01:03.250 ","End":"01:05.580","Text":"Next, the full constants I need to identify,"},{"Start":"01:05.580 ","End":"01:09.600","Text":"I don\u0027t need the c\u0027s but I need both a and b and here they are a_1,"},{"Start":"01:09.600 ","End":"01:11.335","Text":"b_1, a_2, b_2."},{"Start":"01:11.335 ","End":"01:13.970","Text":"Then we need to compute both these products and"},{"Start":"01:13.970 ","End":"01:17.045","Text":"I see they\u0027re different because a_1, b_2,"},{"Start":"01:17.045 ","End":"01:18.830","Text":"this times this is negative 2,"},{"Start":"01:18.830 ","End":"01:21.907","Text":"this times this negative 1 which is not equal so"},{"Start":"01:21.907 ","End":"01:25.410","Text":"we\u0027re going to use what we called Case 2 in the tutorial."},{"Start":"01:25.410 ","End":"01:26.725","Text":"In Case 2,"},{"Start":"01:26.725 ","End":"01:31.925","Text":"what we first do is to solve a pair of linear equations in 2 unknowns."},{"Start":"01:31.925 ","End":"01:36.170","Text":"This equation comes from setting this bit equals 0,"},{"Start":"01:36.170 ","End":"01:39.010","Text":"and the second is from this bit equals 0."},{"Start":"01:39.010 ","End":"01:42.500","Text":"Now, I\u0027m not going to go through all the steps of the solution."},{"Start":"01:42.500 ","End":"01:45.380","Text":"I\u0027ll give you the answer right away because I know"},{"Start":"01:45.380 ","End":"01:48.515","Text":"you know how to do these things and you can always check by substitution."},{"Start":"01:48.515 ","End":"01:52.130","Text":"Then we use these 2 values to do"},{"Start":"01:52.130 ","End":"01:57.380","Text":"a substitution in the differential equation that have original x and y."},{"Start":"01:57.380 ","End":"02:00.889","Text":"We introduce X and Y and this is a substitution,"},{"Start":"02:00.889 ","End":"02:04.640","Text":"x equals X plus whatever came out here for the solution,"},{"Start":"02:04.640 ","End":"02:06.110","Text":"and we let y=Y-2,"},{"Start":"02:06.110 ","End":"02:11.195","Text":"and again plus negative 2, dx=dX and dy=dY."},{"Start":"02:11.195 ","End":"02:13.975","Text":"When we do it this way, it\u0027s always plus or minus a constant."},{"Start":"02:13.975 ","End":"02:18.335","Text":"That\u0027s what we have, that substitute these and here this is what we get."},{"Start":"02:18.335 ","End":"02:21.815","Text":"It looks a mess, but it\u0027s really quite simple because wherever we see,"},{"Start":"02:21.815 ","End":"02:24.365","Text":"I guess it\u0027s not in the same order. It was an original order."},{"Start":"02:24.365 ","End":"02:29.425","Text":"We have x+y+1,1+x+y and then the dx is dx and so on."},{"Start":"02:29.425 ","End":"02:31.475","Text":"Here we have minus x,"},{"Start":"02:31.475 ","End":"02:33.320","Text":"minus 2y, minus 3, and so on."},{"Start":"02:33.320 ","End":"02:34.655","Text":"This is just substitution,"},{"Start":"02:34.655 ","End":"02:36.800","Text":"but the order got a bit mixed up."},{"Start":"02:36.800 ","End":"02:39.710","Text":"The next thing to do is to tidy it up."},{"Start":"02:39.710 ","End":"02:41.930","Text":"This is always what we get is we get"},{"Start":"02:41.930 ","End":"02:44.795","Text":"actually the same coefficients as in the original x,"},{"Start":"02:44.795 ","End":"02:49.660","Text":"y, but there\u0027s no constant term here and that makes this homogeneous."},{"Start":"02:49.660 ","End":"02:52.190","Text":"I already know you could probably skip the checking."},{"Start":"02:52.190 ","End":"02:54.470","Text":"It\u0027s homogeneous step,"},{"Start":"02:54.470 ","End":"02:55.880","Text":"but we all know but I\u0027m just skipping that step."},{"Start":"02:55.880 ","End":"02:57.980","Text":"This is homogeneous of Degree 1."},{"Start":"02:57.980 ","End":"02:59.945","Text":"This is homogeneous of Degree 1,"},{"Start":"02:59.945 ","End":"03:04.780","Text":"same degree so the whole equation is called homogeneous of Degree 1."},{"Start":"03:04.780 ","End":"03:07.100","Text":"The usual strategy for homogeneous,"},{"Start":"03:07.100 ","End":"03:12.080","Text":"which you should remember is to let y equals some new letter v times X,"},{"Start":"03:12.080 ","End":"03:15.080","Text":"sometimes the other way round but here this will be okay,"},{"Start":"03:15.080 ","End":"03:20.900","Text":"y=vX and then we need also dy so product rule gives us this."},{"Start":"03:20.900 ","End":"03:22.010","Text":"It\u0027s always the same."},{"Start":"03:22.010 ","End":"03:23.450","Text":"Now, we substitute this in here."},{"Start":"03:23.450 ","End":"03:27.485","Text":"Here\u0027s X, here\u0027s X, here\u0027s Y. Y is Vx minus x."},{"Start":"03:27.485 ","End":"03:30.930","Text":"Again, Y is Vx and dX is dX,"},{"Start":"03:30.930 ","End":"03:33.810","Text":"but dy is this expression from here,"},{"Start":"03:33.810 ","End":"03:37.190","Text":"and now we don\u0027t have value y anymore."},{"Start":"03:37.190 ","End":"03:41.915","Text":"We just have x and v. Collect the dXs separately and the dvs separately."},{"Start":"03:41.915 ","End":"03:44.585","Text":"Open up the brackets here, here and here."},{"Start":"03:44.585 ","End":"03:49.700","Text":"Some stuff cancels and I rewrote it after the cancel it,"},{"Start":"03:49.700 ","End":"03:51.965","Text":"moved over to a new page, copied it,"},{"Start":"03:51.965 ","End":"03:55.010","Text":"took a big x outside the brackets here,"},{"Start":"03:55.010 ","End":"04:00.170","Text":"divide both sides by x. I\u0027ll put a line through here and the line through this squared."},{"Start":"04:00.170 ","End":"04:05.555","Text":"Of course, we just make note that we can\u0027t let x be 0 because we divided by it."},{"Start":"04:05.555 ","End":"04:07.610","Text":"Getting ready for separation of variables,"},{"Start":"04:07.610 ","End":"04:10.640","Text":"just bring this to the other side by changing the signs here."},{"Start":"04:10.640 ","End":"04:12.485","Text":"Now we can divide,"},{"Start":"04:12.485 ","End":"04:17.165","Text":"bring the v\u0027s over to this side and the xs to the other side."},{"Start":"04:17.165 ","End":"04:18.575","Text":"This is what we get."},{"Start":"04:18.575 ","End":"04:20.930","Text":"Now we already mentioned that x is not equal to 0,"},{"Start":"04:20.930 ","End":"04:22.160","Text":"but we also have to,"},{"Start":"04:22.160 ","End":"04:24.935","Text":"because we divided by 1-2v^2,"},{"Start":"04:24.935 ","End":"04:27.515","Text":"we have to make sure we\u0027re not dividing by 0."},{"Start":"04:27.515 ","End":"04:31.640","Text":"For this to be 0, we would have to be plus or minus the square root of 1/2."},{"Start":"04:31.640 ","End":"04:36.770","Text":"We state here that this is not the case and we\u0027ll return to this later."},{"Start":"04:36.770 ","End":"04:38.584","Text":"I\u0027ll put an asterisk here meanwhile,"},{"Start":"04:38.584 ","End":"04:40.250","Text":"and after we\u0027ve separated,"},{"Start":"04:40.250 ","End":"04:43.550","Text":"the usual thing is to put the integral sign in front of each."},{"Start":"04:43.550 ","End":"04:44.749","Text":"This is the integration."},{"Start":"04:44.749 ","End":"04:46.840","Text":"The left-hand side is straightforward enough,"},{"Start":"04:46.840 ","End":"04:49.595","Text":"1 over x natural log of absolute value of x."},{"Start":"04:49.595 ","End":"04:51.065","Text":"For the right-hand side,"},{"Start":"04:51.065 ","End":"04:55.280","Text":"I\u0027m just going to write the answer and I\u0027ll postpone till later"},{"Start":"04:55.280 ","End":"04:59.750","Text":"the work I don\u0027t want to interrupt the flow of how I got from here to here,"},{"Start":"04:59.750 ","End":"05:02.155","Text":"but I didn\u0027t put the plus c at the end."},{"Start":"05:02.155 ","End":"05:07.505","Text":"Now, we have to substitute back for v. V was Y over X because Y is Xv,"},{"Start":"05:07.505 ","End":"05:10.745","Text":"where I see v like here and here I put Y over X."},{"Start":"05:10.745 ","End":"05:13.400","Text":"This is still not the original variables we had x and"},{"Start":"05:13.400 ","End":"05:16.610","Text":"y and I\u0027m not going to scroll back up, but if you remember,"},{"Start":"05:16.610 ","End":"05:24.065","Text":"X=x-1, or perhaps it was x=X+1."},{"Start":"05:24.065 ","End":"05:26.270","Text":"But either way. Also,"},{"Start":"05:26.270 ","End":"05:32.410","Text":"we have to substitute instead of Y=y+2, or y=Y-2."},{"Start":"05:32.410 ","End":"05:35.040","Text":"But this is the way round we need it."},{"Start":"05:35.040 ","End":"05:40.190","Text":"Now, we have a solution in the original x and y,"},{"Start":"05:40.190 ","End":"05:43.250","Text":"but we have to take care of what are called the singular solutions."},{"Start":"05:43.250 ","End":"05:46.310","Text":"Remember we said v mustn\u0027t be plus or minus the square root"},{"Start":"05:46.310 ","End":"05:49.535","Text":"of 5 and we put an asterisk there. Let\u0027s take one of them."},{"Start":"05:49.535 ","End":"05:51.260","Text":"I take the plus root 5,"},{"Start":"05:51.260 ","End":"05:54.035","Text":"then that gives us Y over X is this."},{"Start":"05:54.035 ","End":"05:57.210","Text":"Then we go back here and isolate little y,"},{"Start":"05:57.210 ","End":"06:03.620","Text":"and this is one singular solution that we get and it satisfies the original equation."},{"Start":"06:03.620 ","End":"06:05.315","Text":"Here we have solutions,"},{"Start":"06:05.315 ","End":"06:08.480","Text":"infinitely many depending on c. Here we have another one,"},{"Start":"06:08.480 ","End":"06:10.715","Text":"singular one, and there\u0027s another singular one,"},{"Start":"06:10.715 ","End":"06:16.430","Text":"which we\u0027re going to get if we take the minus root of 1/2 very similar to before,"},{"Start":"06:16.430 ","End":"06:19.085","Text":"if you just look at it as almost the same thing,"},{"Start":"06:19.085 ","End":"06:23.060","Text":"the same steps, and we get to v, we get Y over X."},{"Start":"06:23.060 ","End":"06:24.440","Text":"We finally go back to little y,"},{"Start":"06:24.440 ","End":"06:26.240","Text":"little x and at the end,"},{"Start":"06:26.240 ","End":"06:28.505","Text":"we just have y in terms of x."},{"Start":"06:28.505 ","End":"06:30.740","Text":"That\u0027s the second singular solution."},{"Start":"06:30.740 ","End":"06:34.775","Text":"The solutions are here with a variable constant c,"},{"Start":"06:34.775 ","End":"06:39.050","Text":"and we have this one singular solution and this singular solution."},{"Start":"06:39.050 ","End":"06:41.120","Text":"There\u0027s 3 places we have a solution,"},{"Start":"06:41.120 ","End":"06:42.150","Text":"but here there\u0027s infinite."},{"Start":"06:42.150 ","End":"06:44.060","Text":"I still owe you that integral."},{"Start":"06:44.060 ","End":"06:46.520","Text":"Well, I didn\u0027t really want to waste too much time"},{"Start":"06:46.520 ","End":"06:49.010","Text":"showing you how I got from here to here."},{"Start":"06:49.010 ","End":"06:50.240","Text":"I\u0027ll just give you some hints."},{"Start":"06:50.240 ","End":"06:52.925","Text":"We\u0027ve done this thing before using partial fractions."},{"Start":"06:52.925 ","End":"06:58.070","Text":"What we do is we write the 1+2v over 1-2v^2."},{"Start":"06:58.070 ","End":"07:00.590","Text":"We factorize the denominator,"},{"Start":"07:00.590 ","End":"07:07.635","Text":"which comes out to be 1 minus root 2v times 1 plus root 2v."},{"Start":"07:07.635 ","End":"07:09.630","Text":"You put some constant A here,"},{"Start":"07:09.630 ","End":"07:12.290","Text":"some constant B here and then we"},{"Start":"07:12.290 ","End":"07:16.505","Text":"cross-multiply and we make certain assignments and so on."},{"Start":"07:16.505 ","End":"07:18.260","Text":"We find what A and B are,"},{"Start":"07:18.260 ","End":"07:21.050","Text":"don\u0027t want to make it a whole lesson in partial fractions."},{"Start":"07:21.050 ","End":"07:22.230","Text":"We integrate this,"},{"Start":"07:22.230 ","End":"07:23.745","Text":"put the integral sign here and here,"},{"Start":"07:23.745 ","End":"07:27.560","Text":"and this comes out to be some variation of natural logarithm."},{"Start":"07:27.560 ","End":"07:30.980","Text":"I\u0027m going to skip this part and I\u0027m just going to say that we\u0027re done."},{"Start":"07:30.980 ","End":"07:32.210","Text":"Where were our solutions?"},{"Start":"07:32.210 ","End":"07:33.530","Text":"Here, they were, here,"},{"Start":"07:33.530 ","End":"07:38.460","Text":"here and here and I\u0027ll leave the partial fractions alone. That\u0027s it."}],"Thumbnail":null,"ID":7666},{"Watched":false,"Name":"Exercise 5","Duration":"7m 14s","ChapterTopicVideoID":7593,"CourseChapterTopicPlaylistID":4220,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.000","Text":"We have here a differential equation and this should look familiar to you because it\u0027s of"},{"Start":"00:06.000 ","End":"00:13.080","Text":"the form some linear expression in x and y. A_1x plus b_1y plus c1,"},{"Start":"00:13.080 ","End":"00:18.285","Text":"dx plus a similar thing just with different numbers, dy."},{"Start":"00:18.285 ","End":"00:20.400","Text":"We know how to solve this."},{"Start":"00:20.400 ","End":"00:22.080","Text":"We write down the coefficients,"},{"Start":"00:22.080 ","End":"00:23.130","Text":"the a\u0027s and the b\u0027s."},{"Start":"00:23.130 ","End":"00:24.690","Text":"We don\u0027t care about the c\u0027s."},{"Start":"00:24.690 ","End":"00:29.240","Text":"In general, there are 2 cases depending on if a_1 b_2 is equal to a_2,"},{"Start":"00:29.240 ","End":"00:30.860","Text":"b_1, or it\u0027s not equal."},{"Start":"00:30.860 ","End":"00:34.060","Text":"In our case, we get the not equal because this times this is 2,"},{"Start":"00:34.060 ","End":"00:35.655","Text":"but this times this is 1."},{"Start":"00:35.655 ","End":"00:37.460","Text":"And in the case of not equal,"},{"Start":"00:37.460 ","End":"00:40.505","Text":"the way we do it is to get it to be"},{"Start":"00:40.505 ","End":"00:43.490","Text":"a homogeneous equation as a trick to get rid"},{"Start":"00:43.490 ","End":"00:46.625","Text":"of the minus 3 and the minus 1 here using a substitution."},{"Start":"00:46.625 ","End":"00:47.960","Text":"But before the substitution,"},{"Start":"00:47.960 ","End":"00:52.730","Text":"we have to solve a pair of equations in 2 unknowns."},{"Start":"00:52.730 ","End":"00:57.050","Text":"Basically we set this to be 0 and this to be 0."},{"Start":"00:57.050 ","End":"00:58.280","Text":"I\u0027m going to give you the solution."},{"Start":"00:58.280 ","End":"01:00.065","Text":"We\u0027re not going to spend time solving it."},{"Start":"01:00.065 ","End":"01:01.700","Text":"This is what it comes out to."},{"Start":"01:01.700 ","End":"01:04.715","Text":"X is 2 and y is minus 1."},{"Start":"01:04.715 ","End":"01:07.880","Text":"These two numbers give us what to substitute."},{"Start":"01:07.880 ","End":"01:10.400","Text":"We need to substitute instead of X and Y,"},{"Start":"01:10.400 ","End":"01:12.110","Text":"I call them big X and big Y,"},{"Start":"01:12.110 ","End":"01:13.670","Text":"X is big X plus 2."},{"Start":"01:13.670 ","End":"01:14.903","Text":"The 2 is from here."},{"Start":"01:14.903 ","End":"01:17.300","Text":"Similarly here we\u0027ve got the minus 1 from here."},{"Start":"01:17.300 ","End":"01:18.890","Text":"But besides x and y,"},{"Start":"01:18.890 ","End":"01:21.080","Text":"we also need to know what dx and dy are."},{"Start":"01:21.080 ","End":"01:22.835","Text":"They always come out to be the same?"},{"Start":"01:22.835 ","End":"01:24.650","Text":"Because it\u0027s always 1x plus something,"},{"Start":"01:24.650 ","End":"01:27.545","Text":"1y minus some other constant."},{"Start":"01:27.545 ","End":"01:31.130","Text":"If I put x and y in here,"},{"Start":"01:31.130 ","End":"01:34.385","Text":"you can see we have x here and here,"},{"Start":"01:34.385 ","End":"01:39.005","Text":"and we put it as X plus 2 here and here for the dx,"},{"Start":"01:39.005 ","End":"01:40.460","Text":"we put dX,"},{"Start":"01:40.460 ","End":"01:42.980","Text":"we have y here and here,"},{"Start":"01:42.980 ","End":"01:45.710","Text":"and that\u0027s what we substitute here and here."},{"Start":"01:45.710 ","End":"01:48.440","Text":"And the dy is the same as dy."},{"Start":"01:48.440 ","End":"01:50.525","Text":"This is after substitution."},{"Start":"01:50.525 ","End":"01:54.980","Text":"And now I want to simplify this and all the constants cancel out."},{"Start":"01:54.980 ","End":"01:59.765","Text":"For example, here I have 2 times 2 is 4 minus 1 minus 3 and so on."},{"Start":"01:59.765 ","End":"02:07.700","Text":"In fact, what we get is the same as basically with the big X and big Y,"},{"Start":"02:07.700 ","End":"02:11.090","Text":"it\u0027s the same as that these constants are now missing."},{"Start":"02:11.090 ","End":"02:14.330","Text":"In fact, if you\u0027ve got to do a lot of these exercises,"},{"Start":"02:14.330 ","End":"02:16.565","Text":"once you know what to substitute,"},{"Start":"02:16.565 ","End":"02:19.955","Text":"you wouldn\u0027t have to actually go through all this and then simplify it."},{"Start":"02:19.955 ","End":"02:22.880","Text":"You could know that the answer straightaway is the same"},{"Start":"02:22.880 ","End":"02:26.135","Text":"as this with big X and big Y and the constants removed."},{"Start":"02:26.135 ","End":"02:29.555","Text":"So that\u0027s how it always happens and it always turns out"},{"Start":"02:29.555 ","End":"02:33.660","Text":"that this is homogeneous of degree 1."},{"Start":"02:33.660 ","End":"02:35.910","Text":"I\u0027m not going to do the thing with the Lambda where we check it."},{"Start":"02:35.910 ","End":"02:39.170","Text":"This function m is homogeneous of degree 1 and is"},{"Start":"02:39.170 ","End":"02:43.505","Text":"homogeneous of degree 1 since both functions are homogeneous with the same degree."},{"Start":"02:43.505 ","End":"02:46.100","Text":"This is a homogeneous equation and we solve"},{"Start":"02:46.100 ","End":"02:50.600","Text":"homogeneous equations the same way with yet another substitution."},{"Start":"02:50.600 ","End":"02:52.970","Text":"This time we substitute instead of big Y,"},{"Start":"02:52.970 ","End":"02:55.985","Text":"usually the letter v is used times X."},{"Start":"02:55.985 ","End":"02:58.970","Text":"Later on we\u0027ll say that v is Y over X."},{"Start":"02:58.970 ","End":"03:00.545","Text":"After we\u0027ve solved everything,"},{"Start":"03:00.545 ","End":"03:03.380","Text":"we\u0027re going to get an equation in Y and v. Then we\u0027re going to"},{"Start":"03:03.380 ","End":"03:06.470","Text":"substitute from v back to a big X and big Y."},{"Start":"03:06.470 ","End":"03:07.550","Text":"And from big X and big Y,"},{"Start":"03:07.550 ","End":"03:10.230","Text":"will go back to little x and little y. Anyway, I\u0027ll just do it."},{"Start":"03:10.270 ","End":"03:15.050","Text":"This is what Y is and dY we get from the product rule,"},{"Start":"03:15.050 ","End":"03:19.770","Text":"derivative of d times this with just this and then d product rule."},{"Start":"03:19.770 ","End":"03:24.395","Text":"Just substituting, we have Y here and here."},{"Start":"03:24.395 ","End":"03:27.710","Text":"That becomes v times X, v times X."},{"Start":"03:27.710 ","End":"03:30.575","Text":"The dY is this whole bit."},{"Start":"03:30.575 ","End":"03:31.910","Text":"We start simplifying."},{"Start":"03:31.910 ","End":"03:34.100","Text":"First of all, expanding. Where we\u0027re heading to is this is"},{"Start":"03:34.100 ","End":"03:37.295","Text":"guaranteed to give us a separable equation."},{"Start":"03:37.295 ","End":"03:41.030","Text":"Let\u0027s see, the first square bracket just opened everything up,"},{"Start":"03:41.030 ","End":"03:45.635","Text":"expanded by multiplying v by this and by this."},{"Start":"03:45.635 ","End":"03:48.890","Text":"Over here, I actually didn\u0027t expand the opposite."},{"Start":"03:48.890 ","End":"03:52.910","Text":"I took an extra X outside the brackets to combine with this X,"},{"Start":"03:52.910 ","End":"03:55.010","Text":"I\u0027m leaving 1 plus v here."},{"Start":"03:55.010 ","End":"03:56.915","Text":"There\u0027s many ways of simplifying this."},{"Start":"03:56.915 ","End":"04:01.640","Text":"Next step is to collect off wherever it goes with dX and whatever it goes with"},{"Start":"04:01.640 ","End":"04:06.605","Text":"dv just simplified and we\u0027re getting very close to separation of variables,"},{"Start":"04:06.605 ","End":"04:07.895","Text":"a jump to a new page."},{"Start":"04:07.895 ","End":"04:09.170","Text":"I basically copied it,"},{"Start":"04:09.170 ","End":"04:12.800","Text":"but I\u0027ve started to do some canceling this X."},{"Start":"04:12.800 ","End":"04:17.180","Text":"One of these X\u0027s will go presuming that X is not 0."},{"Start":"04:17.180 ","End":"04:18.815","Text":"Made a note of that here,"},{"Start":"04:18.815 ","End":"04:20.690","Text":"and X is not going to be 0."},{"Start":"04:20.690 ","End":"04:25.820","Text":"And then I bring this second term until the other side of the plus becomes a minus."},{"Start":"04:25.820 ","End":"04:29.135","Text":"Then this goes to the right-hand side,"},{"Start":"04:29.135 ","End":"04:31.505","Text":"and this will go over to the left-hand side."},{"Start":"04:31.505 ","End":"04:34.445","Text":"Will be left with dX over X equals,"},{"Start":"04:34.445 ","End":"04:35.930","Text":"and I put the minus in front,"},{"Start":"04:35.930 ","End":"04:37.370","Text":"1 plus v over this."},{"Start":"04:37.370 ","End":"04:40.070","Text":"Now we have the variables separated."},{"Start":"04:40.070 ","End":"04:44.044","Text":"Sometimes we have to make sure that the denominator is not 0,"},{"Start":"04:44.044 ","End":"04:48.320","Text":"but this quadratic in v is never 0."},{"Start":"04:48.320 ","End":"04:49.400","Text":"I could get into this."},{"Start":"04:49.400 ","End":"04:53.690","Text":"You could check the discriminant b^2 minus 4ac is negative."},{"Start":"04:53.690 ","End":"04:55.099","Text":"This has no solutions,"},{"Start":"04:55.099 ","End":"04:57.755","Text":"or I could write it as (v+1)^ 2 plus 1."},{"Start":"04:57.755 ","End":"04:58.925","Text":"Anyway, think about it."},{"Start":"04:58.925 ","End":"05:03.680","Text":"This is never 0. So I don\u0027t have to watch out because I\u0027m used to saying,"},{"Start":"05:03.680 ","End":"05:06.440","Text":"this is true as long as v is not equal to this or this."},{"Start":"05:06.440 ","End":"05:08.180","Text":"But in this case, no worry."},{"Start":"05:08.180 ","End":"05:09.350","Text":"Once we\u0027ve had it separated,"},{"Start":"05:09.350 ","End":"05:12.715","Text":"all we have to do now is take an integral sign in front of each, like so."},{"Start":"05:12.715 ","End":"05:16.880","Text":"What I\u0027m going to do next is I\u0027m going to just do a little bit of"},{"Start":"05:16.880 ","End":"05:20.360","Text":"algebra to get this integral to be in a convenient form."},{"Start":"05:20.360 ","End":"05:24.020","Text":"See the numerator is roughly the derivative of denominator."},{"Start":"05:24.020 ","End":"05:30.105","Text":"The derivative of the denominator would be 2 plus 2v."},{"Start":"05:30.105 ","End":"05:34.115","Text":"But here I have 1 plus v. Now it\u0027s not the same thing,"},{"Start":"05:34.115 ","End":"05:37.300","Text":"but if I multiply by 2 and divide by 2,"},{"Start":"05:37.300 ","End":"05:39.240","Text":"like so and put an extra 2 here,"},{"Start":"05:39.240 ","End":"05:40.500","Text":"an extra 2 here."},{"Start":"05:40.500 ","End":"05:44.240","Text":"This of course is twice 1 plus v. Then I have here"},{"Start":"05:44.240 ","End":"05:48.305","Text":"the derivative of the denominator in the numerator a 2 here and here,"},{"Start":"05:48.305 ","End":"05:49.550","Text":"balance each other out."},{"Start":"05:49.550 ","End":"05:55.370","Text":"So now I can easily do the integral on the left natural log with the absolute value."},{"Start":"05:55.370 ","End":"05:58.355","Text":"Also here, natural log of the denominator."},{"Start":"05:58.355 ","End":"06:00.680","Text":"I didn\u0027t need an absolute value here,"},{"Start":"06:00.680 ","End":"06:02.600","Text":"because like I said, the denominator here,"},{"Start":"06:02.600 ","End":"06:04.220","Text":"this is always positive."},{"Start":"06:04.220 ","End":"06:07.265","Text":"You could have put absolute value, but no need."},{"Start":"06:07.265 ","End":"06:09.350","Text":"The half, of course, just stays."},{"Start":"06:09.350 ","End":"06:13.340","Text":"Then we start back substituting because we want to get back to little x, little y."},{"Start":"06:13.340 ","End":"06:19.585","Text":"First of all, we remembered that Y was vX or v was Y over X."},{"Start":"06:19.585 ","End":"06:24.930","Text":"This v is replaced by this Y over X and this v by"},{"Start":"06:24.930 ","End":"06:30.770","Text":"this Y over X. I want to remind you what we substituted earlier for x and y,"},{"Start":"06:30.770 ","End":"06:32.615","Text":"but we can also do it the other way."},{"Start":"06:32.615 ","End":"06:35.780","Text":"What I\u0027m really doing is doing the opposite is saying that big X is"},{"Start":"06:35.780 ","End":"06:40.745","Text":"little x minus 2 and big Y is little y plus 1."},{"Start":"06:40.745 ","End":"06:44.510","Text":"If we substitute for y and for x here,"},{"Start":"06:44.510 ","End":"06:48.005","Text":"you can see y is y plus 1 and x is x minus 2,"},{"Start":"06:48.005 ","End":"06:50.345","Text":"big X, little x minus 2 and so on."},{"Start":"06:50.345 ","End":"06:53.585","Text":"That\u0027s near general solution."},{"Start":"06:53.585 ","End":"06:57.380","Text":"The only thing is we have just mentioned that,"},{"Start":"06:57.380 ","End":"06:59.629","Text":"we had before that x not equal to 0,"},{"Start":"06:59.629 ","End":"07:05.405","Text":"that translates as x minus 2 not equal to 0 or x not equal to 2."},{"Start":"07:05.405 ","End":"07:10.010","Text":"We need that because the solution does not have in its domain x equals to 2,"},{"Start":"07:10.010 ","End":"07:11.810","Text":"because then x minus 2 would be 0."},{"Start":"07:11.810 ","End":"07:15.060","Text":"Anyway, that\u0027s the answer and we\u0027re done."}],"Thumbnail":null,"ID":7667}],"ID":4220},{"Name":"Exact Equations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exact Equations","Duration":"4m 49s","ChapterTopicVideoID":7596,"CourseChapterTopicPlaylistID":4221,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"This exercise is not really an exercise,"},{"Start":"00:02.640 ","End":"00:05.370","Text":"it\u0027s just a tutorial disguised as an exercise,"},{"Start":"00:05.370 ","End":"00:06.600","Text":"just as a format."},{"Start":"00:06.600 ","End":"00:10.350","Text":"It\u0027s like to explain what is an exact differential equation,"},{"Start":"00:10.350 ","End":"00:12.165","Text":"the keyword here is exact,"},{"Start":"00:12.165 ","End":"00:16.350","Text":"and what is the method for solving it? That\u0027s 2 parts."},{"Start":"00:16.350 ","End":"00:17.955","Text":"First, for what it is,"},{"Start":"00:17.955 ","End":"00:20.340","Text":"and later we\u0027ll talk about how to solve it."},{"Start":"00:20.340 ","End":"00:22.410","Text":"An exact equation,"},{"Start":"00:22.410 ","End":"00:24.630","Text":"it looks like this."},{"Start":"00:24.630 ","End":"00:26.080","Text":"Some function of (x,"},{"Start":"00:26.080 ","End":"00:28.550","Text":"y)dx and another function of (x,"},{"Start":"00:28.550 ","End":"00:34.620","Text":"y)dy equals 0 when added with an important condition that"},{"Start":"00:34.620 ","End":"00:37.890","Text":"the derivative of this part with respect to"},{"Start":"00:37.890 ","End":"00:42.210","Text":"y equals the derivative of this part with respect to x."},{"Start":"00:42.210 ","End":"00:45.410","Text":"Maybe not the order you would expect this with respect to y,"},{"Start":"00:45.410 ","End":"00:47.345","Text":"this with respect to x."},{"Start":"00:47.345 ","End":"00:52.895","Text":"The theory here won\u0027t necessarily make too much sense without an example,"},{"Start":"00:52.895 ","End":"00:55.880","Text":"I\u0027m going to give the recipe for how to solve it on here,"},{"Start":"00:55.880 ","End":"00:57.860","Text":"the definition of what it is."},{"Start":"00:57.860 ","End":"01:00.755","Text":"As soon as you see your first solved example,"},{"Start":"01:00.755 ","End":"01:02.330","Text":"you\u0027ll understand it better,"},{"Start":"01:02.330 ","End":"01:03.740","Text":"and then you can come back here."},{"Start":"01:03.740 ","End":"01:05.315","Text":"You can go back and forth."},{"Start":"01:05.315 ","End":"01:08.045","Text":"I just want to have it written down organized,"},{"Start":"01:08.045 ","End":"01:12.200","Text":"though it won\u0027t click until you get to the solved examples."},{"Start":"01:12.200 ","End":"01:14.044","Text":"Here are the main steps."},{"Start":"01:14.044 ","End":"01:17.690","Text":"The solution we\u0027re looking for is going to be an implicit form."},{"Start":"01:17.690 ","End":"01:23.044","Text":"It\u0027s going to be some function of x and y equals a general constant,"},{"Start":"01:23.044 ","End":"01:28.280","Text":"C will stay as a C and our job is to find F(x, y)."},{"Start":"01:28.280 ","End":"01:31.640","Text":"Now what we have to go on is that the derivative of"},{"Start":"01:31.640 ","End":"01:34.910","Text":"this function with respect to x is going to be the M,"},{"Start":"01:34.910 ","End":"01:40.790","Text":"that\u0027s the M from here and the derivative with respect to y will be the N from here."},{"Start":"01:40.790 ","End":"01:43.265","Text":"Using these 2 bits of information,"},{"Start":"01:43.265 ","End":"01:49.610","Text":"we\u0027ll be able to find F. I\u0027m continuing with the steps for finding F."},{"Start":"01:49.610 ","End":"01:56.735","Text":"The first step is to use this equation and perform an integration on this."},{"Start":"01:56.735 ","End":"01:58.645","Text":"Need some more space here."},{"Start":"01:58.645 ","End":"02:01.895","Text":"From the derivative with respect to x,"},{"Start":"02:01.895 ","End":"02:05.675","Text":"we can get F by taking the integral of this with respect to x."},{"Start":"02:05.675 ","End":"02:08.975","Text":"Put an asterisk here because it\u0027s something I want to tell you later."},{"Start":"02:08.975 ","End":"02:13.160","Text":"The integral is also going to be some function of x and y."},{"Start":"02:13.160 ","End":"02:14.555","Text":"We just do the integration,"},{"Start":"02:14.555 ","End":"02:18.380","Text":"but we don\u0027t just put plus c here because"},{"Start":"02:18.380 ","End":"02:22.219","Text":"when you do an integral with respect to x of a partial derivative,"},{"Start":"02:22.219 ","End":"02:26.818","Text":"a constant is replaced by a general function of y."},{"Start":"02:26.818 ","End":"02:28.100","Text":"In the next step,"},{"Start":"02:28.100 ","End":"02:33.470","Text":"we use the other equation that the derivative of F with respect to y is N. Now we have,"},{"Start":"02:33.470 ","End":"02:35.125","Text":"on the 1 hand,"},{"Start":"02:35.125 ","End":"02:36.440","Text":"and is N of course,"},{"Start":"02:36.440 ","End":"02:40.055","Text":"but we do have the form of F as this."},{"Start":"02:40.055 ","End":"02:43.910","Text":"We can say more precisely what is the partial derivative?"},{"Start":"02:43.910 ","End":"02:45.770","Text":"It\u0027s the partial derivative of this,"},{"Start":"02:45.770 ","End":"02:49.860","Text":"which I\u0027ll just write as G(y) lazy to write the x comma y"},{"Start":"02:49.860 ","End":"02:55.550","Text":"and the derivative of this bit is a function of 1 variable y,"},{"Start":"02:55.550 ","End":"02:57.965","Text":"so it\u0027s just h\u0027(y),"},{"Start":"02:57.965 ","End":"03:00.085","Text":"so that we get this equation."},{"Start":"03:00.085 ","End":"03:04.550","Text":"Then of course we\u0027ll be given or you\u0027ll see in the examples how it works out."},{"Start":"03:04.550 ","End":"03:08.135","Text":"I\u0027m just giving you, as I said, the general recipe."},{"Start":"03:08.135 ","End":"03:09.755","Text":"Here, I\u0027ve done 2 steps in 1."},{"Start":"03:09.755 ","End":"03:17.345","Text":"On the 1 hand, I\u0027ve taken the g_y and moved it to the right-hand side so it\u0027s N minus it."},{"Start":"03:17.345 ","End":"03:19.355","Text":"Then from the derivative,"},{"Start":"03:19.355 ","End":"03:25.285","Text":"I conclude that h is the integral of this dy."},{"Start":"03:25.285 ","End":"03:29.580","Text":"At this point, I have N explicitly and I have g_y explicitly."},{"Start":"03:29.580 ","End":"03:31.665","Text":"I just need to do an integral,"},{"Start":"03:31.665 ","End":"03:33.285","Text":"h is what I don\u0027t know."},{"Start":"03:33.285 ","End":"03:35.205","Text":"Once we have h(y),"},{"Start":"03:35.205 ","End":"03:43.470","Text":"we then put it back in here and we have that F = g(x,y) + h(y)."},{"Start":"03:43.470 ","End":"03:46.790","Text":"The solution will be this thing equals"},{"Start":"03:46.790 ","End":"03:51.150","Text":"C. That\u0027s what we said is the general shape of things."},{"Start":"03:51.150 ","End":"03:53.090","Text":"I repeat, in the exercises,"},{"Start":"03:53.090 ","End":"03:58.220","Text":"it will become much clearer now about the asterisk. The asterisk here."},{"Start":"03:58.220 ","End":"04:02.450","Text":"Sometimes this turns out to be a difficult integral and we can"},{"Start":"04:02.450 ","End":"04:09.695","Text":"actually do a variation because we have these 2 equations, F_x and F_y."},{"Start":"04:09.695 ","End":"04:15.080","Text":"Sometimes it\u0027s easier to start with this bit just swapping"},{"Start":"04:15.080 ","End":"04:21.080","Text":"over the roles of M and N get a similar approach by starting with this 1,"},{"Start":"04:21.080 ","End":"04:24.740","Text":"because the integral of N with respect to y,"},{"Start":"04:24.740 ","End":"04:27.670","Text":"might turn out to be easier to compute."},{"Start":"04:27.670 ","End":"04:31.400","Text":"That\u0027s just a variation in case you have difficulty with this integral,"},{"Start":"04:31.400 ","End":"04:34.700","Text":"but this one is easy. That\u0027s all I\u0027m going to say."},{"Start":"04:34.700 ","End":"04:37.580","Text":"Now you have this written down for reference and you have"},{"Start":"04:37.580 ","End":"04:41.135","Text":"a general idea of what to expect in the exercises."},{"Start":"04:41.135 ","End":"04:43.550","Text":"Now, I suggest you go and look at"},{"Start":"04:43.550 ","End":"04:50.010","Text":"the solved exercises and come back to this for reference if need be. I\u0027m done."}],"Thumbnail":null,"ID":7668},{"Watched":false,"Name":"Exercise 1","Duration":"4m 20s","ChapterTopicVideoID":7597,"CourseChapterTopicPlaylistID":4221,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"We have here a differential equation to solve,"},{"Start":"00:02.940 ","End":"00:06.900","Text":"and it happens to be in the chapter on exact equations."},{"Start":"00:06.900 ","End":"00:10.575","Text":"Let\u0027s see that this really is an exact equation."},{"Start":"00:10.575 ","End":"00:16.245","Text":"Copied it. We\u0027ll let this part be M and this part N, just to give them names."},{"Start":"00:16.245 ","End":"00:19.110","Text":"In order for it to be exact,"},{"Start":"00:19.110 ","End":"00:22.890","Text":"what we have to have is that the derivative of this part with respect"},{"Start":"00:22.890 ","End":"00:26.865","Text":"to y and the derivative of this part with respect to x are equal."},{"Start":"00:26.865 ","End":"00:30.000","Text":"Well, this with respect to y is 3."},{"Start":"00:30.000 ","End":"00:33.218","Text":"This respect to x is 3 and they are indeed equal,"},{"Start":"00:33.218 ","End":"00:35.225","Text":"so it\u0027s an exact equation."},{"Start":"00:35.225 ","End":"00:39.425","Text":"We have the recipe for how to solve the exact equations."},{"Start":"00:39.425 ","End":"00:41.750","Text":"The general form of the solution will be"},{"Start":"00:41.750 ","End":"00:47.015","Text":"some function of variables x and y equals a constant."},{"Start":"00:47.015 ","End":"00:49.340","Text":"We get the solution in implicit form."},{"Start":"00:49.340 ","End":"00:54.380","Text":"The F has the properties that its derivative with respect to x is going to"},{"Start":"00:54.380 ","End":"00:57.230","Text":"be M and the derivative with respect to"},{"Start":"00:57.230 ","End":"01:00.824","Text":"y is going to be N. Don\u0027t get these things backward,"},{"Start":"01:00.824 ","End":"01:03.110","Text":"when we do the check, it\u0027s opposite."},{"Start":"01:03.110 ","End":"01:05.945","Text":"It\u0027s M with respect to y and then with respect to x."},{"Start":"01:05.945 ","End":"01:09.035","Text":"But here we\u0027re looking for F with respect to x."},{"Start":"01:09.035 ","End":"01:12.200","Text":"It\u0027s the right way round and this is the opposite way round."},{"Start":"01:12.200 ","End":"01:15.710","Text":"Anyway, you have it written down somewhere so you won\u0027t get confused."},{"Start":"01:15.710 ","End":"01:18.050","Text":"To find this F, I start with one of these."},{"Start":"01:18.050 ","End":"01:20.195","Text":"Usually, I\u0027ll take the first one."},{"Start":"01:20.195 ","End":"01:22.760","Text":"That partial derivative with respect to x is"},{"Start":"01:22.760 ","End":"01:26.615","Text":"M. But if this happens to turn out particularly difficult,"},{"Start":"01:26.615 ","End":"01:31.400","Text":"you can do it the other way round by just reversing the roles of x and y,"},{"Start":"01:31.400 ","End":"01:33.320","Text":"and starting with this one,"},{"Start":"01:33.320 ","End":"01:35.360","Text":"because this one gives us this integral."},{"Start":"01:35.360 ","End":"01:36.620","Text":"If it\u0027s difficult to compute,"},{"Start":"01:36.620 ","End":"01:40.192","Text":"it may be that the integral of N. dy is easier, I\u0027m just mentioning."},{"Start":"01:40.192 ","End":"01:43.630","Text":"It doesn\u0027t always have to be that you start with this one."},{"Start":"01:43.630 ","End":"01:46.365","Text":"In our case, M is just here,"},{"Start":"01:46.365 ","End":"01:48.470","Text":"so I will replace it,"},{"Start":"01:48.470 ","End":"01:50.795","Text":"and we do the integral."},{"Start":"01:50.795 ","End":"01:55.355","Text":"But the thing to notice is that this is an easy integral with respect to x,"},{"Start":"01:55.355 ","End":"01:58.100","Text":"we raise the power by 1, 2/4 is 1/2."},{"Start":"01:58.100 ","End":"02:00.410","Text":"Also, here,"},{"Start":"02:00.410 ","End":"02:04.115","Text":"3y is just a constant as far as x goes."},{"Start":"02:04.115 ","End":"02:06.800","Text":"The thing to watch out for us is not to put plus"},{"Start":"02:06.800 ","End":"02:09.920","Text":"c. That\u0027s when we have a partial derivative."},{"Start":"02:09.920 ","End":"02:11.990","Text":"The function of two variables,"},{"Start":"02:11.990 ","End":"02:17.104","Text":"constant is really any function of y."},{"Start":"02:17.104 ","End":"02:21.590","Text":"Because any function of y is a constant as far x goes."},{"Start":"02:21.590 ","End":"02:26.509","Text":"The way we\u0027re going to find g(y) is by using the other equation that we haven\u0027t used."},{"Start":"02:26.509 ","End":"02:27.905","Text":"We use this one,"},{"Start":"02:27.905 ","End":"02:29.750","Text":"now we\u0027re going to use this one."},{"Start":"02:29.750 ","End":"02:32.075","Text":"I need some more space here."},{"Start":"02:32.075 ","End":"02:36.020","Text":"Now, if this is F, and F with respect to y is"},{"Start":"02:36.020 ","End":"02:40.639","Text":"just the derivative of this gives me nothing because this is a constant,"},{"Start":"02:40.639 ","End":"02:44.240","Text":"and this part gives me 3x."},{"Start":"02:44.240 ","End":"02:50.660","Text":"It\u0027s a constant times y, and this one gives me g\u0027 the derivative with respect to y."},{"Start":"02:50.660 ","End":"02:52.130","Text":"On the other hand,"},{"Start":"02:52.130 ","End":"02:56.756","Text":"the other equation was F with respect to y equals N. I\u0027ll just show you,"},{"Start":"02:56.756 ","End":"03:04.120","Text":"here it is, and N was equal to 3x plus y minus 1."},{"Start":"03:04.120 ","End":"03:12.860","Text":"What we get is that this from this part from here is equal to 3x plus y minus 1."},{"Start":"03:12.860 ","End":"03:15.575","Text":"Both of them are equal to F with respect to y."},{"Start":"03:15.575 ","End":"03:18.935","Text":"We get from here, the 3x cancels with the 3x,"},{"Start":"03:18.935 ","End":"03:22.820","Text":"and so we get g\u0027(y) is y minus 1."},{"Start":"03:22.820 ","End":"03:24.980","Text":"Now we\u0027re dealing in the function in one variable,"},{"Start":"03:24.980 ","End":"03:26.420","Text":"we know the derivative,"},{"Start":"03:26.420 ","End":"03:30.680","Text":"so we\u0027re just looking for any function g. One way is to"},{"Start":"03:30.680 ","End":"03:35.480","Text":"take an indefinite integral of y minus 1."},{"Start":"03:35.480 ","End":"03:37.790","Text":"The indefinite integral is an anti-derivative."},{"Start":"03:37.790 ","End":"03:39.890","Text":"The integral of this comes out to be this."},{"Start":"03:39.890 ","End":"03:41.960","Text":"We don\u0027t need the c because, in the end,"},{"Start":"03:41.960 ","End":"03:43.940","Text":"we have the c on the right-hand side."},{"Start":"03:43.940 ","End":"03:46.045","Text":"We just need a solution."},{"Start":"03:46.045 ","End":"03:48.320","Text":"Now that we have g(y),"},{"Start":"03:48.320 ","End":"03:54.665","Text":"then we can plug that into here because now we can fully know what F is."},{"Start":"03:54.665 ","End":"03:57.215","Text":"This bit is copied here,"},{"Start":"03:57.215 ","End":"03:59.930","Text":"and the g(y) copied from here."},{"Start":"03:59.930 ","End":"04:01.565","Text":"This is what it is."},{"Start":"04:01.565 ","End":"04:06.830","Text":"Finally, you remember we had that F equals a constant with our general solution,"},{"Start":"04:06.830 ","End":"04:08.890","Text":"f(x,y) equal c,"},{"Start":"04:08.890 ","End":"04:10.130","Text":"and in our case,"},{"Start":"04:10.130 ","End":"04:11.825","Text":"the function is this."},{"Start":"04:11.825 ","End":"04:16.847","Text":"This is our general solution in implicit form."},{"Start":"04:16.847 ","End":"04:21.090","Text":"That\u0027s okay and we\u0027re done."}],"Thumbnail":null,"ID":7669},{"Watched":false,"Name":"Exercise 2","Duration":"6m 45s","ChapterTopicVideoID":7595,"CourseChapterTopicPlaylistID":4221,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.120","Text":"We have here this differential equation to solve,"},{"Start":"00:03.120 ","End":"00:06.705","Text":"and I\u0027m going to show that this is an exact equation."},{"Start":"00:06.705 ","End":"00:08.700","Text":"We first copy it. Here we are."},{"Start":"00:08.700 ","End":"00:11.400","Text":"I just labeled the first function of x and y,"},{"Start":"00:11.400 ","End":"00:14.715","Text":"I call that M and the second one I\u0027ll call N."},{"Start":"00:14.715 ","End":"00:18.540","Text":"If you remember the condition for being exact is"},{"Start":"00:18.540 ","End":"00:21.990","Text":"that the derivative of M partial with respect to"},{"Start":"00:21.990 ","End":"00:26.150","Text":"y has got to equal the derivative of N with respect to x."},{"Start":"00:26.150 ","End":"00:29.370","Text":"Let\u0027s start computing, and we\u0027ll start with the left-hand side."},{"Start":"00:29.370 ","End":"00:33.010","Text":"I guess I should put a question mark here because I have to show this."},{"Start":"00:33.010 ","End":"00:35.510","Text":"The derivative of M with respect to y,"},{"Start":"00:35.510 ","End":"00:37.325","Text":"M is here, y^2."},{"Start":"00:37.325 ","End":"00:41.045","Text":"No problem, that\u0027s 2y the 2 gives nothing,"},{"Start":"00:41.045 ","End":"00:42.155","Text":"but the middle bit,"},{"Start":"00:42.155 ","End":"00:45.410","Text":"we\u0027re going to have to use the quotient rule."},{"Start":"00:45.410 ","End":"00:49.100","Text":"You should remember the quotient rule by heart on the denominator,"},{"Start":"00:49.100 ","End":"00:56.030","Text":"we have this denominator squared and then we have the derivative of the numerator."},{"Start":"00:56.030 ","End":"00:59.615","Text":"I\u0027m just writing it, derivative with respect to y of the numerator"},{"Start":"00:59.615 ","End":"01:04.040","Text":"times the denominator minus the numerator as is,"},{"Start":"01:04.040 ","End":"01:07.280","Text":"which is here, times the derivative of the denominator."},{"Start":"01:07.280 ","End":"01:10.630","Text":"I haven\u0027t actually done the derivatives, I\u0027m just indicating."},{"Start":"01:10.630 ","End":"01:14.780","Text":"I took the derivative with respect to y(y), that\u0027s just 1."},{"Start":"01:14.780 ","End":"01:16.340","Text":"Haven\u0027t done this derivative yet."},{"Start":"01:16.340 ","End":"01:22.010","Text":"I just took x in front of the derivative because it\u0027s a constant as far as y goes,"},{"Start":"01:22.010 ","End":"01:24.275","Text":"so we have this expression and"},{"Start":"01:24.275 ","End":"01:28.670","Text":"this derivative with respect to y is 1 because the x disappears."},{"Start":"01:28.670 ","End":"01:31.115","Text":"Then just simplify this a bit."},{"Start":"01:31.115 ","End":"01:34.925","Text":"x times x is x^2, x times y is xy minus xy,"},{"Start":"01:34.925 ","End":"01:36.920","Text":"the xy cancels,"},{"Start":"01:36.920 ","End":"01:40.250","Text":"and after that, the x^2 cancels with the x^2."},{"Start":"01:40.250 ","End":"01:42.035","Text":"This is the partial derivative."},{"Start":"01:42.035 ","End":"01:45.410","Text":"We\u0027ve computed M with respect to y as this."},{"Start":"01:45.410 ","End":"01:47.330","Text":"I will go and do the other one,"},{"Start":"01:47.330 ","End":"01:48.995","Text":"N with respect to x,"},{"Start":"01:48.995 ","End":"01:50.420","Text":"can\u0027t see N anywhere,"},{"Start":"01:50.420 ","End":"01:53.225","Text":"so just go back up and see what N was."},{"Start":"01:53.225 ","End":"01:56.515","Text":"This is N. I\u0027ll go back down now."},{"Start":"01:56.515 ","End":"01:59.955","Text":"I\u0027ll just copy that so we have it in front of our eyes."},{"Start":"01:59.955 ","End":"02:03.740","Text":"Now, I can do the partial derivative with respect to x."},{"Start":"02:03.740 ","End":"02:07.100","Text":"It\u0027s going to be using the quotient rule."},{"Start":"02:07.100 ","End":"02:09.350","Text":"Here I put the denominator squared,"},{"Start":"02:09.350 ","End":"02:12.040","Text":"the derivative of the numerator is nothing,"},{"Start":"02:12.040 ","End":"02:14.735","Text":"so I get the second bit with the minus,"},{"Start":"02:14.735 ","End":"02:19.280","Text":"the derivative of the denominator times the numerator."},{"Start":"02:19.280 ","End":"02:23.870","Text":"That\u0027s for the second bit 2y is a constant as far as x goes."},{"Start":"02:23.870 ","End":"02:27.205","Text":"It stays and I just have to differentiate the second bit."},{"Start":"02:27.205 ","End":"02:30.785","Text":"The derivative of this with respect to x is just 1,"},{"Start":"02:30.785 ","End":"02:34.930","Text":"and the derivative of this with respect to x is also 1."},{"Start":"02:34.930 ","End":"02:36.995","Text":"We do have that this equals this."},{"Start":"02:36.995 ","End":"02:40.325","Text":"Well, I should\u0027ve written this as M with respect to y."},{"Start":"02:40.325 ","End":"02:42.110","Text":"This and this are equal."},{"Start":"02:42.110 ","End":"02:43.940","Text":"It\u0027s just a slight rearrangement."},{"Start":"02:43.940 ","End":"02:46.820","Text":"This is 2y(2y) minus this,"},{"Start":"02:46.820 ","End":"02:47.900","Text":"so they are equal."},{"Start":"02:47.900 ","End":"02:52.655","Text":"We have an exact equation and that was the first step, just verifying."},{"Start":"02:52.655 ","End":"02:55.910","Text":"Then the second step is we declare that our solution is going to"},{"Start":"02:55.910 ","End":"03:01.115","Text":"be some function of x and y is equal to general constant."},{"Start":"03:01.115 ","End":"03:05.615","Text":"But we have conditions on f that its partial derivative with respect to x is M,"},{"Start":"03:05.615 ","End":"03:09.050","Text":"with respect to y, it\u0027s N. If I started with this one,"},{"Start":"03:09.050 ","End":"03:11.885","Text":"we usually start with this though we could have started with the other."},{"Start":"03:11.885 ","End":"03:14.630","Text":"If the derivative of f with respect to x is M,"},{"Start":"03:14.630 ","End":"03:18.570","Text":"then f is just the integral of M with respect to x. I\u0027ll just"},{"Start":"03:18.570 ","End":"03:23.150","Text":"copy the function M over here or this bit in the brackets is M. Now,"},{"Start":"03:23.150 ","End":"03:25.370","Text":"this integral is not immediately obvious."},{"Start":"03:25.370 ","End":"03:28.070","Text":"I mean the first part y^2x is okay,"},{"Start":"03:28.070 ","End":"03:29.735","Text":"but how did I get to this?"},{"Start":"03:29.735 ","End":"03:31.550","Text":"Let me just do that at the side."},{"Start":"03:31.550 ","End":"03:34.955","Text":"The idea is to write this part,"},{"Start":"03:34.955 ","End":"03:40.430","Text":"y over x times x plus y as a partial fraction."},{"Start":"03:40.430 ","End":"03:41.840","Text":"I\u0027m not going to do the whole work,"},{"Start":"03:41.840 ","End":"03:44.405","Text":"I\u0027m just going to show you part of it."},{"Start":"03:44.405 ","End":"03:52.510","Text":"We know that it\u0027s going to be something like A/x plus B/x plus y."},{"Start":"03:52.510 ","End":"03:55.909","Text":"Then we start doing the technique of partial fractions,"},{"Start":"03:55.909 ","End":"03:58.685","Text":"cross multiplying, and substituting values."},{"Start":"03:58.685 ","End":"04:06.885","Text":"I\u0027ll just you the solution that A turns out to be 1 there and B turns out to be minus 1,"},{"Start":"04:06.885 ","End":"04:08.910","Text":"I made that a minus, minus 1."},{"Start":"04:08.910 ","End":"04:10.790","Text":"I\u0027m not going to show you"},{"Start":"04:10.790 ","End":"04:13.970","Text":"the steps because I don\u0027t want to spend time on partial fractions,"},{"Start":"04:13.970 ","End":"04:16.715","Text":"but you could check it by multiplying out."},{"Start":"04:16.715 ","End":"04:18.360","Text":"If you evaluated this,"},{"Start":"04:18.360 ","End":"04:23.540","Text":"you would put a common denominator of x times x plus y,"},{"Start":"04:23.540 ","End":"04:26.525","Text":"and then we get this times this is x plus y."},{"Start":"04:26.525 ","End":"04:28.880","Text":"This times this gives me minus x."},{"Start":"04:28.880 ","End":"04:35.360","Text":"The x\u0027s cancel and we do indeed get y over x times x plus y."},{"Start":"04:35.360 ","End":"04:37.325","Text":"This thing is with a minus."},{"Start":"04:37.325 ","End":"04:38.690","Text":"If we have a minus,"},{"Start":"04:38.690 ","End":"04:41.000","Text":"it\u0027s going to be a minus this plus this,"},{"Start":"04:41.000 ","End":"04:45.020","Text":"this one gives me the natural log of x."},{"Start":"04:45.020 ","End":"04:48.545","Text":"This one gives me natural log of x plus y."},{"Start":"04:48.545 ","End":"04:50.030","Text":"Then you might say, well,"},{"Start":"04:50.030 ","End":"04:55.080","Text":"I was expecting to get from the 2 plus 2y,"},{"Start":"04:55.080 ","End":"05:00.230","Text":"but I don\u0027t need to write this because I have to add a general function of y when I\u0027m"},{"Start":"05:00.230 ","End":"05:05.990","Text":"integrating with respect to x and the 2y get swallowed up in the general function of y."},{"Start":"05:05.990 ","End":"05:08.975","Text":"This is the results so far for f,"},{"Start":"05:08.975 ","End":"05:13.280","Text":"but we still have to find g. I just copied that results,"},{"Start":"05:13.280 ","End":"05:14.875","Text":"started a new page."},{"Start":"05:14.875 ","End":"05:17.900","Text":"Now, I\u0027m differentiating f with respect to y."},{"Start":"05:17.900 ","End":"05:23.555","Text":"The reason is that we have to satisfy the condition that f with respect to y is N,"},{"Start":"05:23.555 ","End":"05:26.960","Text":"f with respect to y from here, 2yx,"},{"Start":"05:26.960 ","End":"05:31.172","Text":"from here, I get nothing because this just contains x,"},{"Start":"05:31.172 ","End":"05:33.905","Text":"and from here I get 1 over x plus y."},{"Start":"05:33.905 ","End":"05:38.280","Text":"g just gives me g\u0027 derivative of g with respect to y."},{"Start":"05:38.280 ","End":"05:41.600","Text":"From this equation that f with respect to y is N,"},{"Start":"05:41.600 ","End":"05:47.210","Text":"I get f with respect to y I copied from here and I just copied it from what was above."},{"Start":"05:47.210 ","End":"05:48.650","Text":"I\u0027m not going to scroll backup."},{"Start":"05:48.650 ","End":"05:50.570","Text":"This was the function N,"},{"Start":"05:50.570 ","End":"05:54.155","Text":"just open the brackets here and that gives me this,"},{"Start":"05:54.155 ","End":"05:55.396","Text":"and this is what we get."},{"Start":"05:55.396 ","End":"06:01.110","Text":"The reason we get that is this cancels with this and this one cancels with this one,"},{"Start":"06:01.110 ","End":"06:03.105","Text":"and we\u0027re left with this."},{"Start":"06:03.105 ","End":"06:05.975","Text":"Now, we need to integrate with respect to y,"},{"Start":"06:05.975 ","End":"06:09.035","Text":"so g(y) is the integral of this."},{"Start":"06:09.035 ","End":"06:11.105","Text":"Integral of 2y is just y^2."},{"Start":"06:11.105 ","End":"06:12.425","Text":"We don\u0027t need the constant."},{"Start":"06:12.425 ","End":"06:16.430","Text":"Then I substitute g(y) in the expression for f. I\u0027ve lost it,"},{"Start":"06:16.430 ","End":"06:18.920","Text":"let\u0027s see if we can go back up and find it."},{"Start":"06:18.920 ","End":"06:25.915","Text":"Here is f, I took this expression and instead of g(y), we put y^2."},{"Start":"06:25.915 ","End":"06:28.610","Text":"Finally, don\u0027t forget the last step."},{"Start":"06:28.610 ","End":"06:30.995","Text":"We have to say that f equals constant."},{"Start":"06:30.995 ","End":"06:33.350","Text":"Here we are f(x, y) equals a constant."},{"Start":"06:33.350 ","End":"06:36.380","Text":"This is the solution because"},{"Start":"06:36.380 ","End":"06:39.470","Text":"different c\u0027s give me different solutions that instead of f,"},{"Start":"06:39.470 ","End":"06:41.270","Text":"I write what it is from here,"},{"Start":"06:41.270 ","End":"06:46.440","Text":"so this is the answer to the question and we are done."}],"Thumbnail":null,"ID":7670},{"Watched":false,"Name":"Exercise 3","Duration":"7m 51s","ChapterTopicVideoID":7599,"CourseChapterTopicPlaylistID":4221,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"In this exercise, we\u0027re given a differential equation to"},{"Start":"00:03.060 ","End":"00:06.480","Text":"solve and it\u0027s in the chapter on exact equations,"},{"Start":"00:06.480 ","End":"00:08.880","Text":"so that\u0027s what it\u0027s going to be probably."},{"Start":"00:08.880 ","End":"00:11.490","Text":"Let\u0027s check that this is an exact equation."},{"Start":"00:11.490 ","End":"00:15.090","Text":"Just copied it, labeled the first function M,"},{"Start":"00:15.090 ","End":"00:18.180","Text":"the one that goes with dx and the one that goes with dy I call"},{"Start":"00:18.180 ","End":"00:22.080","Text":"it N and I hope you remember that the condition for it"},{"Start":"00:22.080 ","End":"00:26.040","Text":"to be exact is that the partial derivative of M with respect to"},{"Start":"00:26.040 ","End":"00:31.395","Text":"y has got to equal the partial derivative of N with respect to x."},{"Start":"00:31.395 ","End":"00:37.670","Text":"Let\u0027s go and check that this is indeed so the derivative of M with respect to y,"},{"Start":"00:37.670 ","End":"00:39.200","Text":"so we\u0027ll use this form,"},{"Start":"00:39.200 ","End":"00:43.940","Text":"the d by dy notation on this function M and there\u0027s a sum,"},{"Start":"00:43.940 ","End":"00:46.625","Text":"so we take for each piece separately,"},{"Start":"00:46.625 ","End":"00:50.165","Text":"and this bit is not going to count towards anything."},{"Start":"00:50.165 ","End":"00:53.810","Text":"This one is going to be 0 because it\u0027s with respect to y,"},{"Start":"00:53.810 ","End":"00:55.760","Text":"so it\u0027s as if this wasn\u0027t here."},{"Start":"00:55.760 ","End":"00:58.520","Text":"Now I just have this and we\u0027re going to use the product rule."},{"Start":"00:58.520 ","End":"01:00.570","Text":"This is like u and this is like v,"},{"Start":"01:00.570 ","End":"01:04.310","Text":"so it\u0027s the derivative of the first times"},{"Start":"01:04.310 ","End":"01:08.240","Text":"the second plus the first that is derivative of the second."},{"Start":"01:08.240 ","End":"01:11.930","Text":"I\u0027m just indicating that I have to do a derivative haven\u0027t done the derivative yet."},{"Start":"01:11.930 ","End":"01:13.370","Text":"That\u0027s what I\u0027ll do now."},{"Start":"01:13.370 ","End":"01:19.520","Text":"This derivative is 2y times this and then the y^2 over here and the derivative of this,"},{"Start":"01:19.520 ","End":"01:21.455","Text":"maybe it needs a word of explanation."},{"Start":"01:21.455 ","End":"01:24.470","Text":"When I have e to the power of something and"},{"Start":"01:24.470 ","End":"01:27.620","Text":"I\u0027m differentiating it in this case with respect to y,"},{"Start":"01:27.620 ","End":"01:29.861","Text":"but it doesn\u0027t matter with respect to something,"},{"Start":"01:29.861 ","End":"01:33.575","Text":"this is like a template where we say according to the chain rule,"},{"Start":"01:33.575 ","End":"01:38.030","Text":"first of all, derivative of e to the something is just e to the something."},{"Start":"01:38.030 ","End":"01:40.205","Text":"But because it\u0027s a something,"},{"Start":"01:40.205 ","End":"01:43.130","Text":"then we need also the internal derivative,"},{"Start":"01:43.130 ","End":"01:46.850","Text":"which is box prime and then this case,"},{"Start":"01:46.850 ","End":"01:48.725","Text":"this is like the box,"},{"Start":"01:48.725 ","End":"01:53.720","Text":"so what we get a box in this case would be this bit here,"},{"Start":"01:53.720 ","End":"02:00.785","Text":"so e to the power of box is this part here and the box prime,"},{"Start":"02:00.785 ","End":"02:04.130","Text":"which would be the derivative of this with respect to y,"},{"Start":"02:04.130 ","End":"02:06.035","Text":"is going to be just 2xy rather,"},{"Start":"02:06.035 ","End":"02:09.380","Text":"x is a constant that bring the 2 in front and that\u0027s this 2xy,"},{"Start":"02:09.380 ","End":"02:11.045","Text":"that\u0027s this part here."},{"Start":"02:11.045 ","End":"02:14.735","Text":"Now I just have to simplify this by taking out a common factor."},{"Start":"02:14.735 ","End":"02:17.440","Text":"I can take 2ye to this thing."},{"Start":"02:17.440 ","End":"02:19.130","Text":"Here I have the 2y,"},{"Start":"02:19.130 ","End":"02:22.495","Text":"here I have 2 and i also have a y here,"},{"Start":"02:22.495 ","End":"02:27.140","Text":"so what we\u0027re left here is just 1 and here the missing bits after the 2y."},{"Start":"02:27.140 ","End":"02:31.190","Text":"We have the y^2 and we have the x, so that\u0027s xy^2."},{"Start":"02:31.190 ","End":"02:37.550","Text":"This bit is the M with respect to y and now we have to do the other bit,"},{"Start":"02:37.550 ","End":"02:42.250","Text":"N with respect to x. N with respect to x. N is what"},{"Start":"02:42.250 ","End":"02:47.225","Text":"is written here and I want the derivative with respect to x using this notation,"},{"Start":"02:47.225 ","End":"02:50.780","Text":"the 3y minus 3y^2 just drops out."},{"Start":"02:50.780 ","End":"02:53.315","Text":"It gives me nothing because it doesn\u0027t contain x,"},{"Start":"02:53.315 ","End":"02:55.490","Text":"so we\u0027re going to do this with the product rule."},{"Start":"02:55.490 ","End":"02:57.230","Text":"This is going to be one part of the product."},{"Start":"02:57.230 ","End":"02:58.820","Text":"This is the other part of the product,"},{"Start":"02:58.820 ","End":"03:02.915","Text":"so actually I couldn\u0027t take the 2y upfront."},{"Start":"03:02.915 ","End":"03:07.250","Text":"Here, the 2 with the y is what goes upfront and then x is"},{"Start":"03:07.250 ","End":"03:11.710","Text":"the other part of x times e to this thing as a product,"},{"Start":"03:11.710 ","End":"03:13.520","Text":"so first differentiate the x,"},{"Start":"03:13.520 ","End":"03:14.870","Text":"the other one untouched,"},{"Start":"03:14.870 ","End":"03:19.430","Text":"x as is the derivative of this with respect to x."},{"Start":"03:19.430 ","End":"03:22.550","Text":"Derivative of x with respect to x is just 1."},{"Start":"03:22.550 ","End":"03:23.600","Text":"I didn\u0027t write anything,"},{"Start":"03:23.600 ","End":"03:26.960","Text":"just this thing as is, x is here."},{"Start":"03:26.960 ","End":"03:29.570","Text":"Here we\u0027re going to use this template once again."},{"Start":"03:29.570 ","End":"03:31.865","Text":"We\u0027ve got e to the power of something."},{"Start":"03:31.865 ","End":"03:36.260","Text":"Here\u0027s that e to the power of something and then this was the box."},{"Start":"03:36.260 ","End":"03:39.620","Text":"Derivative of box with respect to x is the y^2."},{"Start":"03:39.620 ","End":"03:42.200","Text":"Then we just have to simplify a bit."},{"Start":"03:42.200 ","End":"03:48.665","Text":"I can take an extra e^xy^2 outside the brackets from here and from here,"},{"Start":"03:48.665 ","End":"03:54.240","Text":"and then all we\u0027re left with is 1 plus xy^2 and I\u0027ll highlight this."},{"Start":"03:54.240 ","End":"03:59.790","Text":"Look, these two are exactly the same and so this thing is equal to M_y,"},{"Start":"03:59.790 ","End":"04:03.770","Text":"so we do have the condition for exact was that the M_y"},{"Start":"04:03.770 ","End":"04:08.675","Text":"equals N_x and we do have that and so it is an exact equation."},{"Start":"04:08.675 ","End":"04:10.550","Text":"Now that\u0027s just verifying that it\u0027s exact."},{"Start":"04:10.550 ","End":"04:12.590","Text":"Now we\u0027re going to have to actually solve it."},{"Start":"04:12.590 ","End":"04:15.500","Text":"The next step, which is solving it involves solving"},{"Start":"04:15.500 ","End":"04:18.725","Text":"these two equations for some function f I should have mentioned."},{"Start":"04:18.725 ","End":"04:21.290","Text":"We\u0027re looking for a solution in the form f(x,"},{"Start":"04:21.290 ","End":"04:26.570","Text":"y) equals a constant and these are the two equations it has to satisfy,"},{"Start":"04:26.570 ","End":"04:28.235","Text":"so we start with one of them,"},{"Start":"04:28.235 ","End":"04:30.590","Text":"whichever one is easier to integrate."},{"Start":"04:30.590 ","End":"04:33.860","Text":"Let\u0027s see if I can see M and N somewhere."},{"Start":"04:33.860 ","End":"04:35.390","Text":"Taught to see everything."},{"Start":"04:35.390 ","End":"04:40.295","Text":"But I don\u0027t see any major difference in integrating this with respect to x,"},{"Start":"04:40.295 ","End":"04:41.600","Text":"so this with respect to y,"},{"Start":"04:41.600 ","End":"04:44.274","Text":"so I will go with this with respect to x."},{"Start":"04:44.274 ","End":"04:48.280","Text":"We\u0027re going to start by solving this."},{"Start":"04:48.280 ","End":"04:50.240","Text":"To solve this for another derivative,"},{"Start":"04:50.240 ","End":"04:52.025","Text":"and I want to get back to the original,"},{"Start":"04:52.025 ","End":"04:54.620","Text":"I do an antiderivative or the integral,"},{"Start":"04:54.620 ","End":"05:01.025","Text":"so I need the integral of M and there\u0027s copied M from above as this."},{"Start":"05:01.025 ","End":"05:03.785","Text":"We need the integral of this with respect to x."},{"Start":"05:03.785 ","End":"05:08.855","Text":"Now, I need maybe a word of explanation as to how I got from this part to this part."},{"Start":"05:08.855 ","End":"05:11.130","Text":"Remembering when we\u0027re integrating with respect to x,"},{"Start":"05:11.130 ","End":"05:12.380","Text":"y is a constant,"},{"Start":"05:12.380 ","End":"05:14.060","Text":"so if I just looked at this bit,"},{"Start":"05:14.060 ","End":"05:18.845","Text":"it\u0027s of the form e^a times x,"},{"Start":"05:18.845 ","End":"05:22.940","Text":"where y^2 is like my a to emphasize is a constant and the"},{"Start":"05:22.940 ","End":"05:28.540","Text":"integral of this would be 1/a e^ax constant,"},{"Start":"05:28.540 ","End":"05:30.015","Text":"which we don\u0027t need yet."},{"Start":"05:30.015 ","End":"05:32.190","Text":"With a being y^2,"},{"Start":"05:32.190 ","End":"05:38.285","Text":"we get 1/y^2 e^xy^2 and this y^2 just sticks because it\u0027s a constant,"},{"Start":"05:38.285 ","End":"05:40.760","Text":"the multiplicative constant, that\u0027s the first bit."},{"Start":"05:40.760 ","End":"05:42.065","Text":"Then the 4x^3,"},{"Start":"05:42.065 ","End":"05:45.950","Text":"that is just x^3 gives me x^4 over 4."},{"Start":"05:45.950 ","End":"05:48.890","Text":"Of course, the 4 is going to cancel with the 4 and the y^2"},{"Start":"05:48.890 ","End":"05:52.070","Text":"with the y^2 and so after we do that so this with this,"},{"Start":"05:52.070 ","End":"05:54.945","Text":"and this with this, we get something much simpler,"},{"Start":"05:54.945 ","End":"05:58.490","Text":"e^xy^2 plus x^4 plus g(y)."},{"Start":"05:58.490 ","End":"06:00.485","Text":"I should have mentioned, well, you know this already."},{"Start":"06:00.485 ","End":"06:03.410","Text":"When we integrate with respect to x we don\u0027t just have a constant,"},{"Start":"06:03.410 ","End":"06:09.425","Text":"we have a general function of y because this is like a constant as far as x goes."},{"Start":"06:09.425 ","End":"06:11.120","Text":"That\u0027s this part."},{"Start":"06:11.120 ","End":"06:13.205","Text":"Now we still have an unknown function g,"},{"Start":"06:13.205 ","End":"06:15.320","Text":"and that\u0027s where we use the second equation."},{"Start":"06:15.320 ","End":"06:18.680","Text":"Now we differentiate this, this is F,"},{"Start":"06:18.680 ","End":"06:23.420","Text":"and if I do F with respect to y and equate that to N and on the one hand,"},{"Start":"06:23.420 ","End":"06:29.015","Text":"F with respect to y is just what we get from differentiating this,"},{"Start":"06:29.015 ","End":"06:31.055","Text":"this one gives me this,"},{"Start":"06:31.055 ","End":"06:36.245","Text":"again, using that template with e to the power of a derivative of xy^2 is 2xy,"},{"Start":"06:36.245 ","End":"06:39.620","Text":"x^4 gives nothing g\u0027(y)."},{"Start":"06:39.620 ","End":"06:41.000","Text":"That\u0027s on the one hand."},{"Start":"06:41.000 ","End":"06:42.095","Text":"On the other hand,"},{"Start":"06:42.095 ","End":"06:46.600","Text":"this is just got to equal N and so this,"},{"Start":"06:46.600 ","End":"06:51.315","Text":"which is F with respect to y and N is just copied from above,"},{"Start":"06:51.315 ","End":"06:54.485","Text":"so this is the equation that we have now."},{"Start":"06:54.485 ","End":"06:57.800","Text":"Let\u0027s see. This cancels with this,"},{"Start":"06:57.800 ","End":"07:01.880","Text":"so we\u0027re left with just the derivative of g with respect to y as this"},{"Start":"07:01.880 ","End":"07:06.310","Text":"and if g\u0027(y) is equal to this, this is written here,"},{"Start":"07:06.310 ","End":"07:11.555","Text":"then g is just the integral of this function of y with respect to y,"},{"Start":"07:11.555 ","End":"07:13.340","Text":"which is y^2,"},{"Start":"07:13.340 ","End":"07:15.080","Text":"becomes y^3 over 3,"},{"Start":"07:15.080 ","End":"07:17.525","Text":"so it\u0027s minus y^3."},{"Start":"07:17.525 ","End":"07:21.140","Text":"It didn\u0027t really have to put the c here because we have a c at the end,"},{"Start":"07:21.140 ","End":"07:24.965","Text":"but never mind, I just erased it because we have a constant at the end."},{"Start":"07:24.965 ","End":"07:27.590","Text":"Now that we found g(y),"},{"Start":"07:27.590 ","End":"07:30.200","Text":"we just substitute that here."},{"Start":"07:30.200 ","End":"07:34.490","Text":"I\u0027ve got my F here is just the same as this,"},{"Start":"07:34.490 ","End":"07:35.960","Text":"but instead of g(y),"},{"Start":"07:35.960 ","End":"07:38.000","Text":"I put the minus y^3."},{"Start":"07:38.000 ","End":"07:41.980","Text":"Then finally, we just say that this function equals the constant,"},{"Start":"07:41.980 ","End":"07:43.250","Text":"F equals a constant,"},{"Start":"07:43.250 ","End":"07:47.000","Text":"which in our case here\u0027s our F equals a constant and that\u0027s"},{"Start":"07:47.000 ","End":"07:52.440","Text":"the general solution where c is any constant and we are done."}],"Thumbnail":null,"ID":7671},{"Watched":false,"Name":"Exercise 4","Duration":"5m 18s","ChapterTopicVideoID":7600,"CourseChapterTopicPlaylistID":4221,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.505","Text":"Here we\u0027re given this differential equation to"},{"Start":"00:02.505 ","End":"00:06.525","Text":"solve and because we\u0027re in the chapter on exact equations,"},{"Start":"00:06.525 ","End":"00:08.460","Text":"that\u0027s what it\u0027s going to probably be."},{"Start":"00:08.460 ","End":"00:11.970","Text":"Let\u0027s check for that and you probably remember how to do that."},{"Start":"00:11.970 ","End":"00:16.230","Text":"We label the first function M and the other one N,"},{"Start":"00:16.230 ","End":"00:17.520","Text":"that\u0027s the one with dx,"},{"Start":"00:17.520 ","End":"00:18.630","Text":"the one with dy."},{"Start":"00:18.630 ","End":"00:21.330","Text":"We have to show that the partial derivative of this with respect to"},{"Start":"00:21.330 ","End":"00:24.840","Text":"y equals the partial derivative of this with respect to x."},{"Start":"00:24.840 ","End":"00:28.380","Text":"Start with this one with respect to y. I\u0027ll use the d"},{"Start":"00:28.380 ","End":"00:32.210","Text":"by dy notation and what we get is we have a sum,"},{"Start":"00:32.210 ","End":"00:36.669","Text":"so we differentiate each bit separately."},{"Start":"00:36.669 ","End":"00:39.115","Text":"What we get is this."},{"Start":"00:39.115 ","End":"00:45.410","Text":"See what I did was this is a constant and this is a constant as far as y goes."},{"Start":"00:45.410 ","End":"00:48.350","Text":"These constants just stick and we just have"},{"Start":"00:48.350 ","End":"00:51.545","Text":"to take the partial derivative of y in the first case,"},{"Start":"00:51.545 ","End":"00:54.125","Text":"and of e^y in the second case,"},{"Start":"00:54.125 ","End":"00:57.710","Text":"and the cosine x and the 2x just stick to them."},{"Start":"00:57.710 ","End":"00:59.570","Text":"This derivative is 1,"},{"Start":"00:59.570 ","End":"01:03.425","Text":"so we\u0027ve got cosine x and this derivative of e^y."},{"Start":"01:03.425 ","End":"01:05.270","Text":"We have 2xe^y."},{"Start":"01:05.270 ","End":"01:06.560","Text":"That\u0027s 1 side."},{"Start":"01:06.560 ","End":"01:09.320","Text":"Now we need N with respect to x. I\u0027ll just"},{"Start":"01:09.320 ","End":"01:12.895","Text":"highlight this first and then we\u0027ll see what we get for the other."},{"Start":"01:12.895 ","End":"01:17.660","Text":"N with respect to x is just d by dx of this,"},{"Start":"01:17.660 ","End":"01:21.050","Text":"and I just take the derivative of the first term,"},{"Start":"01:21.050 ","End":"01:22.280","Text":"it\u0027s the sine x,"},{"Start":"01:22.280 ","End":"01:23.660","Text":"and then in the second term,"},{"Start":"01:23.660 ","End":"01:25.370","Text":"e^y is a constant,"},{"Start":"01:25.370 ","End":"01:28.370","Text":"so I just have to differentiate the x squared and the left term is a"},{"Start":"01:28.370 ","End":"01:31.645","Text":"real constant and so just disappears."},{"Start":"01:31.645 ","End":"01:34.970","Text":"What we get is just derivative of sine x is cosine x,"},{"Start":"01:34.970 ","End":"01:37.940","Text":"derivative of x squared is 2x and this is"},{"Start":"01:37.940 ","End":"01:42.305","Text":"what we get for the derivative of N with respect to x."},{"Start":"01:42.305 ","End":"01:48.080","Text":"Therefore this is equal to M with respect to y. N with respect to these 2 are equal."},{"Start":"01:48.080 ","End":"01:53.560","Text":"We have an exact equation and now we have to go about solving it."},{"Start":"01:53.560 ","End":"01:57.530","Text":"The way we solve an exact equation should\u0027ve mentioned that we assume that"},{"Start":"01:57.530 ","End":"02:02.135","Text":"the solution in implicit form is some function of x and y"},{"Start":"02:02.135 ","End":"02:04.610","Text":"equals C. The solution will always turn out to be of"},{"Start":"02:04.610 ","End":"02:07.220","Text":"this form and the conditions on function"},{"Start":"02:07.220 ","End":"02:13.430","Text":"F is that it\u0027s 2 partial derivatives with respect to x and y or M and N respectively."},{"Start":"02:13.430 ","End":"02:15.455","Text":"We start with one of them."},{"Start":"02:15.455 ","End":"02:17.090","Text":"I\u0027ll just take the first usually,"},{"Start":"02:17.090 ","End":"02:20.975","Text":"but if it happens to be difficult and you could always start with the second,"},{"Start":"02:20.975 ","End":"02:22.830","Text":"so I start with this one."},{"Start":"02:22.830 ","End":"02:25.760","Text":"If I know the partial derivative with respect to x,"},{"Start":"02:25.760 ","End":"02:27.350","Text":"then to get back to F,"},{"Start":"02:27.350 ","End":"02:30.230","Text":"I just take the integral with respect to x,"},{"Start":"02:30.230 ","End":"02:36.375","Text":"and that\u0027s the integral of M I just copied from above is this function here,"},{"Start":"02:36.375 ","End":"02:39.140","Text":"but yeah, we have to put the dx here and that for one thing"},{"Start":"02:39.140 ","End":"02:42.200","Text":"means that y is a constant as far as that goes."},{"Start":"02:42.200 ","End":"02:46.220","Text":"The integral of this y just sticks."},{"Start":"02:46.220 ","End":"02:53.160","Text":"The integral of cosine is sine and then here also the e^y bit is constant,"},{"Start":"02:53.160 ","End":"02:55.975","Text":"so we just need the integral of 2x,"},{"Start":"02:55.975 ","End":"02:58.940","Text":"which is integral of x is x squared over 2."},{"Start":"02:58.940 ","End":"03:00.800","Text":"We could have said straight away x squared anyway."},{"Start":"03:00.800 ","End":"03:02.900","Text":"This is going to cancel with this and of course,"},{"Start":"03:02.900 ","End":"03:06.840","Text":"we have this plus a constant only in the case of integration with respect to x."},{"Start":"03:06.840 ","End":"03:09.245","Text":"A constant means any function of y,"},{"Start":"03:09.245 ","End":"03:11.455","Text":"constant as far as x is go."},{"Start":"03:11.455 ","End":"03:14.720","Text":"That\u0027s simplification just means I can cross out those 2s."},{"Start":"03:14.720 ","End":"03:18.095","Text":"Now our task is to find g(y)."},{"Start":"03:18.095 ","End":"03:20.254","Text":"We have that this is F,"},{"Start":"03:20.254 ","End":"03:22.085","Text":"but we don\u0027t know what g(y) is."},{"Start":"03:22.085 ","End":"03:25.290","Text":"What we do is we use the other parts of the equation."},{"Start":"03:25.290 ","End":"03:30.455","Text":"This time we use this one to help us to get g. Remember I\u0027m looking at this."},{"Start":"03:30.455 ","End":"03:33.010","Text":"I need F with respect to y."},{"Start":"03:33.010 ","End":"03:38.930","Text":"This is F. The derivative with respect to y is d by dy of this thing here."},{"Start":"03:38.930 ","End":"03:44.720","Text":"Now what we get is x is a constant this time like sine x is a constant also."},{"Start":"03:44.720 ","End":"03:49.685","Text":"This is a constant and x squared would be a constant as far as y goes."},{"Start":"03:49.685 ","End":"03:52.545","Text":"Constant times y is just the constant."},{"Start":"03:52.545 ","End":"03:57.510","Text":"Then we have constant times e^y is just that constant times e^y."},{"Start":"03:57.510 ","End":"03:58.820","Text":"If this was like 3^y,"},{"Start":"03:58.820 ","End":"04:01.935","Text":"the derivative is also 3^y."},{"Start":"04:01.935 ","End":"04:05.330","Text":"For g, we just get g\u0027(y), its derivative."},{"Start":"04:05.330 ","End":"04:07.175","Text":"The next thing we\u0027re going to do is compare,"},{"Start":"04:07.175 ","End":"04:11.190","Text":"this is F with respect to y and we\u0027re going to compare"},{"Start":"04:11.190 ","End":"04:15.425","Text":"that to N and I just went and copied it from the top."},{"Start":"04:15.425 ","End":"04:21.230","Text":"This in fact is N. What is written here is N just trust me or scroll back or whatever."},{"Start":"04:21.230 ","End":"04:22.970","Text":"I hope there\u0027s a typo,"},{"Start":"04:22.970 ","End":"04:24.800","Text":"there shouldn\u0027t be a minus here,"},{"Start":"04:24.800 ","End":"04:26.839","Text":"sorry, and that stuff cancels."},{"Start":"04:26.839 ","End":"04:31.940","Text":"This will cancel with this from both sides and this with this and what we\u0027re going to"},{"Start":"04:31.940 ","End":"04:37.445","Text":"end up with is g\u0027(y) equals minus 1 and from here,"},{"Start":"04:37.445 ","End":"04:41.900","Text":"we d o an anti-derivative or an integral of minus 1,"},{"Start":"04:41.900 ","End":"04:43.940","Text":"which is just minus y."},{"Start":"04:43.940 ","End":"04:47.020","Text":"We don\u0027t need the constants here because we have a constant at the end."},{"Start":"04:47.020 ","End":"04:51.425","Text":"What I do with this minus y is that this is g(y)."},{"Start":"04:51.425 ","End":"04:53.120","Text":"I can put that in here."},{"Start":"04:53.120 ","End":"04:58.460","Text":"I\u0027ve got now F is equal to this and just replace g(y) by minus y,"},{"Start":"04:58.460 ","End":"05:01.040","Text":"so here is our function F,"},{"Start":"05:01.040 ","End":"05:05.120","Text":"and now all we have to do is compare this to see because in general,"},{"Start":"05:05.120 ","End":"05:07.880","Text":"the solution is F equals c, but in our case,"},{"Start":"05:07.880 ","End":"05:11.690","Text":"F is the solution or solutions if you like,"},{"Start":"05:11.690 ","End":"05:13.490","Text":"because c can be many things,"},{"Start":"05:13.490 ","End":"05:19.140","Text":"is this which from here is equal to c. That\u0027s the answer."}],"Thumbnail":null,"ID":7672},{"Watched":false,"Name":"Exercise 5","Duration":"6m 36s","ChapterTopicVideoID":7601,"CourseChapterTopicPlaylistID":4221,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.580","Text":"Here we have this differential equation to solve and it\u0027s an exact equation."},{"Start":"00:04.580 ","End":"00:06.210","Text":"That\u0027s the first thing I\u0027m going to show you."},{"Start":"00:06.210 ","End":"00:13.005","Text":"I start out by labeling the first portion M and the other bit N, including the minus."},{"Start":"00:13.005 ","End":"00:16.455","Text":"Where we show an equation like this is an exact one,"},{"Start":"00:16.455 ","End":"00:20.205","Text":"is by showing that the derivative partial of"},{"Start":"00:20.205 ","End":"00:25.335","Text":"M with respect to y is equal to the partial derivative of N with respect to x."},{"Start":"00:25.335 ","End":"00:27.120","Text":"Notice it\u0027s backwards,"},{"Start":"00:27.120 ","End":"00:30.000","Text":"not the way you might think M goes with the dx,"},{"Start":"00:30.000 ","End":"00:31.845","Text":"but it\u0027s M with respect to y."},{"Start":"00:31.845 ","End":"00:34.160","Text":"Sometimes people get it the wrong way around."},{"Start":"00:34.160 ","End":"00:36.600","Text":"Let\u0027s compute each one of them."},{"Start":"00:36.600 ","End":"00:38.350","Text":"Start with M,"},{"Start":"00:38.350 ","End":"00:40.880","Text":"its derivative with respect to y."},{"Start":"00:40.880 ","End":"00:47.190","Text":"The one gives me nothing and sine 2x is a constant as far as y goes,"},{"Start":"00:47.190 ","End":"00:52.650","Text":"so I just have to differentiate the y^2 and sine 2x sticks."},{"Start":"00:52.650 ","End":"00:54.060","Text":"This is 2y,"},{"Start":"00:54.060 ","End":"00:56.400","Text":"so it\u0027s 2y sine 2x."},{"Start":"00:56.400 ","End":"01:01.010","Text":"Now the other one, and with respect to x this time y is the constant."},{"Start":"01:01.010 ","End":"01:06.205","Text":"The minus 2y is the constant and I just need to differentiate the cosine squared x."},{"Start":"01:06.205 ","End":"01:09.930","Text":"Again, minus 2y just carries something squared,"},{"Start":"01:09.930 ","End":"01:12.380","Text":"so it\u0027s twice that something times"},{"Start":"01:12.380 ","End":"01:15.905","Text":"the inner derivative from the chain rule, x there, sorry."},{"Start":"01:15.905 ","End":"01:17.510","Text":"The derivative of cosine x,"},{"Start":"01:17.510 ","End":"01:21.070","Text":"with expect to x is just minus sine x,"},{"Start":"01:21.070 ","End":"01:23.535","Text":"so we get minus 2y,"},{"Start":"01:23.535 ","End":"01:28.880","Text":"2 cosine x minus sine x and the minus and the minus cancels."},{"Start":"01:28.880 ","End":"01:32.870","Text":"But what I have doesn\u0027t quite look like what I want it to be,"},{"Start":"01:32.870 ","End":"01:36.770","Text":"which is this, so let\u0027s use a bit of trigonometry here."},{"Start":"01:36.770 ","End":"01:42.020","Text":"There\u0027s a famous trigonometric identity, well,"},{"Start":"01:42.020 ","End":"01:44.660","Text":"it\u0027s usually given in terms of Alpha, but it could be x,"},{"Start":"01:44.660 ","End":"01:50.390","Text":"2 cosine Alpha sine Alpha is equal to sine of 2 Alpha."},{"Start":"01:50.390 ","End":"01:52.040","Text":"If I use that here,"},{"Start":"01:52.040 ","End":"02:00.000","Text":"it\u0027s going to come out as sine of 2x and so this is 2y sine 2x."},{"Start":"02:00.000 ","End":"02:04.685","Text":"We do indeed have that this is equal to this."},{"Start":"02:04.685 ","End":"02:08.010","Text":"N with respect to x equals M with respect to y."},{"Start":"02:08.010 ","End":"02:11.780","Text":"That means that we\u0027re really dealing with an exact equation."},{"Start":"02:11.780 ","End":"02:13.790","Text":"Now that we\u0027ve identified it,"},{"Start":"02:13.790 ","End":"02:15.550","Text":"we want to try and solve it."},{"Start":"02:15.550 ","End":"02:18.965","Text":"As usual, these are the equations."},{"Start":"02:18.965 ","End":"02:22.190","Text":"But I should have mentioned again that this is in"},{"Start":"02:22.190 ","End":"02:26.750","Text":"the context that the solution we\u0027re looking for is going to be of the form F (x,"},{"Start":"02:26.750 ","End":"02:30.860","Text":"y) equals general constant C and F has to"},{"Start":"02:30.860 ","End":"02:35.615","Text":"satisfy these 2 equations involving partial derivatives."},{"Start":"02:35.615 ","End":"02:37.325","Text":"Let\u0027s start with one of them."},{"Start":"02:37.325 ","End":"02:40.095","Text":"Usually I\u0027ll start with the first one and"},{"Start":"02:40.095 ","End":"02:43.400","Text":"if it turns out to be difficult then I\u0027ll try the other one."},{"Start":"02:43.400 ","End":"02:44.840","Text":"Let\u0027s start with this one."},{"Start":"02:44.840 ","End":"02:48.485","Text":"If the derivative of F with respect to x is M,"},{"Start":"02:48.485 ","End":"02:52.400","Text":"then it means that F is the integral of M with respect to x."},{"Start":"02:52.400 ","End":"02:53.750","Text":"This is what we get."},{"Start":"02:53.750 ","End":"02:58.820","Text":"But you have to remember that y is a constant so far as x is concerned."},{"Start":"02:58.820 ","End":"03:01.445","Text":"Remember we\u0027re integrating with respect to x,"},{"Start":"03:01.445 ","End":"03:04.040","Text":"so 1 gives me x."},{"Start":"03:04.040 ","End":"03:06.200","Text":"Now y^2 is a constant,"},{"Start":"03:06.200 ","End":"03:10.010","Text":"so y^2 just sticks and the integral of sine 2x."},{"Start":"03:10.010 ","End":"03:13.280","Text":"Integral of sine in general is minus cosine because"},{"Start":"03:13.280 ","End":"03:16.655","Text":"it\u0027s 2x and not x you have divide by 2."},{"Start":"03:16.655 ","End":"03:22.085","Text":"Then, whereas we usually say plus C with integration when it\u0027s partial integration,"},{"Start":"03:22.085 ","End":"03:24.710","Text":"I put a general function of y because"},{"Start":"03:24.710 ","End":"03:28.355","Text":"all that will be a constant as far as x is concerned."},{"Start":"03:28.355 ","End":"03:33.020","Text":"Then just slightly to simplify the middle term and we get this."},{"Start":"03:33.020 ","End":"03:34.730","Text":"That\u0027s F with respect to x,"},{"Start":"03:34.730 ","End":"03:36.960","Text":"but we still don\u0027t know what g is,"},{"Start":"03:36.960 ","End":"03:38.945","Text":"so how do we find g(y)?"},{"Start":"03:38.945 ","End":"03:42.155","Text":"Well, for this we use the other equation that we have."},{"Start":"03:42.155 ","End":"03:45.860","Text":"This is now my F. Differentiate this with"},{"Start":"03:45.860 ","End":"03:49.955","Text":"respect to y and then compare it to the N that we have above."},{"Start":"03:49.955 ","End":"03:54.605","Text":"First from differentiating this with respect to y,"},{"Start":"03:54.605 ","End":"03:55.940","Text":"where x is a constant,"},{"Start":"03:55.940 ","End":"03:59.195","Text":"now we get x gives me nothing."},{"Start":"03:59.195 ","End":"04:02.225","Text":"The y^2 gives me 2y,"},{"Start":"04:02.225 ","End":"04:09.715","Text":"but the cosine 2x over 2 just stays and g (y) its derivative is g\u0027 (y)."},{"Start":"04:09.715 ","End":"04:12.000","Text":"Just canceling the 2s here,"},{"Start":"04:12.000 ","End":"04:13.665","Text":"you write it in this form."},{"Start":"04:13.665 ","End":"04:14.960","Text":"That\u0027s on the one hand."},{"Start":"04:14.960 ","End":"04:16.175","Text":"On the other hand,"},{"Start":"04:16.175 ","End":"04:20.450","Text":"we know that F with respect to y equals N. Well, that\u0027s what we\u0027re looking for."},{"Start":"04:20.450 ","End":"04:24.410","Text":"I take this and equate it to the function N,"},{"Start":"04:24.410 ","End":"04:26.930","Text":"which has somehow disappeared off screen,"},{"Start":"04:26.930 ","End":"04:28.970","Text":"but I copied it from above."},{"Start":"04:28.970 ","End":"04:30.625","Text":"It\u0027s equal to this."},{"Start":"04:30.625 ","End":"04:32.775","Text":"We end up with this,"},{"Start":"04:32.775 ","End":"04:34.190","Text":"but when I compare them,"},{"Start":"04:34.190 ","End":"04:37.190","Text":"and I\u0027m going to need a trigonometrical identity to help me"},{"Start":"04:37.190 ","End":"04:40.340","Text":"here because it doesn\u0027t quite look like I can combine these,"},{"Start":"04:40.340 ","End":"04:41.360","Text":"to do anything useful."},{"Start":"04:41.360 ","End":"04:45.185","Text":"I\u0027m going to need the trigonometrical identity for cosine of twice the angle."},{"Start":"04:45.185 ","End":"04:50.595","Text":"It\u0027s usually written in terms of an angle Alpha or Theta cosine 2 Alpha."},{"Start":"04:50.595 ","End":"04:52.325","Text":"There\u0027s actually 3 formulas."},{"Start":"04:52.325 ","End":"04:56.740","Text":"One of them says cosine squared Alpha minus sine squared Alpha."},{"Start":"04:56.740 ","End":"05:01.300","Text":"One of them says 2 cosine squared Alpha minus 1."},{"Start":"05:01.300 ","End":"05:05.130","Text":"The third possibility is 1 minus 2 sine squared Alpha."},{"Start":"05:05.130 ","End":"05:08.210","Text":"Anyway, we get g\u0027(y) equals,"},{"Start":"05:08.210 ","End":"05:11.180","Text":"and I\u0027ll just bring this to the other side."},{"Start":"05:11.180 ","End":"05:15.440","Text":"I\u0027ve got minus 2y cosine squared x plus y cosine 2x."},{"Start":"05:15.440 ","End":"05:19.085","Text":"At this point, I\u0027m going need to replace cosine 2x."},{"Start":"05:19.085 ","End":"05:20.930","Text":"Well, x will be in place of Alpha."},{"Start":"05:20.930 ","End":"05:25.595","Text":"I\u0027m going to replace it by use the middle formula, this one."},{"Start":"05:25.595 ","End":"05:27.609","Text":"We take the y out front,"},{"Start":"05:27.609 ","End":"05:30.660","Text":"and then minus 2 cosine squared x is here."},{"Start":"05:30.660 ","End":"05:32.400","Text":"Instead of the cosine 2x,"},{"Start":"05:32.400 ","End":"05:35.220","Text":"I put this 2 cosine squared x minus 1."},{"Start":"05:35.220 ","End":"05:37.260","Text":"I know x instead of Alpha."},{"Start":"05:37.260 ","End":"05:44.390","Text":"Luckily minus 2 cosine squared x and 2 cosine squared x cancel y times minus 1,"},{"Start":"05:44.390 ","End":"05:46.820","Text":"we\u0027re down to just minus y,"},{"Start":"05:46.820 ","End":"05:48.830","Text":"and that\u0027s what g\u0027(y) is."},{"Start":"05:48.830 ","End":"05:52.430","Text":"We have that the derivative of g is minus y,"},{"Start":"05:52.430 ","End":"05:56.495","Text":"so g is just the anti-derivative or the integral of minus y,"},{"Start":"05:56.495 ","End":"05:58.685","Text":"which is minus y^2 over 2."},{"Start":"05:58.685 ","End":"06:04.715","Text":"What I do with this is I take this g(y) and I put it in here,"},{"Start":"06:04.715 ","End":"06:07.010","Text":"and that will give me what F is."},{"Start":"06:07.010 ","End":"06:08.960","Text":"F turns out to be what is here,"},{"Start":"06:08.960 ","End":"06:11.420","Text":"x minus y^2 cosine 2x over 2,"},{"Start":"06:11.420 ","End":"06:12.875","Text":"and the g(y) from here,"},{"Start":"06:12.875 ","End":"06:15.300","Text":"which is minus y^2 over 2."},{"Start":"06:15.300 ","End":"06:16.985","Text":"Now we\u0027re almost at the end,"},{"Start":"06:16.985 ","End":"06:18.815","Text":"because once we have what F is,"},{"Start":"06:18.815 ","End":"06:21.620","Text":"the general solution is F equals a constant."},{"Start":"06:21.620 ","End":"06:24.350","Text":"F equals a constant means that"},{"Start":"06:24.350 ","End":"06:29.479","Text":"this whole expression on the right is just equal to C and that\u0027s the general solution."},{"Start":"06:29.479 ","End":"06:31.655","Text":"Or if you\u0027d like plural solutions,"},{"Start":"06:31.655 ","End":"06:33.815","Text":"because there\u0027s many C\u0027s possible."},{"Start":"06:33.815 ","End":"06:36.150","Text":"We\u0027re done."}],"Thumbnail":null,"ID":7673},{"Watched":false,"Name":"Exercise 6","Duration":"5m 15s","ChapterTopicVideoID":7598,"CourseChapterTopicPlaylistID":4221,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"We have here a differential equation to solve and we\u0027re going"},{"Start":"00:03.150 ","End":"00:06.345","Text":"to see that it\u0027s an exact equation and we\u0027ll solve it as such."},{"Start":"00:06.345 ","End":"00:08.985","Text":"But there\u0027s a difference than what you\u0027re used to."},{"Start":"00:08.985 ","End":"00:12.660","Text":"Usually the variables dx and dy,"},{"Start":"00:12.660 ","End":"00:17.700","Text":"meaning variables are x and y and here we have a function of x and y and here also,"},{"Start":"00:17.700 ","End":"00:22.200","Text":"but this time it\u0027s t and x whenever think otherwise works the same."},{"Start":"00:22.200 ","End":"00:24.120","Text":"This function that goes with the t,"},{"Start":"00:24.120 ","End":"00:25.740","Text":"we will call it M,"},{"Start":"00:25.740 ","End":"00:27.915","Text":"M(x and t) or t and x."},{"Start":"00:27.915 ","End":"00:33.300","Text":"Here we have N(t and x) to show that it\u0027s exact just like before."},{"Start":"00:33.300 ","End":"00:38.230","Text":"But we\u0027ve got to remember that this time we need the partial derivative of M_x,"},{"Start":"00:38.750 ","End":"00:43.475","Text":"has got to equal the partial derivative of N_t."},{"Start":"00:43.475 ","End":"00:48.095","Text":"It\u0027s the other way round if N is next to the dx,"},{"Start":"00:48.095 ","End":"00:51.575","Text":"then it\u0027s the derivative with respect to t and vice versa."},{"Start":"00:51.575 ","End":"00:53.920","Text":"Let\u0027s start with the M_x."},{"Start":"00:53.920 ","End":"00:58.280","Text":"That\u0027s this one, remember that in this case t is a constant,"},{"Start":"00:58.280 ","End":"01:05.126","Text":"so it\u0027s 2t for this term 2t and the derivative of x^2 with respect to x."},{"Start":"01:05.126 ","End":"01:06.410","Text":"For the other one,"},{"Start":"01:06.410 ","End":"01:09.950","Text":"the minus 2 is a constant and we have to differentiate x^3."},{"Start":"01:09.950 ","End":"01:13.185","Text":"So we have 2t times 2x from here,"},{"Start":"01:13.185 ","End":"01:15.075","Text":"2t times 2x is 4xt."},{"Start":"01:15.075 ","End":"01:18.930","Text":"From here we have minus 2 times 3x^2,"},{"Start":"01:18.930 ","End":"01:21.135","Text":"that gives us the 6x^2."},{"Start":"01:21.135 ","End":"01:22.695","Text":"That\u0027s the left side."},{"Start":"01:22.695 ","End":"01:25.120","Text":"Now the right side, N_t."},{"Start":"01:25.120 ","End":"01:28.885","Text":"So x is a constant, this disappears."},{"Start":"01:28.885 ","End":"01:30.410","Text":"6x^2 is a constant."},{"Start":"01:30.410 ","End":"01:32.570","Text":"We just differentiate the t and"},{"Start":"01:32.570 ","End":"01:37.565","Text":"2x times the derivative of t^2 with respect to t this time,"},{"Start":"01:37.565 ","End":"01:43.070","Text":"minus 6x^2 times 1 plus 2x times 2t, which is 4xt."},{"Start":"01:43.070 ","End":"01:45.365","Text":"And if we notice,"},{"Start":"01:45.365 ","End":"01:49.355","Text":"I\u0027m going to highlight here we got 4xt minus 6x^2."},{"Start":"01:49.355 ","End":"01:52.205","Text":"Here it doesn\u0027t quite look the same."},{"Start":"01:52.205 ","End":"01:54.740","Text":"It\u0027s minus 6x^2 plus 4xt,"},{"Start":"01:54.740 ","End":"01:56.720","Text":"but it\u0027s just a change of order,"},{"Start":"01:56.720 ","End":"01:57.950","Text":"so it is the same."},{"Start":"01:57.950 ","End":"02:01.220","Text":"So we do get the N_t equals M_x,"},{"Start":"02:01.220 ","End":"02:04.025","Text":"so the equation is exact."},{"Start":"02:04.025 ","End":"02:07.190","Text":"We\u0027ve identified it, but we still have to solve it."},{"Start":"02:07.190 ","End":"02:10.400","Text":"It\u0027s the same technique with this exact equation,"},{"Start":"02:10.400 ","End":"02:12.335","Text":"we\u0027d look for a general function."},{"Start":"02:12.335 ","End":"02:13.445","Text":"I should have written it,"},{"Start":"02:13.445 ","End":"02:17.210","Text":"especially since it\u0027s not x and y, it\u0027s t and x,"},{"Start":"02:17.210 ","End":"02:19.460","Text":"we look for a solution in implicit form,"},{"Start":"02:19.460 ","End":"02:22.835","Text":"some function of t and x equals a constant."},{"Start":"02:22.835 ","End":"02:25.820","Text":"For the exact equation for this one,"},{"Start":"02:25.820 ","End":"02:28.355","Text":"it has to satisfy these 2 conditions."},{"Start":"02:28.355 ","End":"02:31.115","Text":"Derivative with respect to t has to be M,"},{"Start":"02:31.115 ","End":"02:32.360","Text":"and with respect to x,"},{"Start":"02:32.360 ","End":"02:36.030","Text":"it has to be N. We\u0027ll start out with one of them,"},{"Start":"02:36.030 ","End":"02:38.840","Text":"let\u0027s start with the derivative with respect to t is"},{"Start":"02:38.840 ","End":"02:41.810","Text":"M. If we have the derivative, find the function."},{"Start":"02:41.810 ","End":"02:43.760","Text":"We do the anti-derivative only with"},{"Start":"02:43.760 ","End":"02:46.460","Text":"respect to t because there\u0027s more than one variable here."},{"Start":"02:46.460 ","End":"02:48.200","Text":"We take this M,"},{"Start":"02:48.200 ","End":"02:50.810","Text":"which is somehow scrolled by,"},{"Start":"02:50.810 ","End":"02:58.040","Text":"but this was M 2x^2t minus 2x^3 with respect to t. This is just equal to,"},{"Start":"02:58.040 ","End":"03:00.515","Text":"remember that x^2 is a constant."},{"Start":"03:00.515 ","End":"03:05.375","Text":"So the x^2 here stays and 2t the integral of 2t is t^2,"},{"Start":"03:05.375 ","End":"03:09.320","Text":"that\u0027s the first term minus and then the second term."},{"Start":"03:09.320 ","End":"03:11.870","Text":"This whole thing is a constant with respect to t,"},{"Start":"03:11.870 ","End":"03:15.590","Text":"so we just add a t to it, add means multiplier,"},{"Start":"03:15.590 ","End":"03:22.470","Text":"but the constant is not C it\u0027s a whole function of x because as far as t goes,"},{"Start":"03:22.470 ","End":"03:24.840","Text":"a constant is a function of x."},{"Start":"03:24.840 ","End":"03:28.610","Text":"Now what we have to do is find what is g(x)"},{"Start":"03:28.610 ","End":"03:32.270","Text":"and way we do that is another piece of information we haven\u0027t used."},{"Start":"03:32.270 ","End":"03:35.885","Text":"Is that the derivative with respect to x is n. So we take this,"},{"Start":"03:35.885 ","End":"03:38.480","Text":"this is equal to F, and I\u0027ll take F_x,"},{"Start":"03:38.480 ","End":"03:40.445","Text":"which means from here,"},{"Start":"03:40.445 ","End":"03:44.435","Text":"differentiating so we get 2xt^2 from here,"},{"Start":"03:44.435 ","End":"03:47.225","Text":"6x^2t, and from g, we get g\u0027."},{"Start":"03:47.225 ","End":"03:49.105","Text":"That\u0027s on the one hand."},{"Start":"03:49.105 ","End":"03:53.095","Text":"On the other hand is got to equal N. And so F_x,"},{"Start":"03:53.095 ","End":"03:54.980","Text":"which are just copy from here,"},{"Start":"03:54.980 ","End":"03:56.540","Text":"that\u0027s this bit has got to equal"},{"Start":"03:56.540 ","End":"03:59.780","Text":"the original function N which is written somewhere anyway,"},{"Start":"03:59.780 ","End":"04:02.120","Text":"I just copied here is equal to this."},{"Start":"04:02.120 ","End":"04:04.175","Text":"A lot of stuff cancels,"},{"Start":"04:04.175 ","End":"04:06.650","Text":"this cancels with this,"},{"Start":"04:06.650 ","End":"04:09.215","Text":"and this goes with this,"},{"Start":"04:09.215 ","End":"04:14.600","Text":"and so we end up with g\u0027(x) is 4x^3."},{"Start":"04:14.600 ","End":"04:16.385","Text":"Now this is just in one variable,"},{"Start":"04:16.385 ","End":"04:19.805","Text":"the derivative of a function is 4x^3."},{"Start":"04:19.805 ","End":"04:23.165","Text":"The function g is going to be the anti-derivative."},{"Start":"04:23.165 ","End":"04:25.490","Text":"So g(x) is a typo here,"},{"Start":"04:25.490 ","End":"04:27.665","Text":"this should be an x,"},{"Start":"04:27.665 ","End":"04:30.140","Text":"not a y sorry about that."},{"Start":"04:30.140 ","End":"04:34.670","Text":"Is equal to, the anti-derivative or primitive or indefinite integral,"},{"Start":"04:34.670 ","End":"04:38.615","Text":"or the same thing of 4x^3 is x^4."},{"Start":"04:38.615 ","End":"04:42.840","Text":"What I do with this x^4 is put it here instead"},{"Start":"04:42.840 ","End":"04:47.310","Text":"of g(x) is founded and that will give me what F is."},{"Start":"04:47.310 ","End":"04:54.170","Text":"So I can now write that F is x^2t^2 plus 2x^3t,"},{"Start":"04:54.170 ","End":"04:57.119","Text":"and then plus the g which is x^4."},{"Start":"04:57.119 ","End":"05:03.210","Text":"The final step is to say that F(t and x)is a constant."},{"Start":"05:03.640 ","End":"05:06.379","Text":"So use 2x and y,"},{"Start":"05:06.379 ","End":"05:08.090","Text":"to give me, it\u0027s t and x."},{"Start":"05:08.090 ","End":"05:11.090","Text":"Don\u0027t you make the same mistake as I did learn from my mistakes,"},{"Start":"05:11.090 ","End":"05:16.930","Text":"t and x is this thing equals c, and we\u0027re done."}],"Thumbnail":null,"ID":7674},{"Watched":false,"Name":"Exercise 7","Duration":"6m 27s","ChapterTopicVideoID":7602,"CourseChapterTopicPlaylistID":4221,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.680","Text":"In this exercise, we\u0027re given"},{"Start":"00:01.680 ","End":"00:06.420","Text":"a differential equation as follows and there\u0027s a slight difference"},{"Start":"00:06.420 ","End":"00:13.065","Text":"than the previous exercises because notice that there\u0027s a constant k that\u0027s here."},{"Start":"00:13.065 ","End":"00:16.515","Text":"You\u0027ll see what this means after we\u0027ve seen the first question."},{"Start":"00:16.515 ","End":"00:18.570","Text":"We have to find the constant k,"},{"Start":"00:18.570 ","End":"00:20.760","Text":"which makes the equation exact."},{"Start":"00:20.760 ","End":"00:25.020","Text":"Let\u0027s just concentrate on part A and later we\u0027ll see what part B is."},{"Start":"00:25.020 ","End":"00:29.085","Text":"We know what the formula for an equation to be exact is."},{"Start":"00:29.085 ","End":"00:32.310","Text":"If we label these 2 functions, M and N,"},{"Start":"00:32.310 ","End":"00:38.025","Text":"then we know that the exact means that M_y partial derivative that is,"},{"Start":"00:38.025 ","End":"00:41.460","Text":"is equal to N_x and don\u0027t get it backwards."},{"Start":"00:41.460 ","End":"00:46.355","Text":"The 1 next to dx is the partial derivative with respect to y and vice versa."},{"Start":"00:46.355 ","End":"00:50.660","Text":"Let\u0027s compute each of these and then assign them to be equal to each other."},{"Start":"00:50.660 ","End":"00:54.155","Text":"We\u0027ll start with M_y derivative with respect to y,"},{"Start":"00:54.155 ","End":"00:57.800","Text":"3x^2 is a constant as far as y goes, so that gives nothing."},{"Start":"00:57.800 ","End":"00:59.360","Text":"Here we have the product rule,"},{"Start":"00:59.360 ","End":"01:01.745","Text":"we have y and we have e^xy."},{"Start":"01:01.745 ","End":"01:04.565","Text":"Take the derivative of the first times the second,"},{"Start":"01:04.565 ","End":"01:05.855","Text":"and then vice versa."},{"Start":"01:05.855 ","End":"01:08.350","Text":"The first as is and derivative of the second."},{"Start":"01:08.350 ","End":"01:09.635","Text":"What we get from here,"},{"Start":"01:09.635 ","End":"01:10.940","Text":"derivative here is 1,"},{"Start":"01:10.940 ","End":"01:12.455","Text":"so that\u0027s just e^xy."},{"Start":"01:12.455 ","End":"01:14.600","Text":"Then this bit here,"},{"Start":"01:14.600 ","End":"01:17.540","Text":"e^xy, let me just a side, show you how I got that."},{"Start":"01:17.540 ","End":"01:19.250","Text":"There is a template that if I have e to"},{"Start":"01:19.250 ","End":"01:22.340","Text":"the something derivative with respect to whatever,"},{"Start":"01:22.340 ","End":"01:27.320","Text":"it\u0027s just e to that something you might expect but with an internal derivative."},{"Start":"01:27.320 ","End":"01:31.310","Text":"In our case, this xy is our box."},{"Start":"01:31.310 ","End":"01:34.370","Text":"We get e^xy and the internal"},{"Start":"01:34.370 ","End":"01:38.330","Text":"derivative is just x because we\u0027re differentiating with respect to y."},{"Start":"01:38.330 ","End":"01:40.940","Text":"That\u0027s the x here and I pulled it out in front."},{"Start":"01:40.940 ","End":"01:44.525","Text":"Finally, we take e^xy outside the brackets."},{"Start":"01:44.525 ","End":"01:47.780","Text":"This is what we get for M_y."},{"Start":"01:47.780 ","End":"01:51.805","Text":"I\u0027ll highlight this, that is M_y."},{"Start":"01:51.805 ","End":"01:53.010","Text":"Now the next 1."},{"Start":"01:53.010 ","End":"01:55.940","Text":"N_x is equal to,"},{"Start":"01:55.940 ","End":"01:59.540","Text":"well, this first bit 2y^3 goes to 0."},{"Start":"01:59.540 ","End":"02:01.970","Text":"It\u0027s a function of y. It\u0027s a constant."},{"Start":"02:01.970 ","End":"02:03.935","Text":"This time, remember x is the variable,"},{"Start":"02:03.935 ","End":"02:05.495","Text":"and here we have a product."},{"Start":"02:05.495 ","End":"02:08.150","Text":"The k comes out because it\u0027s a constant."},{"Start":"02:08.150 ","End":"02:10.790","Text":"We take the derivative of the first,"},{"Start":"02:10.790 ","End":"02:16.430","Text":"which is x times the second e^xy plus x times the derivative of the second."},{"Start":"02:16.430 ","End":"02:18.635","Text":"That\u0027s similar to what we had up here."},{"Start":"02:18.635 ","End":"02:22.640","Text":"Well, first part gives me 1 times e^xy and the derivative of"},{"Start":"02:22.640 ","End":"02:26.600","Text":"this is just e^xy times the inner derivative."},{"Start":"02:26.600 ","End":"02:28.820","Text":"The box is this in this case and"},{"Start":"02:28.820 ","End":"02:33.590","Text":"the inner derivative with respect to x is y and that\u0027s this y here."},{"Start":"02:33.590 ","End":"02:38.300","Text":"We got this and then we take another thing out the bracket, that\u0027s e^xy."},{"Start":"02:38.300 ","End":"02:41.675","Text":"We\u0027ve got ke^xy,1 plus xy."},{"Start":"02:41.675 ","End":"02:44.525","Text":"N_x is this."},{"Start":"02:44.525 ","End":"02:46.580","Text":"Now these 2 have to be equal."},{"Start":"02:46.580 ","End":"02:48.560","Text":"If this is going to equal this,"},{"Start":"02:48.560 ","End":"02:53.390","Text":"I think it\u0027s fairly clear that the only way this is going to happen is if k=1."},{"Start":"02:53.390 ","End":"02:57.275","Text":"That answers the first part, k=1."},{"Start":"02:57.275 ","End":"03:02.040","Text":"Now I\u0027ll go to a new page and do the second part."},{"Start":"03:02.040 ","End":"03:04.470","Text":"Now we\u0027re at part b,"},{"Start":"03:04.470 ","End":"03:07.080","Text":"and in part b we already solved a."},{"Start":"03:07.080 ","End":"03:11.370","Text":"We found that k=1 was what solved a."},{"Start":"03:11.370 ","End":"03:13.740","Text":"Now we have to solve the equation."},{"Start":"03:13.740 ","End":"03:18.320","Text":"Now we can replace k by 1 and the only change"},{"Start":"03:18.320 ","End":"03:22.865","Text":"I made was in this term here you see the k is vanished because it\u0027s equal to 1."},{"Start":"03:22.865 ","End":"03:27.290","Text":"Let\u0027s label this M and this N. We already know its exact from part a,"},{"Start":"03:27.290 ","End":"03:29.360","Text":"so we just have to solve it now."},{"Start":"03:29.360 ","End":"03:34.759","Text":"If you recall, we\u0027re looking for a function f which satisfies these 2 equations."},{"Start":"03:34.759 ","End":"03:38.615","Text":"I should\u0027ve said that f(x,y) equals,"},{"Start":"03:38.615 ","End":"03:40.820","Text":"a constant is the general solution and f is"},{"Start":"03:40.820 ","End":"03:43.730","Text":"a function of 2 variables which satisfies these 2."},{"Start":"03:43.730 ","End":"03:45.200","Text":"We\u0027ll start with 1 of them."},{"Start":"03:45.200 ","End":"03:51.170","Text":"Let\u0027s start with F_x equals M. To get from F_x back to F,"},{"Start":"03:51.170 ","End":"03:54.560","Text":"we take the integral with respect to x."},{"Start":"03:54.560 ","End":"03:56.520","Text":"We take the integral of Mdx,"},{"Start":"03:56.520 ","End":"03:58.215","Text":"but M is this."},{"Start":"03:58.215 ","End":"04:00.405","Text":"I need the integral of this dx."},{"Start":"04:00.405 ","End":"04:04.730","Text":"An integral in the sense of indefinite integral, antiderivative."},{"Start":"04:04.730 ","End":"04:10.415","Text":"Just don\u0027t forget that y is a constant because we\u0027re doing it dx."},{"Start":"04:10.415 ","End":"04:13.160","Text":"This with respect to x is x^3."},{"Start":"04:13.160 ","End":"04:15.320","Text":"Now this y is a constant,"},{"Start":"04:15.320 ","End":"04:19.895","Text":"it stays and the integral of e^xy is almost e^xy,"},{"Start":"04:19.895 ","End":"04:24.425","Text":"but we have to divide by the inner derivative of xy, which is y."},{"Start":"04:24.425 ","End":"04:27.185","Text":"You can do this when this is a linear function of x."},{"Start":"04:27.185 ","End":"04:30.710","Text":"The other thing to remember is that we don\u0027t add a constant."},{"Start":"04:30.710 ","End":"04:33.440","Text":"We add a constant as far as x goes,"},{"Start":"04:33.440 ","End":"04:35.690","Text":"we add a general function of y."},{"Start":"04:35.690 ","End":"04:38.465","Text":"The only thing to do now is just to simplify this,"},{"Start":"04:38.465 ","End":"04:40.160","Text":"y over y is 1."},{"Start":"04:40.160 ","End":"04:41.975","Text":"This is what we get."},{"Start":"04:41.975 ","End":"04:45.245","Text":"This is our function f(x,y)."},{"Start":"04:45.245 ","End":"04:49.310","Text":"That\u0027s this here, but we don\u0027t have it yet because we don\u0027t have"},{"Start":"04:49.310 ","End":"04:53.915","Text":"g. Now we have to start finding g. To do that,"},{"Start":"04:53.915 ","End":"04:56.870","Text":"we can use the other part of the equation where"},{"Start":"04:56.870 ","End":"05:02.330","Text":"F_y is N. We\u0027ll take this and differentiate it with respect to y."},{"Start":"05:02.330 ","End":"05:04.520","Text":"The x^3 gives nothing,"},{"Start":"05:04.520 ","End":"05:06.200","Text":"gets constant as far as y goes,"},{"Start":"05:06.200 ","End":"05:10.370","Text":"e^xy is e^xy times the derivative of xy,"},{"Start":"05:10.370 ","End":"05:13.970","Text":"which is x and g(y) just gives us g\u0027(y)."},{"Start":"05:13.970 ","End":"05:15.335","Text":"That\u0027s on the 1 hand,"},{"Start":"05:15.335 ","End":"05:18.650","Text":"and on the other hand we have N. I take this F_y from"},{"Start":"05:18.650 ","End":"05:22.660","Text":"here and let it equal N. It\u0027s still visible up here."},{"Start":"05:22.660 ","End":"05:24.680","Text":"If this equals this, well,"},{"Start":"05:24.680 ","End":"05:26.525","Text":"this will cancel with this,"},{"Start":"05:26.525 ","End":"05:28.939","Text":"and we\u0027ll be left with this equation."},{"Start":"05:28.939 ","End":"05:32.220","Text":"The derivative of g is 2y^3."},{"Start":"05:32.220 ","End":"05:37.670","Text":"What we need to do now is to find g. We do this by integration."},{"Start":"05:37.670 ","End":"05:40.010","Text":"I\u0027m here reminding you that the function is the"},{"Start":"05:40.010 ","End":"05:43.025","Text":"integral of its derivative because they are opposite operations."},{"Start":"05:43.025 ","End":"05:45.800","Text":"I\u0027ve got the integral of 2y^3dy,"},{"Start":"05:45.800 ","End":"05:49.040","Text":"and that is just 2y^4/4,"},{"Start":"05:49.040 ","End":"05:51.950","Text":"but 2/4 is 1/2. This is what we get."},{"Start":"05:51.950 ","End":"05:53.300","Text":"We don\u0027t need the constant here."},{"Start":"05:53.300 ","End":"05:54.485","Text":"We have 1 at the end."},{"Start":"05:54.485 ","End":"05:57.605","Text":"Now I found that g(y) is this,"},{"Start":"05:57.605 ","End":"05:59.690","Text":"so I can substitute it."},{"Start":"05:59.690 ","End":"06:04.025","Text":"Let\u0027s see, I think I\u0027ve lost definition of F. Here it is."},{"Start":"06:04.025 ","End":"06:06.335","Text":"I put g(y) in here."},{"Start":"06:06.335 ","End":"06:11.675","Text":"F is x^3 plus e^xy plus y^4/2."},{"Start":"06:11.675 ","End":"06:15.035","Text":"Finally, we need to set F=c,"},{"Start":"06:15.035 ","End":"06:18.800","Text":"like so F=c but our particular F, which is this."},{"Start":"06:18.800 ","End":"06:21.410","Text":"We have this equals c and that\u0027s the solution."},{"Start":"06:21.410 ","End":"06:23.735","Text":"You could say solutions is in plural because"},{"Start":"06:23.735 ","End":"06:27.930","Text":"each c gives a different solution. That\u0027s it."}],"Thumbnail":null,"ID":7675}],"ID":4221},{"Name":"Integration Factor","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Integration Factor","Duration":"5m 29s","ChapterTopicVideoID":7634,"CourseChapterTopicPlaylistID":4222,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"This exercise is not really an exercise,"},{"Start":"00:02.820 ","End":"00:09.255","Text":"it\u0027s a tutorial just disguised as an exercise just to use the same format."},{"Start":"00:09.255 ","End":"00:14.010","Text":"What it really is is a tutorial on the integration factor,"},{"Start":"00:14.010 ","End":"00:17.070","Text":"the concept and to phrase this as a question,"},{"Start":"00:17.070 ","End":"00:19.874","Text":"explain how a non-exact differential equation"},{"Start":"00:19.874 ","End":"00:23.670","Text":"might be made exact by means of an integration factor."},{"Start":"00:23.670 ","End":"00:26.450","Text":"That already gives you an idea what it\u0027s about."},{"Start":"00:26.450 ","End":"00:28.490","Text":"When I say non-exact,"},{"Start":"00:28.490 ","End":"00:33.125","Text":"I don\u0027t just mean anything that\u0027s totally wild but not an exact equation."},{"Start":"00:33.125 ","End":"00:36.380","Text":"I mean, that it\u0027s in the form of an exact equation,"},{"Start":"00:36.380 ","End":"00:40.250","Text":"in a sense it\u0027s still some function of (x,y)dx,"},{"Start":"00:40.250 ","End":"00:42.080","Text":"another function of (x,y)dy,"},{"Start":"00:42.080 ","End":"00:48.980","Text":"the non just applies to this equality that this happens not to equal this,"},{"Start":"00:48.980 ","End":"00:50.990","Text":"but it\u0027s in the right format."},{"Start":"00:50.990 ","End":"00:53.150","Text":"The idea of an integration factor,"},{"Start":"00:53.150 ","End":"00:56.135","Text":"someone thought that maybe it isn\u0027t exact,"},{"Start":"00:56.135 ","End":"01:00.815","Text":"but suppose we multiplied this whole equation by something,"},{"Start":"01:00.815 ","End":"01:03.475","Text":"maybe that could make it exact."},{"Start":"01:03.475 ","End":"01:08.730","Text":"In this case, we call such a function or we label it h(x,y),"},{"Start":"01:08.730 ","End":"01:13.310","Text":"and if we take this function and multiply it here and here,"},{"Start":"01:13.310 ","End":"01:15.050","Text":"and on the other side by 0,"},{"Start":"01:15.050 ","End":"01:17.545","Text":"then we get an exact equation,"},{"Start":"01:17.545 ","End":"01:20.389","Text":"then this is called the integrating factor."},{"Start":"01:20.389 ","End":"01:21.920","Text":"Let me spell it out."},{"Start":"01:21.920 ","End":"01:24.290","Text":"So we take this equation, it\u0027s not exact."},{"Start":"01:24.290 ","End":"01:25.955","Text":"But we multiply everything,"},{"Start":"01:25.955 ","End":"01:29.720","Text":"we take h and multiply it by m and we multiply it by n,"},{"Start":"01:29.720 ","End":"01:33.739","Text":"and of course, we multiply it by 0 also, but that stays 0."},{"Start":"01:33.739 ","End":"01:36.365","Text":"Just supposing that this comes out to be exact,"},{"Start":"01:36.365 ","End":"01:38.450","Text":"then this function, like I said,"},{"Start":"01:38.450 ","End":"01:39.485","Text":"I\u0027m repeating myself,"},{"Start":"01:39.485 ","End":"01:41.275","Text":"is an integration factor."},{"Start":"01:41.275 ","End":"01:44.030","Text":"Now you might say this sounds like a great idea,"},{"Start":"01:44.030 ","End":"01:47.960","Text":"but how do we find such a function h?"},{"Start":"01:47.960 ","End":"01:50.440","Text":"How do we find an integration factor?"},{"Start":"01:50.440 ","End":"01:53.115","Text":"There are several ways, I\u0027ll just, as a remark,"},{"Start":"01:53.115 ","End":"01:58.025","Text":"say that h(x,y) doesn\u0027t have to actually be a function of two variables."},{"Start":"01:58.025 ","End":"02:01.580","Text":"It could just be a function of x or just a function of y."},{"Start":"02:01.580 ","End":"02:05.465","Text":"One of the variables doesn\u0027t have to explicitly appear and that often happens."},{"Start":"02:05.465 ","End":"02:08.030","Text":"Now, how do we find the integration factor?"},{"Start":"02:08.030 ","End":"02:12.259","Text":"So there are several different cases that might happen."},{"Start":"02:12.259 ","End":"02:14.315","Text":"The first scenario is the best."},{"Start":"02:14.315 ","End":"02:16.580","Text":"It\u0027s simply provided in the exercise."},{"Start":"02:16.580 ","End":"02:18.440","Text":"The next slides might say show that"},{"Start":"02:18.440 ","End":"02:22.970","Text":"this non-exact equation becomes exact with the following integration factor,"},{"Start":"02:22.970 ","End":"02:26.570","Text":"and it\u0027s given specifically in the question, that\u0027s good."},{"Start":"02:26.570 ","End":"02:28.820","Text":"But if it\u0027s not given in the exercise,"},{"Start":"02:28.820 ","End":"02:31.270","Text":"there are techniques that you could try."},{"Start":"02:31.270 ","End":"02:35.015","Text":"Nothing\u0027s going to be guaranteed that you will find one."},{"Start":"02:35.015 ","End":"02:37.340","Text":"Because if you\u0027re given an exercise to solve,"},{"Start":"02:37.340 ","End":"02:42.785","Text":"then most likely or almost certainly one of these techniques will work for you."},{"Start":"02:42.785 ","End":"02:44.395","Text":"The first one is this."},{"Start":"02:44.395 ","End":"02:47.760","Text":"Now, my is not equal to nx."},{"Start":"02:47.760 ","End":"02:50.390","Text":"My minus nx is not 0."},{"Start":"02:50.390 ","End":"02:53.060","Text":"We try computing this expression,"},{"Start":"02:53.060 ","End":"02:58.280","Text":"this minus this over this and if it turns out to be a function of x,"},{"Start":"02:58.280 ","End":"02:59.440","Text":"when I say equals f(x),"},{"Start":"02:59.440 ","End":"03:02.600","Text":"it\u0027s just my shorthand way of saying that this thing is"},{"Start":"03:02.600 ","End":"03:06.230","Text":"a function of x alone and no y\u0027s appear in it."},{"Start":"03:06.230 ","End":"03:11.090","Text":"Then this function, if you take e to the power of the integral of this function,"},{"Start":"03:11.090 ","End":"03:12.560","Text":"in other words, you compute this,"},{"Start":"03:12.560 ","End":"03:14.960","Text":"will be an integration factor."},{"Start":"03:14.960 ","End":"03:17.600","Text":"Now Case Number 2 is very similar to this,"},{"Start":"03:17.600 ","End":"03:20.680","Text":"just with x and y\u0027s roles reversed."},{"Start":"03:20.680 ","End":"03:22.460","Text":"Instead of dividing by n,"},{"Start":"03:22.460 ","End":"03:23.740","Text":"we divide by m,"},{"Start":"03:23.740 ","End":"03:28.339","Text":"and if this happens to be a function just the y and no x\u0027s,"},{"Start":"03:28.339 ","End":"03:31.950","Text":"then this e to the power of,"},{"Start":"03:31.950 ","End":"03:34.055","Text":"notice there\u0027s a slight difference also,"},{"Start":"03:34.055 ","End":"03:37.370","Text":"there\u0027s a minus here but there isn\u0027t a minus here,"},{"Start":"03:37.370 ","End":"03:39.910","Text":"is going be an integration factor."},{"Start":"03:39.910 ","End":"03:43.460","Text":"In case you\u0027re wondering why is there a minus here and not here,"},{"Start":"03:43.460 ","End":"03:45.095","Text":"what breaks the symmetry,"},{"Start":"03:45.095 ","End":"03:49.070","Text":"it\u0027s because we do my minus nx,"},{"Start":"03:49.070 ","End":"03:50.690","Text":"if you wanted to reverse everything,"},{"Start":"03:50.690 ","End":"03:53.395","Text":"you\u0027d also reverse the order of this subtraction,"},{"Start":"03:53.395 ","End":"03:57.305","Text":"and then g would come out negative and you wouldn\u0027t need this minus."},{"Start":"03:57.305 ","End":"04:00.500","Text":"Now what if these two techniques don\u0027t work? Let\u0027s see."},{"Start":"04:00.500 ","End":"04:02.270","Text":"Do we have any other tricks?"},{"Start":"04:02.270 ","End":"04:03.975","Text":"Yeah, okay,"},{"Start":"04:03.975 ","End":"04:05.930","Text":"I\u0027ve just uncovered everything."},{"Start":"04:05.930 ","End":"04:08.300","Text":"There\u0027s a table here and I\u0027ll relate to it,"},{"Start":"04:08.300 ","End":"04:10.055","Text":"we just read this paragraph."},{"Start":"04:10.055 ","End":"04:12.080","Text":"Suppose that we tidy it up,"},{"Start":"04:12.080 ","End":"04:14.720","Text":"collect like terms and so on and so on."},{"Start":"04:14.720 ","End":"04:19.760","Text":"If we identify an expression that looks something like in the left column,"},{"Start":"04:19.760 ","End":"04:23.750","Text":"if we see an (x)dy minus (y)dx, we do one thing."},{"Start":"04:23.750 ","End":"04:28.070","Text":"If we see that we have in our non-exact equation this,"},{"Start":"04:28.070 ","End":"04:29.600","Text":"we try something else."},{"Start":"04:29.600 ","End":"04:31.505","Text":"What this is saying is, okay,"},{"Start":"04:31.505 ","End":"04:36.965","Text":"I spot (x)dy minus (y)dx in my equation in some prominent part of it."},{"Start":"04:36.965 ","End":"04:39.500","Text":"Then here are some recommendations for"},{"Start":"04:39.500 ","End":"04:43.655","Text":"trying an integration factor, there\u0027s nothing guaranteed."},{"Start":"04:43.655 ","End":"04:46.910","Text":"But these are things you might try and one of these,"},{"Start":"04:46.910 ","End":"04:48.655","Text":"is very likely will work."},{"Start":"04:48.655 ","End":"04:52.470","Text":"If we see (x)dy plus (y)dx,"},{"Start":"04:52.470 ","End":"04:56.470","Text":"then we try and integration factor of 1 over"},{"Start":"04:56.470 ","End":"05:00.830","Text":"xy to the power of n. We can try different values of n also,"},{"Start":"05:00.830 ","End":"05:02.120","Text":"n could be 1,2,3,"},{"Start":"05:02.120 ","End":"05:03.620","Text":"some whole number,"},{"Start":"05:03.620 ","End":"05:05.195","Text":"could even be negative."},{"Start":"05:05.195 ","End":"05:08.405","Text":"If we see (x)dx plus (y)dy,"},{"Start":"05:08.405 ","End":"05:12.080","Text":"then we try an expression of this form for"},{"Start":"05:12.080 ","End":"05:16.410","Text":"some value of n. This won\u0027t make sense until the examples,"},{"Start":"05:16.410 ","End":"05:22.550","Text":"but at least I\u0027ve recorded the steps that we do to try and find an integration factor."},{"Start":"05:22.550 ","End":"05:25.730","Text":"Let\u0027s leave that here and"},{"Start":"05:25.730 ","End":"05:29.990","Text":"suggest you go on to the examples where everything will become clearer."}],"Thumbnail":null,"ID":7691},{"Watched":false,"Name":"Exercise 1","Duration":"5m 45s","ChapterTopicVideoID":13833,"CourseChapterTopicPlaylistID":4222,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.830","Text":"In this exercise, we have"},{"Start":"00:01.830 ","End":"00:07.245","Text":"a differential equation and we have to show that it\u0027s not exact,"},{"Start":"00:07.245 ","End":"00:15.135","Text":"but that we can use this as an integration factor and make it exact and then solve it."},{"Start":"00:15.135 ","End":"00:18.555","Text":"What I want to do with this equation I just copied it,"},{"Start":"00:18.555 ","End":"00:23.865","Text":"is to bring it into standard form something dx plus something dy equals 0."},{"Start":"00:23.865 ","End":"00:28.200","Text":"I\u0027m going to replace y\u0027 by dy over dx."},{"Start":"00:28.200 ","End":"00:30.870","Text":"Then I can multiply both sides by dx."},{"Start":"00:30.870 ","End":"00:32.940","Text":"That gives us this,"},{"Start":"00:32.940 ","End":"00:35.310","Text":"just putting the dx here basically,"},{"Start":"00:35.310 ","End":"00:39.800","Text":"and also label these functions m and n. I want to remind"},{"Start":"00:39.800 ","End":"00:45.410","Text":"you that what we do is compare partial derivative of m with respect to y,"},{"Start":"00:45.410 ","End":"00:49.070","Text":"with partial derivative of n with respect to x."},{"Start":"00:49.070 ","End":"00:52.480","Text":"If it\u0027s equal, then they are exact."},{"Start":"00:52.480 ","End":"00:58.190","Text":"Actually we\u0027re hoping here for not exact meaning we get a not equal here,"},{"Start":"00:58.190 ","End":"00:59.900","Text":"and that\u0027s straightforward enough."},{"Start":"00:59.900 ","End":"01:03.515","Text":"Here\u0027s m, its derivative with respect to y is this."},{"Start":"01:03.515 ","End":"01:05.760","Text":"I won\u0027t go into the details obvious."},{"Start":"01:05.760 ","End":"01:07.273","Text":"As for n,"},{"Start":"01:07.273 ","End":"01:11.435","Text":"its derivative with respect to x is just this 1 plus y^2."},{"Start":"01:11.435 ","End":"01:13.925","Text":"These are two different functions."},{"Start":"01:13.925 ","End":"01:17.330","Text":"You can\u0027t just algebraically make one into the other."},{"Start":"01:17.330 ","End":"01:19.360","Text":"They are not equal,"},{"Start":"01:19.360 ","End":"01:23.210","Text":"and so this equation is not exact at this one."},{"Start":"01:23.210 ","End":"01:26.120","Text":"This is where the integration factor comes in."},{"Start":"01:26.120 ","End":"01:30.755","Text":"The idea is that if I take this and multiply it by everything,"},{"Start":"01:30.755 ","End":"01:33.440","Text":"and it\u0027s in a different color so you can see it."},{"Start":"01:33.440 ","End":"01:36.320","Text":"Hopefully this will be exact."},{"Start":"01:36.320 ","End":"01:41.370","Text":"What we\u0027ll do is we\u0027ll label this and this."},{"Start":"01:41.370 ","End":"01:43.545","Text":"After the multiplication,"},{"Start":"01:43.545 ","End":"01:47.715","Text":"the y^3 cancels and x^2 over x is x."},{"Start":"01:47.715 ","End":"01:51.195","Text":"Here x and the x cancel."},{"Start":"01:51.195 ","End":"01:54.570","Text":"What I get is 1 plus y^2 over y^3."},{"Start":"01:54.570 ","End":"01:58.205","Text":"I\u0027m not really supposed to use the same letter again,"},{"Start":"01:58.205 ","End":"02:01.310","Text":"but I think we can safely reuse m and n,"},{"Start":"02:01.310 ","End":"02:03.530","Text":"If I use a different letter it\u0027s going to be confusing."},{"Start":"02:03.530 ","End":"02:08.389","Text":"So I think there\u0027s no possibility of confusion from this point onwards."},{"Start":"02:08.389 ","End":"02:15.905","Text":"m and n, and now new m and n are not the same as the old ones."},{"Start":"02:15.905 ","End":"02:19.355","Text":"Now use the same test for exactness on"},{"Start":"02:19.355 ","End":"02:24.770","Text":"this new m and n. This turns out to be easier than we might have expected,"},{"Start":"02:24.770 ","End":"02:28.115","Text":"because you see m is just a function of x."},{"Start":"02:28.115 ","End":"02:30.230","Text":"With respect to y at 0."},{"Start":"02:30.230 ","End":"02:34.730","Text":"Similarly, n is just a function of y so with respect to x is 0."},{"Start":"02:34.730 ","End":"02:37.310","Text":"If each of these is 0, then they\u0027re equal to each other."},{"Start":"02:37.310 ","End":"02:40.855","Text":"We do have my equals nx."},{"Start":"02:40.855 ","End":"02:44.060","Text":"The new equation is indeed exact."},{"Start":"02:44.060 ","End":"02:46.595","Text":"Now comes the hard part,"},{"Start":"02:46.595 ","End":"02:48.845","Text":"you actually have to solve the equation."},{"Start":"02:48.845 ","End":"02:51.380","Text":"Sorry, I shouldn\u0027t have said hard now it\u0027s not hard, it\u0027s easy."},{"Start":"02:51.380 ","End":"02:52.715","Text":"You just a bit lengthy."},{"Start":"02:52.715 ","End":"02:55.355","Text":"Move to a new page here."},{"Start":"02:55.355 ","End":"02:58.385","Text":"We know the general form of the solution."},{"Start":"02:58.385 ","End":"03:03.110","Text":"It\u0027s of the form f(x,y) equals c. The conditions are that"},{"Start":"03:03.110 ","End":"03:09.540","Text":"the partial derivative of f with respect to x is the new m. With respect to y,"},{"Start":"03:09.540 ","End":"03:12.690","Text":"it\u0027s this new n. Let\u0027s start with this one."},{"Start":"03:12.690 ","End":"03:15.035","Text":"If we know the derivative with respect to x,"},{"Start":"03:15.035 ","End":"03:18.575","Text":"we just take m and integrate it with respect to x."},{"Start":"03:18.575 ","End":"03:23.895","Text":"What m is, is x. we need the integral of xdx,"},{"Start":"03:23.895 ","End":"03:27.850","Text":"and this part is 1.5x^2."},{"Start":"03:28.040 ","End":"03:31.260","Text":"We don\u0027t add a constant."},{"Start":"03:31.260 ","End":"03:34.715","Text":"We add a general function of y because as far as"},{"Start":"03:34.715 ","End":"03:38.435","Text":"integration with respect to x with two variables,"},{"Start":"03:38.435 ","End":"03:41.705","Text":"y is a constant as far as x goes."},{"Start":"03:41.705 ","End":"03:47.180","Text":"Any function of y here will do its derivative with respect to x would be 0."},{"Start":"03:47.180 ","End":"03:48.440","Text":"Well, you\u0027ve seen all this before."},{"Start":"03:48.440 ","End":"03:50.285","Text":"I\u0027m just reiterating."},{"Start":"03:50.285 ","End":"03:55.650","Text":"Now, we don\u0027t have f exactly because we don\u0027t know what g is."},{"Start":"03:55.650 ","End":"03:59.435","Text":"What we do is we use the other piece of information,"},{"Start":"03:59.435 ","End":"04:08.130","Text":"this part just contains x."},{"Start":"04:08.130 ","End":"04:09.480","Text":"This gives me nothing."},{"Start":"04:09.480 ","End":"04:12.215","Text":"It\u0027s 0 plus the derivative of this."},{"Start":"04:12.215 ","End":"04:14.015","Text":"It\u0027s g\u0027(y)."},{"Start":"04:14.015 ","End":"04:17.720","Text":"On the other hand, it\u0027s going to equal n. We get that"},{"Start":"04:17.720 ","End":"04:22.205","Text":"this g\u0027(y) is equal to the function n,"},{"Start":"04:22.205 ","End":"04:24.215","Text":"which is still here on the board,"},{"Start":"04:24.215 ","End":"04:26.615","Text":"on the screen over here."},{"Start":"04:26.615 ","End":"04:31.370","Text":"Now we can find what g(y) is because we have its derivative."},{"Start":"04:31.370 ","End":"04:34.490","Text":"We can take the antiderivative or the integral"},{"Start":"04:34.490 ","End":"04:37.950","Text":"indefinite integral like so g(y) is this integral."},{"Start":"04:37.950 ","End":"04:39.080","Text":"How do we do this?"},{"Start":"04:39.080 ","End":"04:41.660","Text":"I suggest breaking up the fraction."},{"Start":"04:41.660 ","End":"04:46.410","Text":"We have 1 over y^3 plus y^2 over y^3."},{"Start":"04:46.410 ","End":"04:48.950","Text":"The second one is just 1 over y."},{"Start":"04:48.950 ","End":"04:52.565","Text":"Then we can do each piece separately."},{"Start":"04:52.565 ","End":"04:56.405","Text":"If I use negative exponents this will be y^-3."},{"Start":"04:56.405 ","End":"04:59.600","Text":"It\u0027s integral is y^-2 over minus 2,"},{"Start":"04:59.600 ","End":"05:04.645","Text":"1/y, we know is natural log of y."},{"Start":"05:04.645 ","End":"05:09.000","Text":"Let me just highlight this that we found g(y),"},{"Start":"05:09.000 ","End":"05:14.115","Text":"and f has just gone off screen here. Here it is."},{"Start":"05:14.115 ","End":"05:18.720","Text":"What I\u0027m going to do is replace this g(y) by what we have done here"},{"Start":"05:18.720 ","End":"05:23.780","Text":"to get f. f will be 0.5x^2 plus this."},{"Start":"05:23.780 ","End":"05:29.240","Text":"Here we are. Finally, we just have to let this equal a constant,"},{"Start":"05:29.240 ","End":"05:31.940","Text":"like so this thing equal to constant,"},{"Start":"05:31.940 ","End":"05:36.425","Text":"that\u0027s the general solution to the differential equation."},{"Start":"05:36.425 ","End":"05:39.980","Text":"This shows you how an integration factor can make"},{"Start":"05:39.980 ","End":"05:45.300","Text":"a non-exact into an exact and that we know how to solve. Done here."}],"Thumbnail":null,"ID":14632},{"Watched":false,"Name":"Exercise 2","Duration":"6m 32s","ChapterTopicVideoID":13834,"CourseChapterTopicPlaylistID":4222,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"Here we have this differential equation and we have to show that it\u0027s not exact."},{"Start":"00:04.830 ","End":"00:10.245","Text":"But to solve it using ye^x as an integration factor."},{"Start":"00:10.245 ","End":"00:13.290","Text":"Let\u0027s first of all show that it\u0027s not exact."},{"Start":"00:13.290 ","End":"00:20.490","Text":"Just label the first part M and the second function we\u0027ll call that N. What we have to"},{"Start":"00:20.490 ","End":"00:28.900","Text":"show is that M_y is not equal to N_x."},{"Start":"00:29.060 ","End":"00:33.510","Text":"Let\u0027s first do M_y. That\u0027s this one."},{"Start":"00:33.510 ","End":"00:36.810","Text":"Now the whole second part is a function of x, that doesn\u0027t appear."},{"Start":"00:36.810 ","End":"00:38.460","Text":"All I have is this quotient,"},{"Start":"00:38.460 ","End":"00:39.990","Text":"use the quotient rule."},{"Start":"00:39.990 ","End":"00:42.800","Text":"On the bottom I put this denominator squared."},{"Start":"00:42.800 ","End":"00:48.560","Text":"Now I have the denominator times derivative of numerator so it\u0027s y times"},{"Start":"00:48.560 ","End":"00:55.280","Text":"cosine y minus the derivative of the denominator,"},{"Start":"00:55.280 ","End":"00:57.770","Text":"and that\u0027s just 1 times sine y."},{"Start":"00:57.770 ","End":"01:00.510","Text":"This is what we get for this."},{"Start":"01:00.510 ","End":"01:05.670","Text":"Now N_x and at this time y is the constant so"},{"Start":"01:05.670 ","End":"01:11.360","Text":"this cosine y just drops out and this y on the denominator is a constant, so it stays."},{"Start":"01:11.360 ","End":"01:15.290","Text":"All I have to do is differentiate this part with respect to x"},{"Start":"01:15.290 ","End":"01:19.940","Text":"using the product rule and the 2 is a constant,"},{"Start":"01:19.940 ","End":"01:23.885","Text":"so I can take that separately, now I have a product."},{"Start":"01:23.885 ","End":"01:27.965","Text":"I take e^-x derived,"},{"Start":"01:27.965 ","End":"01:31.655","Text":"that\u0027s minus e^-x cosine x as is,"},{"Start":"01:31.655 ","End":"01:34.835","Text":"and then e^-x as is."},{"Start":"01:34.835 ","End":"01:37.940","Text":"But the derivative of cosine is minus sine,"},{"Start":"01:37.940 ","End":"01:40.250","Text":"so that\u0027s where this minus comes in."},{"Start":"01:40.250 ","End":"01:45.040","Text":"Let me take the e^-x outside the brackets,"},{"Start":"01:45.040 ","End":"01:46.470","Text":"and the minus sine."},{"Start":"01:46.470 ","End":"01:49.970","Text":"In any a event it\u0027s pretty clear that these two functions are not equal."},{"Start":"01:49.970 ","End":"01:55.020","Text":"For one thing, this is just a function of y and this contains x,"},{"Start":"01:55.150 ","End":"01:58.090","Text":"M_y is not equal to N_x,"},{"Start":"01:58.090 ","End":"02:00.335","Text":"so it\u0027s not exact."},{"Start":"02:00.335 ","End":"02:01.580","Text":"That\u0027s the first part."},{"Start":"02:01.580 ","End":"02:04.850","Text":"Now, let\u0027s look at that integration factor."},{"Start":"02:04.850 ","End":"02:09.980","Text":"The ye^x is the integration factor that we were asked to use."},{"Start":"02:09.980 ","End":"02:14.164","Text":"Hopefully, now this is going to be an exact equation."},{"Start":"02:14.164 ","End":"02:16.790","Text":"Bit of algebra, this with this,"},{"Start":"02:16.790 ","End":"02:19.790","Text":"the y\u0027s cancel again, e^x sine y."},{"Start":"02:19.790 ","End":"02:23.585","Text":"Here, e^x and e^-x cancel."},{"Start":"02:23.585 ","End":"02:29.195","Text":"I get the minus 2 still and it\u0027s going to be y sine x."},{"Start":"02:29.195 ","End":"02:32.085","Text":"This with this similarly, y\u0027s cancel,"},{"Start":"02:32.085 ","End":"02:33.990","Text":"e^x goes with the cosine,"},{"Start":"02:33.990 ","End":"02:38.955","Text":"and e^x cancels this e^-x so we get this,"},{"Start":"02:38.955 ","End":"02:41.100","Text":"label the first function M,"},{"Start":"02:41.100 ","End":"02:43.750","Text":"label the second function N. Not"},{"Start":"02:43.750 ","End":"02:47.240","Text":"strictly proper to do that because we had M and N before,"},{"Start":"02:47.240 ","End":"02:51.080","Text":"but like out with the old from this point,"},{"Start":"02:51.080 ","End":"02:53.525","Text":"these are the new M and N. Now,"},{"Start":"02:53.525 ","End":"02:56.975","Text":"this time we\u0027re hoping that we will get an exact equation."},{"Start":"02:56.975 ","End":"03:01.620","Text":"We want M_y to be equal to N_x,"},{"Start":"03:01.620 ","End":"03:03.230","Text":"at least as we\u0027re going to show this time."},{"Start":"03:03.230 ","End":"03:06.410","Text":"First we showed it isn\u0027t and now after the integration factor,"},{"Start":"03:06.410 ","End":"03:08.630","Text":"we hope that it will be equal."},{"Start":"03:08.630 ","End":"03:11.075","Text":"Start with M_y,"},{"Start":"03:11.075 ","End":"03:13.560","Text":"e^x is a constant,"},{"Start":"03:13.560 ","End":"03:15.915","Text":"derivative of sine is cosine."},{"Start":"03:15.915 ","End":"03:22.160","Text":"Here the x is still constant so it\u0027s a constant times y,"},{"Start":"03:22.160 ","End":"03:23.570","Text":"the y just drops out."},{"Start":"03:23.570 ","End":"03:26.780","Text":"This is this one and now let\u0027s do the other one."},{"Start":"03:26.780 ","End":"03:32.570","Text":"N_x and this time the y is constant,"},{"Start":"03:32.570 ","End":"03:37.390","Text":"so the cosine y stays on e^x,"},{"Start":"03:37.390 ","End":"03:41.515","Text":"is differentiated, just happens to be the same as the original."},{"Start":"03:41.515 ","End":"03:43.565","Text":"This with respect to x,"},{"Start":"03:43.565 ","End":"03:47.060","Text":"cosine derivative is minus sine so this is what we"},{"Start":"03:47.060 ","End":"03:50.735","Text":"get and it\u0027s pretty clear that these two are equal."},{"Start":"03:50.735 ","End":"03:53.480","Text":"This is equal to this and that\u0027s the first step in"},{"Start":"03:53.480 ","End":"03:57.785","Text":"solving this differential equation, which is exact."},{"Start":"03:57.785 ","End":"04:00.290","Text":"That was the verification stage."},{"Start":"04:00.290 ","End":"04:04.690","Text":"In the next step, we just declare that we\u0027re looking for a solution of this form F(x,"},{"Start":"04:04.690 ","End":"04:06.815","Text":"y) = c an implicit form,"},{"Start":"04:06.815 ","End":"04:10.205","Text":"and we know that it has to satisfy these two conditions."},{"Start":"04:10.205 ","End":"04:15.230","Text":"As I said, the M and the N are the new M and N after the integration factor."},{"Start":"04:15.230 ","End":"04:17.705","Text":"Let\u0027s take the first one,"},{"Start":"04:17.705 ","End":"04:22.720","Text":"F_x equals M so to get from F_x back to F,"},{"Start":"04:22.720 ","End":"04:26.195","Text":"we do the integral dx, the antiderivative."},{"Start":"04:26.195 ","End":"04:31.170","Text":"Now M, I\u0027m just going to copy it from above and here it is,"},{"Start":"04:31.170 ","End":"04:33.360","Text":"just trust me, I copied it."},{"Start":"04:33.360 ","End":"04:35.905","Text":"Now we do the integral,"},{"Start":"04:35.905 ","End":"04:38.650","Text":"but remember that y is like a constant."},{"Start":"04:38.650 ","End":"04:43.540","Text":"The integral of e^x is e^x and the constant sine y just sticks."},{"Start":"04:43.540 ","End":"04:50.125","Text":"Here the integral of minus sine x is cosine x and the 2y just sticks."},{"Start":"04:50.125 ","End":"04:52.450","Text":"But when we have function of"},{"Start":"04:52.450 ","End":"04:56.080","Text":"two variables and they take the integral with respect to one,"},{"Start":"04:56.080 ","End":"04:57.280","Text":"we don\u0027t add a constant,"},{"Start":"04:57.280 ","End":"05:00.190","Text":"we add a general function of the other variable,"},{"Start":"05:00.190 ","End":"05:01.885","Text":"in this case g(y)."},{"Start":"05:01.885 ","End":"05:07.015","Text":"Now we have F, except that we have to find out what g(y) is."},{"Start":"05:07.015 ","End":"05:12.250","Text":"We\u0027ve used this part and we\u0027ll get g by using the other piece of information."},{"Start":"05:12.250 ","End":"05:15.935","Text":"Differentiating with respect to y,"},{"Start":"05:15.935 ","End":"05:17.270","Text":"it\u0027s pretty straightforward,"},{"Start":"05:17.270 ","End":"05:18.859","Text":"I won\u0027t even go into the details."},{"Start":"05:18.859 ","End":"05:20.450","Text":"This gives this, this gives this,"},{"Start":"05:20.450 ","End":"05:23.690","Text":"and g when you differentiate, it gives you g prime."},{"Start":"05:23.690 ","End":"05:28.085","Text":"The other hand is going to equal N and I\u0027m going to copy N from above."},{"Start":"05:28.085 ","End":"05:32.080","Text":"Just scrolled back and copied it here what N is."},{"Start":"05:32.080 ","End":"05:34.185","Text":"Look, if F_y is going to equal N,"},{"Start":"05:34.185 ","End":"05:40.070","Text":"this is going to equal this and like 2 cosine x would cancel with 2 cosine x,"},{"Start":"05:40.070 ","End":"05:44.405","Text":"e^x cosine y would cancel with e^x cosine y"},{"Start":"05:44.405 ","End":"05:49.325","Text":"and we end up with g prime of y is 0 because there\u0027s nothing left."},{"Start":"05:49.325 ","End":"05:52.025","Text":"I compare this right-hand side to this right-hand side,"},{"Start":"05:52.025 ","End":"05:54.365","Text":"everything\u0027s assigned except for the g prime."},{"Start":"05:54.365 ","End":"06:00.215","Text":"If g\u0027(y)=0, then we can take a solution that g(y)=0."},{"Start":"06:00.215 ","End":"06:02.480","Text":"Of course it could be any constant,"},{"Start":"06:02.480 ","End":"06:06.240","Text":"but we save the constant for the N. Now I take this"},{"Start":"06:06.240 ","End":"06:10.860","Text":"0 from g(y) and I see that I have g(y) here."},{"Start":"06:10.860 ","End":"06:13.935","Text":"That\u0027s all I needed to find what F is."},{"Start":"06:13.935 ","End":"06:17.055","Text":"I put g(y)=0, then F is just this,"},{"Start":"06:17.055 ","End":"06:18.975","Text":"which is written down here."},{"Start":"06:18.975 ","End":"06:21.900","Text":"Here\u0027s the last stage where we let F equals the"},{"Start":"06:21.900 ","End":"06:25.730","Text":"constant so this part is equal to a constant and"},{"Start":"06:25.730 ","End":"06:33.330","Text":"this is the general solution to the differential equation and we\u0027re done."}],"Thumbnail":null,"ID":14633},{"Watched":false,"Name":"Exercise 3","Duration":"6m 52s","ChapterTopicVideoID":13835,"CourseChapterTopicPlaylistID":4222,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"Here we have another one of these questions where we have"},{"Start":"00:03.150 ","End":"00:06.645","Text":"a differential equation which is not exact,"},{"Start":"00:06.645 ","End":"00:11.505","Text":"but with the help of an integration factor we can make it exact and solve it."},{"Start":"00:11.505 ","End":"00:14.235","Text":"First, let\u0027s show that this is not exact."},{"Start":"00:14.235 ","End":"00:16.470","Text":"I just rewrote it here."},{"Start":"00:16.470 ","End":"00:22.080","Text":"Some extra brackets thrown in and call the first function M and the second"},{"Start":"00:22.080 ","End":"00:28.665","Text":"function N. We have to show that M with respect to y not equal to N with respect to x."},{"Start":"00:28.665 ","End":"00:32.370","Text":"Let\u0027s start. This with respect to y."},{"Start":"00:32.370 ","End":"00:36.900","Text":"Then x plus 2 is a constant it just stays and sine becomes cosine."},{"Start":"00:36.900 ","End":"00:39.070","Text":"But here with respect to x,"},{"Start":"00:39.070 ","End":"00:41.359","Text":"the cosine y is a constant,"},{"Start":"00:41.359 ","End":"00:42.530","Text":"so a constant times x,"},{"Start":"00:42.530 ","End":"00:44.350","Text":"the answer is just the constant."},{"Start":"00:44.350 ","End":"00:46.520","Text":"We see that we got 2 different functions."},{"Start":"00:46.520 ","End":"00:48.695","Text":"This not equals means the different functions."},{"Start":"00:48.695 ","End":"00:55.770","Text":"Possible for some values of x and y that these would be equal but as a rule they\u0027re not."},{"Start":"00:56.240 ","End":"01:02.455","Text":"Not exact, so let\u0027s try now to multiply by this integration factor."},{"Start":"01:02.455 ","End":"01:06.650","Text":"Here it is in blue multiplying everything on the right of course is 0,"},{"Start":"01:06.650 ","End":"01:10.960","Text":"so that\u0027s multiplied also but it\u0027s still 0."},{"Start":"01:10.960 ","End":"01:15.185","Text":"Just for emphasis throw these inside the brackets."},{"Start":"01:15.185 ","End":"01:18.770","Text":"The x with the x here becomes x^2."},{"Start":"01:18.770 ","End":"01:23.780","Text":"Now we have a new M and N. Officially we\u0027re not supposed to reuse the same letter,"},{"Start":"01:23.780 ","End":"01:25.310","Text":"but there\u0027s no confusion."},{"Start":"01:25.310 ","End":"01:33.002","Text":"Here we have an M and an N and hopefully this time when we try to do M_y and N_x,"},{"Start":"01:33.002 ","End":"01:34.385","Text":"here we got not equal."},{"Start":"01:34.385 ","End":"01:37.760","Text":"Hopefully this time we will get an equality,"},{"Start":"01:37.760 ","End":"01:41.840","Text":"that\u0027s what the integration factor\u0027s meant to do and then when we get an exact equation."},{"Start":"01:41.840 ","End":"01:43.390","Text":"Well, let\u0027s check that."},{"Start":"01:43.390 ","End":"01:45.860","Text":"This function with respect to y."},{"Start":"01:45.860 ","End":"01:48.290","Text":"Well, all this stuff with x is a constant,"},{"Start":"01:48.290 ","End":"01:51.403","Text":"it stays there and the derivative of sine is cosine,"},{"Start":"01:51.403 ","End":"01:52.700","Text":"so that\u0027s that bit."},{"Start":"01:52.700 ","End":"01:54.370","Text":"Now the next bit."},{"Start":"01:54.370 ","End":"01:56.015","Text":"This with respect to x,"},{"Start":"01:56.015 ","End":"02:00.500","Text":"then the part containing the y is constant,"},{"Start":"02:00.500 ","End":"02:05.723","Text":"so the cosine y I can take out first and then I have a product rule on x^2 e^x."},{"Start":"02:05.723 ","End":"02:09.785","Text":"Product rule differentiate x^2 e^x alone."},{"Start":"02:09.785 ","End":"02:14.450","Text":"Then take x^2 as it is and differentiate e^x,"},{"Start":"02:14.450 ","End":"02:16.205","Text":"but that\u0027s still e^x."},{"Start":"02:16.205 ","End":"02:19.010","Text":"Now, at first sight they don\u0027t quite look the"},{"Start":"02:19.010 ","End":"02:21.980","Text":"same but sometimes you have to work a little bit."},{"Start":"02:21.980 ","End":"02:25.550","Text":"Like here I can take e^x outside the brackets,"},{"Start":"02:25.550 ","End":"02:28.055","Text":"and I can take x outside the brackets."},{"Start":"02:28.055 ","End":"02:29.720","Text":"I have cosine y."},{"Start":"02:29.720 ","End":"02:31.115","Text":"Now I take, as I said,"},{"Start":"02:31.115 ","End":"02:35.060","Text":"the x and e^x."},{"Start":"02:35.060 ","End":"02:38.450","Text":"What am I left with? from xe^x, I\u0027ve got 2."},{"Start":"02:38.450 ","End":"02:41.180","Text":"From xe^x here I have x."},{"Start":"02:41.180 ","End":"02:42.860","Text":"It\u0027s not exactly the same as this,"},{"Start":"02:42.860 ","End":"02:45.560","Text":"but you can already see that all the factors here or here,"},{"Start":"02:45.560 ","End":"02:48.010","Text":"that these really are equal."},{"Start":"02:48.010 ","End":"02:52.970","Text":"Yes, we have equality and the equation becomes exact."},{"Start":"02:52.970 ","End":"02:56.000","Text":"We\u0027ve identified this one as being exact,"},{"Start":"02:56.000 ","End":"02:57.965","Text":"now we have to go about solving it."},{"Start":"02:57.965 ","End":"03:00.280","Text":"I\u0027ll do that on the next page."},{"Start":"03:00.280 ","End":"03:07.145","Text":"The solution to the exact equation is going to be of the form f(x, y)=c."},{"Start":"03:07.145 ","End":"03:09.530","Text":"A function given an implicit form,"},{"Start":"03:09.530 ","End":"03:12.515","Text":"but there are 2 conditions it has to satisfy."},{"Start":"03:12.515 ","End":"03:15.230","Text":"Its derivative with respect to x has to be M,"},{"Start":"03:15.230 ","End":"03:17.000","Text":"and with respect to y it has to be N,"},{"Start":"03:17.000 ","End":"03:20.870","Text":"and I\u0027m talking about the new M and N. In most of the exercises,"},{"Start":"03:20.870 ","End":"03:23.365","Text":"I\u0027ve started with this equation."},{"Start":"03:23.365 ","End":"03:28.025","Text":"Then said, f is going to be the integral of M with respect to x."},{"Start":"03:28.025 ","End":"03:29.600","Text":"But in this case,"},{"Start":"03:29.600 ","End":"03:32.960","Text":"it turns out to be quite difficult and this one is going to be quite easy."},{"Start":"03:32.960 ","End":"03:34.490","Text":"Let me just take you back,"},{"Start":"03:34.490 ","End":"03:37.418","Text":"and here we are, and let\u0027s look."},{"Start":"03:37.418 ","End":"03:42.665","Text":"What would be easier to do to integrate this with respect to x or this with respect to y?"},{"Start":"03:42.665 ","End":"03:46.550","Text":"If you look at it, the x part is quite complicated."},{"Start":"03:46.550 ","End":"03:50.000","Text":"I\u0027ve got some x^2 plus 2x and then e^x"},{"Start":"03:50.000 ","End":"03:54.065","Text":"and it\u0027s going to need integration by parts and maybe more than once,"},{"Start":"03:54.065 ","End":"03:56.780","Text":"it looks like not so easy with respect to x."},{"Start":"03:56.780 ","End":"03:59.450","Text":"On the other hand this one with respect to y,"},{"Start":"03:59.450 ","End":"04:03.815","Text":"the x\u0027s I can ignore they\u0027re constants and I\u0027ll just have to integrate cosine."},{"Start":"04:03.815 ","End":"04:08.180","Text":"I would much rather take the integral of this with respect to y."},{"Start":"04:08.180 ","End":"04:14.195","Text":"Let\u0027s go back to where we were and we won\u0027t do this, we\u0027ll do this."},{"Start":"04:14.195 ","End":"04:16.100","Text":"From here we get to here,"},{"Start":"04:16.100 ","End":"04:18.780","Text":"from here take the integral of N with respect to y."},{"Start":"04:18.780 ","End":"04:21.065","Text":"It\u0027s the integral of this with respect to y."},{"Start":"04:21.065 ","End":"04:22.550","Text":"As we noticed before,"},{"Start":"04:22.550 ","End":"04:25.580","Text":"this x^2 e^x is a constant,"},{"Start":"04:25.580 ","End":"04:26.900","Text":"this is far as y goes."},{"Start":"04:26.900 ","End":"04:32.225","Text":"All I have to do is integrate the cosine and get sine and then we don\u0027t add a constant."},{"Start":"04:32.225 ","End":"04:33.920","Text":"As you might remember,"},{"Start":"04:33.920 ","End":"04:35.780","Text":"if we\u0027re doing an integral with respect to y,"},{"Start":"04:35.780 ","End":"04:38.630","Text":"a constant is just a general function of x."},{"Start":"04:38.630 ","End":"04:42.620","Text":"Because when you differentiate this with respect to y you get 0."},{"Start":"04:42.620 ","End":"04:46.070","Text":"Now what we\u0027re missing is we have f but we don\u0027t have"},{"Start":"04:46.070 ","End":"04:51.850","Text":"g. We get our unknown function by going to the other equation that we didn\u0027t use."},{"Start":"04:51.850 ","End":"04:57.020","Text":"This time we\u0027re going to use this one and differentiate this with respect to x."},{"Start":"04:57.020 ","End":"05:00.320","Text":"The first term, sine y is a constant,"},{"Start":"05:00.320 ","End":"05:01.700","Text":"so I take it first."},{"Start":"05:01.700 ","End":"05:04.609","Text":"This, I do buy the product rule of this times this."},{"Start":"05:04.609 ","End":"05:06.994","Text":"Derivative of the first times the second,"},{"Start":"05:06.994 ","End":"05:15.080","Text":"plus vice versa and in the end I have to also differentiate g with respect to x as g\u0027."},{"Start":"05:15.080 ","End":"05:17.330","Text":"That\u0027s on the 1 hand,"},{"Start":"05:17.330 ","End":"05:19.055","Text":"that\u0027s f with respect to x,"},{"Start":"05:19.055 ","End":"05:24.320","Text":"on the other hand is equal to M. Before that we just do a bit of simplification."},{"Start":"05:24.320 ","End":"05:30.440","Text":"See I have xe^x here and I have an x from here and then e^x here."},{"Start":"05:30.440 ","End":"05:32.090","Text":"After I take that out the brackets,"},{"Start":"05:32.090 ","End":"05:34.295","Text":"I\u0027m left here with just the 2,"},{"Start":"05:34.295 ","End":"05:37.230","Text":"here with x and I reverse the order,"},{"Start":"05:37.230 ","End":"05:38.900","Text":"so this is what we get."},{"Start":"05:38.900 ","End":"05:43.550","Text":"Now when we say that this is equal to M,"},{"Start":"05:43.550 ","End":"05:47.390","Text":"this I\u0027m copying from here and M,"},{"Start":"05:47.390 ","End":"05:50.480","Text":"I just went up and copied what it was."},{"Start":"05:50.480 ","End":"05:55.325","Text":"This is what it was I just don\u0027t want to scroll back up again. Here it is."},{"Start":"05:55.325 ","End":"05:57.335","Text":"If you look at these 2,"},{"Start":"05:57.335 ","End":"06:01.820","Text":"everything is the same except that here there\u0027s a g\u0027(x) and here there isn\u0027t,"},{"Start":"06:01.820 ","End":"06:06.860","Text":"which means that g\u0027(x)=0."},{"Start":"06:06.860 ","End":"06:11.470","Text":"Then we can get g by integrating 0."},{"Start":"06:11.470 ","End":"06:13.335","Text":"The integral of 0 is also 0."},{"Start":"06:13.335 ","End":"06:17.330","Text":"We could add a constant but the constant we take care of ones at the end."},{"Start":"06:17.330 ","End":"06:19.220","Text":"Now that we have g,"},{"Start":"06:19.220 ","End":"06:23.180","Text":"we can plug it back in just off screen here,"},{"Start":"06:23.180 ","End":"06:26.035","Text":"I plugin that g is 0 here,"},{"Start":"06:26.035 ","End":"06:28.755","Text":"this 0 and this g(x)."},{"Start":"06:28.755 ","End":"06:31.955","Text":"F is just x^2 e^x sine y."},{"Start":"06:31.955 ","End":"06:36.665","Text":"Not in quite the same order as I said it but yeah."},{"Start":"06:36.665 ","End":"06:40.940","Text":"Finally we just let this equal to a constant, and this is it."},{"Start":"06:40.940 ","End":"06:44.765","Text":"This is the general solution for whatever constant you put here,"},{"Start":"06:44.765 ","End":"06:46.970","Text":"it\u0027s infinite number of solutions,"},{"Start":"06:46.970 ","End":"06:50.720","Text":"but this is what it is where c is any constant."},{"Start":"06:50.720 ","End":"06:52.740","Text":"We are done."}],"Thumbnail":null,"ID":14634},{"Watched":false,"Name":"Exercise 4","Duration":"6m 29s","ChapterTopicVideoID":13836,"CourseChapterTopicPlaylistID":4222,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.810","Text":"In this exercise, we have to solve this differential equation and we\u0027re"},{"Start":"00:03.810 ","End":"00:07.320","Text":"not told that it\u0027s exact or anything."},{"Start":"00:07.320 ","End":"00:09.285","Text":"Well, let\u0027s see if it is exact."},{"Start":"00:09.285 ","End":"00:13.260","Text":"First thing to do is just to label the functions M and N."},{"Start":"00:13.260 ","End":"00:17.640","Text":"Then we want to see if M with respect to y=N with respect to x,"},{"Start":"00:17.640 ","End":"00:20.280","Text":"partial derivatives I mean, and no,"},{"Start":"00:20.280 ","End":"00:24.045","Text":"they\u0027re not equal because this with respect to y, is to 2y."},{"Start":"00:24.045 ","End":"00:25.695","Text":"y just appears here,"},{"Start":"00:25.695 ","End":"00:27.615","Text":"and here with respect to x,"},{"Start":"00:27.615 ","End":"00:29.480","Text":"x is the variable,"},{"Start":"00:29.480 ","End":"00:31.655","Text":"y is the constant, so we get just the constant."},{"Start":"00:31.655 ","End":"00:34.400","Text":"Anyway, 2y is not equal to y as a function,"},{"Start":"00:34.400 ","End":"00:38.470","Text":"you\u0027re right, they\u0027re not always equal, so not exact."},{"Start":"00:38.470 ","End":"00:44.149","Text":"Our next hope might be to find an integrating factor to make it exact."},{"Start":"00:44.149 ","End":"00:46.550","Text":"If you go back to the tutorial,"},{"Start":"00:46.550 ","End":"00:50.510","Text":"one of the ways is to subtract this minus this,"},{"Start":"00:50.510 ","End":"00:53.510","Text":"and then divide it either by M or by N,"},{"Start":"00:53.510 ","End":"00:54.740","Text":"and if you get just a function of x,"},{"Start":"00:54.740 ","End":"00:56.345","Text":"so just a function of y."},{"Start":"00:56.345 ","End":"00:58.700","Text":"We\u0027ll try 1, if it doesn\u0027t work, we\u0027ll try the other."},{"Start":"00:58.700 ","End":"01:05.060","Text":"We\u0027ll try this minus this over n. What we get is,"},{"Start":"01:05.060 ","End":"01:06.465","Text":"this is 2y,"},{"Start":"01:06.465 ","End":"01:09.690","Text":"this is y, N is xy."},{"Start":"01:09.690 ","End":"01:12.105","Text":"We get this expression."},{"Start":"01:12.105 ","End":"01:14.900","Text":"Then if I just simplify it,"},{"Start":"01:14.900 ","End":"01:17.788","Text":"I see that everything can be divisible by y,"},{"Start":"01:17.788 ","End":"01:19.460","Text":"and that\u0027s what I get."},{"Start":"01:19.460 ","End":"01:25.280","Text":"First of all, do the subtraction and then I get 2y minus y is y over xy."},{"Start":"01:25.280 ","End":"01:27.935","Text":"Then I cancel the y and I\u0027m left with is 1 here."},{"Start":"01:27.935 ","End":"01:32.750","Text":"In short, I get that this over this is a function of x."},{"Start":"01:32.750 ","End":"01:38.675","Text":"When we get that this expression is some function of just x with no ys in it,"},{"Start":"01:38.675 ","End":"01:41.450","Text":"then if you refer to the notes in the tutorial,"},{"Start":"01:41.450 ","End":"01:47.110","Text":"it says that e to the power of the integral of this would be an integrating factor."},{"Start":"01:47.110 ","End":"01:51.050","Text":"We figure out e to the power of integral of f(x),"},{"Start":"01:51.050 ","End":"01:53.150","Text":"e to the power of integral of this,"},{"Start":"01:53.150 ","End":"01:57.515","Text":"e to the power of natural log(x)."},{"Start":"01:57.515 ","End":"02:00.200","Text":"I didn\u0027t want to mess it up with taking the absolute value."},{"Start":"02:00.200 ","End":"02:02.570","Text":"Let\u0027s just say that x is positive."},{"Start":"02:02.570 ","End":"02:04.460","Text":"We get e to the power of natural log."},{"Start":"02:04.460 ","End":"02:05.573","Text":"They cancel each other out,"},{"Start":"02:05.573 ","End":"02:08.540","Text":"so we\u0027ve got x as an integrating factor."},{"Start":"02:08.540 ","End":"02:12.020","Text":"What this means is that I can take this x,"},{"Start":"02:12.020 ","End":"02:13.730","Text":"I\u0027ll go back up and show you."},{"Start":"02:13.730 ","End":"02:17.600","Text":"I\u0027m going to take this equation and"},{"Start":"02:17.600 ","End":"02:22.970","Text":"multiply it by x. I have to get something off the screen."},{"Start":"02:22.970 ","End":"02:25.445","Text":"Anyway, I copied what was up there,"},{"Start":"02:25.445 ","End":"02:28.750","Text":"and then the x here in blue."},{"Start":"02:28.750 ","End":"02:32.540","Text":"This is what we get after multiplying by the integrating factor."},{"Start":"02:32.540 ","End":"02:35.570","Text":"Now I\u0027m going to take this whole thing as my new M,"},{"Start":"02:35.570 ","End":"02:37.850","Text":"not the same M and N as it was before."},{"Start":"02:37.850 ","End":"02:39.320","Text":"It\u0027s like, okay, up to here,"},{"Start":"02:39.320 ","End":"02:40.865","Text":"M and N meant one thing."},{"Start":"02:40.865 ","End":"02:44.900","Text":"Now we have a new M and a new N. Hopefully this"},{"Start":"02:44.900 ","End":"02:50.105","Text":"time it will be an exact equation because that\u0027s the idea of an integrating factor."},{"Start":"02:50.105 ","End":"02:52.280","Text":"Let\u0027s see. First of all,"},{"Start":"02:52.280 ","End":"02:54.545","Text":"I\u0027ll just multiply everything out,"},{"Start":"02:54.545 ","End":"02:56.900","Text":"x times x^2, and so on."},{"Start":"02:56.900 ","End":"03:00.095","Text":"This is x^3, y^2x and so on."},{"Start":"03:00.095 ","End":"03:03.830","Text":"This is my M and this is my N. Now I need to see if this with"},{"Start":"03:03.830 ","End":"03:07.360","Text":"respect to y is equal to this with respect to x,"},{"Start":"03:07.360 ","End":"03:08.630","Text":"and the answer is yes,"},{"Start":"03:08.630 ","End":"03:11.380","Text":"because this with respect to y,"},{"Start":"03:11.380 ","End":"03:13.440","Text":"this and this don\u0027t count, they\u0027re constants,"},{"Start":"03:13.440 ","End":"03:15.585","Text":"and so I just get to 2xy,"},{"Start":"03:15.585 ","End":"03:18.530","Text":"and this if I differentiate with respect to x,"},{"Start":"03:18.530 ","End":"03:20.405","Text":"I also get 2xy."},{"Start":"03:20.405 ","End":"03:25.835","Text":"So yes, this new equation is now exact."},{"Start":"03:25.835 ","End":"03:31.400","Text":"We use our technique for solving exact equations and I\u0027ll continue on the next page."},{"Start":"03:31.400 ","End":"03:39.270","Text":"If you remember, we look for a function of the form f(x,y)=c."},{"Start":"03:39.950 ","End":"03:45.315","Text":"The conditions are that f with respect to x=M,"},{"Start":"03:45.315 ","End":"03:47.415","Text":"that\u0027s the new M,"},{"Start":"03:47.415 ","End":"03:50.865","Text":"and f with respect to y=N."},{"Start":"03:50.865 ","End":"03:53.400","Text":"We\u0027ve done this several times, you should know this."},{"Start":"03:53.400 ","End":"03:55.370","Text":"We\u0027ll start with either one of these,"},{"Start":"03:55.370 ","End":"03:57.290","Text":"but unless we run into any difficulties,"},{"Start":"03:57.290 ","End":"03:58.640","Text":"I\u0027ll start with the first one."},{"Start":"03:58.640 ","End":"04:00.410","Text":"If it gets difficult as an integral,"},{"Start":"04:00.410 ","End":"04:01.785","Text":"we\u0027ll go for the other one."},{"Start":"04:01.785 ","End":"04:03.740","Text":"If this is true,"},{"Start":"04:03.740 ","End":"04:07.485","Text":"then f is the integral of M with respect to x."},{"Start":"04:07.485 ","End":"04:11.380","Text":"But M is this just copied it from what was before."},{"Start":"04:11.380 ","End":"04:13.625","Text":"That\u0027s a straightforward integral."},{"Start":"04:13.625 ","End":"04:16.160","Text":"Use decimals for a change and not fractions,"},{"Start":"04:16.160 ","End":"04:18.215","Text":"x^4 over 4,"},{"Start":"04:18.215 ","End":"04:21.080","Text":"because I\u0027m running a quarter, I\u0027ll write 0.25."},{"Start":"04:21.080 ","End":"04:27.175","Text":"Here, the y^2 is a constant and then I get x^2 over 2 or 0.5x."},{"Start":"04:27.175 ","End":"04:31.580","Text":"Here I will use fractions because 1 over 3 is not very good"},{"Start":"04:31.580 ","End":"04:35.795","Text":"as a decimal and trivial matters don\u0027t really matter."},{"Start":"04:35.795 ","End":"04:37.190","Text":"Also, instead of a constant,"},{"Start":"04:37.190 ","End":"04:39.755","Text":"we have a general function of y,"},{"Start":"04:39.755 ","End":"04:42.470","Text":"because the integral is produced back to x."},{"Start":"04:42.470 ","End":"04:46.325","Text":"Now we need to find what g(y) is."},{"Start":"04:46.325 ","End":"04:51.700","Text":"The way we do that is by utilizing the unused equation, which is this."},{"Start":"04:51.700 ","End":"04:56.090","Text":"I differentiate this with respect to y and then equate it to N,"},{"Start":"04:56.090 ","End":"04:58.895","Text":"the left-hand side, f with respect to y."},{"Start":"04:58.895 ","End":"05:02.315","Text":"This is nothing because it\u0027s just contains x."},{"Start":"05:02.315 ","End":"05:04.250","Text":"This was respect to y,"},{"Start":"05:04.250 ","End":"05:07.115","Text":"2 times 1 over 2 is 1,"},{"Start":"05:07.115 ","End":"05:09.485","Text":"and it\u0027s x^2y, that\u0027s here."},{"Start":"05:09.485 ","End":"05:11.630","Text":"This thing also just contains x,"},{"Start":"05:11.630 ","End":"05:12.875","Text":"so that disappears,"},{"Start":"05:12.875 ","End":"05:15.445","Text":"and this g prime of y."},{"Start":"05:15.445 ","End":"05:18.395","Text":"On the other hand, N from above,"},{"Start":"05:18.395 ","End":"05:20.420","Text":"you might not remember, anyway,"},{"Start":"05:20.420 ","End":"05:25.084","Text":"I can tell you the N was equal to x^2y."},{"Start":"05:25.084 ","End":"05:27.380","Text":"Go back and check if you don\u0027t believe me."},{"Start":"05:27.380 ","End":"05:31.350","Text":"Now, this equation has got to equal this."},{"Start":"05:31.350 ","End":"05:36.875","Text":"We get x^2y plus g prime of y=x^2y."},{"Start":"05:36.875 ","End":"05:38.690","Text":"This cancels with this,"},{"Start":"05:38.690 ","End":"05:41.525","Text":"so g prime of y is just 0."},{"Start":"05:41.525 ","End":"05:42.815","Text":"There, I\u0027ve written it,"},{"Start":"05:42.815 ","End":"05:45.635","Text":"and then g is just the integral of 0."},{"Start":"05:45.635 ","End":"05:48.590","Text":"Here, I don\u0027t need the constants at this stage"},{"Start":"05:48.590 ","End":"05:51.955","Text":"because we\u0027re going to have a constant at the end."},{"Start":"05:51.955 ","End":"05:55.820","Text":"Now that I have g, g is 0,"},{"Start":"05:55.820 ","End":"05:58.490","Text":"I can substitute that in here,"},{"Start":"05:58.490 ","End":"06:01.400","Text":"and that will give me what f is exactly."},{"Start":"06:01.400 ","End":"06:05.270","Text":"It\u0027s just these terms without the g of y because that\u0027s 0,"},{"Start":"06:05.270 ","End":"06:07.100","Text":"so just copy that there."},{"Start":"06:07.100 ","End":"06:13.550","Text":"The final thing to do is to set this f to be equal to a constant c,"},{"Start":"06:13.550 ","End":"06:16.245","Text":"a general constant, and here we are,"},{"Start":"06:16.245 ","End":"06:22.280","Text":"this equals c, and this is our general solution to the differential equation,"},{"Start":"06:22.280 ","End":"06:28.889","Text":"which wasn\u0027t the exact after multiplying by an integrating factor."}],"Thumbnail":null,"ID":14635},{"Watched":false,"Name":"Exercise 5","Duration":"8m 53s","ChapterTopicVideoID":13837,"CourseChapterTopicPlaylistID":4222,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"In this exercise, we have this here differential equation to solve,"},{"Start":"00:04.050 ","End":"00:06.060","Text":"and we might get lucky,"},{"Start":"00:06.060 ","End":"00:08.085","Text":"let\u0027s check if it\u0027s exact."},{"Start":"00:08.085 ","End":"00:12.990","Text":"Let\u0027s start by labeling this part m and the other part n. Now"},{"Start":"00:12.990 ","End":"00:15.090","Text":"we need to check the partial derivatives"},{"Start":"00:15.090 ","End":"00:18.540","Text":"this with respect to y and this with respect to x."},{"Start":"00:18.540 ","End":"00:22.140","Text":"We see that they\u0027re not equal because this with respect to y,"},{"Start":"00:22.140 ","End":"00:24.180","Text":"from here I get minus 2y,"},{"Start":"00:24.180 ","End":"00:27.050","Text":"and from here I get 0 because there is no x here,"},{"Start":"00:27.050 ","End":"00:29.190","Text":"and so these are not the same function,"},{"Start":"00:29.190 ","End":"00:31.305","Text":"so not the exact."},{"Start":"00:31.305 ","End":"00:35.010","Text":"Maybe I\u0027ll try finding integrating factor."},{"Start":"00:35.010 ","End":"00:39.620","Text":"There are several techniques that I mentioned in the tutorial and one of them,"},{"Start":"00:39.620 ","End":"00:44.015","Text":"which I did in an earlier exercise was to compute this expression,"},{"Start":"00:44.015 ","End":"00:47.430","Text":"this minus this over n,"},{"Start":"00:47.430 ","End":"00:51.380","Text":"and if it comes out to be a function just of x without any y,"},{"Start":"00:51.380 ","End":"00:54.965","Text":"then we can use it to find an integrating factor."},{"Start":"00:54.965 ","End":"00:59.240","Text":"Or if I put m here and I get a function of just y,"},{"Start":"00:59.240 ","End":"01:02.960","Text":"that will also be good, so let\u0027s see."},{"Start":"01:02.960 ","End":"01:08.660","Text":"We get this minus this over this,"},{"Start":"01:08.660 ","End":"01:11.525","Text":"so it\u0027s minus 2y-0 over y,"},{"Start":"01:11.525 ","End":"01:14.980","Text":"and the y cancels,"},{"Start":"01:14.980 ","End":"01:18.260","Text":"and it comes out to be the constant minus 2."},{"Start":"01:18.260 ","End":"01:20.900","Text":"But I can consider that a function of x, call it f(x),"},{"Start":"01:20.900 ","End":"01:22.250","Text":"if I want to,"},{"Start":"01:22.250 ","End":"01:24.580","Text":"as long as there is no y here,"},{"Start":"01:24.580 ","End":"01:26.990","Text":"doesn\u0027t matter that x doesn\u0027t explicitly appear in,"},{"Start":"01:26.990 ","End":"01:28.400","Text":"if this really bothers you,"},{"Start":"01:28.400 ","End":"01:31.820","Text":"you could always write minus 2+0x,"},{"Start":"01:31.820 ","End":"01:34.835","Text":"yes sure, it\u0027s a function of x."},{"Start":"01:34.835 ","End":"01:39.050","Text":"Anyway, I don\u0027t think we need to do that, I was just saying."},{"Start":"01:39.050 ","End":"01:43.850","Text":"We can use this to get us an integrating factor not in itself,"},{"Start":"01:43.850 ","End":"01:47.065","Text":"we have to take e the power of the integral of this."},{"Start":"01:47.065 ","End":"01:49.960","Text":"So e to the power of the integral of this,"},{"Start":"01:49.960 ","End":"01:52.865","Text":"is e to the power of the integral of minus 2,"},{"Start":"01:52.865 ","End":"01:55.460","Text":"and that\u0027s e to the minus 2x."},{"Start":"01:55.460 ","End":"01:59.360","Text":"We could use e to the minus 2x as"},{"Start":"01:59.360 ","End":"02:04.130","Text":"an integrating factor and then multiply it by the original equations."},{"Start":"02:04.130 ","End":"02:05.720","Text":"Just copy the original equation,"},{"Start":"02:05.720 ","End":"02:08.150","Text":"then put the e to the minus 2x in here."},{"Start":"02:08.150 ","End":"02:10.100","Text":"Now we have a new m and a new n,"},{"Start":"02:10.100 ","End":"02:15.050","Text":"and this time it really is exact as we expected."},{"Start":"02:15.050 ","End":"02:18.830","Text":"Just look, m with respect to y,"},{"Start":"02:18.830 ","End":"02:22.560","Text":"is this e to the minus 2x is a constant,"},{"Start":"02:22.560 ","End":"02:25.555","Text":"this with respect to y is minus 2y."},{"Start":"02:25.555 ","End":"02:29.195","Text":"But then again, if I differentiate n with respect 2x,"},{"Start":"02:29.195 ","End":"02:35.750","Text":"I also get this expression times the inner derivative of the e to the minus 2x,"},{"Start":"02:35.750 ","End":"02:37.370","Text":"which is minus 2,"},{"Start":"02:37.370 ","End":"02:39.395","Text":"and I can also put the y in front,"},{"Start":"02:39.395 ","End":"02:43.340","Text":"so it is really an exact equation."},{"Start":"02:43.340 ","End":"02:45.455","Text":"We know how to solve this."},{"Start":"02:45.455 ","End":"02:48.230","Text":"I\u0027m not going to do it this time."},{"Start":"02:48.230 ","End":"02:49.880","Text":"I\u0027m going to make it up to you though."},{"Start":"02:49.880 ","End":"02:52.925","Text":"I mean, now you know how to do it and we\u0027re not learning anything new."},{"Start":"02:52.925 ","End":"02:58.140","Text":"What I am going to do is show you an alternative method to teach you something new,"},{"Start":"02:58.140 ","End":"03:02.540","Text":"and I\u0027ll take that new method and we will solve it to the end."},{"Start":"03:02.540 ","End":"03:07.685","Text":"You will walk away from this exercise with the solution of the example."},{"Start":"03:07.685 ","End":"03:10.015","Text":"Let me go to a new page."},{"Start":"03:10.015 ","End":"03:12.470","Text":"Here\u0027s our original exercise,"},{"Start":"03:12.470 ","End":"03:14.750","Text":"and this is the alternative method."},{"Start":"03:14.750 ","End":"03:17.345","Text":"If you remember in the tutorials,"},{"Start":"03:17.345 ","End":"03:24.305","Text":"we had a little table that if you see this in some form,"},{"Start":"03:24.305 ","End":"03:28.770","Text":"you might try 1 of these and so on and so on."},{"Start":"03:28.770 ","End":"03:32.855","Text":"What we can do is if I slightly rewrite this,"},{"Start":"03:32.855 ","End":"03:38.255","Text":"I can take this x with this dx and then plus ydy,"},{"Start":"03:38.255 ","End":"03:43.780","Text":"and what I\u0027m left with is minus x^2 plus y^2 dx."},{"Start":"03:43.780 ","End":"03:46.700","Text":"The thing is I have this expression here,"},{"Start":"03:46.700 ","End":"03:52.070","Text":"which matches with this expression here."},{"Start":"03:52.070 ","End":"03:56.205","Text":"Sorry, not that 1, this 1 here."},{"Start":"03:56.205 ","End":"04:00.255","Text":"What this says is to try 1 over x^2"},{"Start":"04:00.255 ","End":"04:06.005","Text":"plus y^2 to some power of n as the integrating factor."},{"Start":"04:06.005 ","End":"04:08.920","Text":"Now which value of n should I choose?"},{"Start":"04:08.920 ","End":"04:11.510","Text":"I\u0027m going to be multiplying everything by this,"},{"Start":"04:11.510 ","End":"04:16.640","Text":"so my suggestion is that here we have x^2 plus y^2 to the power of 1."},{"Start":"04:16.640 ","End":"04:21.120","Text":"If I put n here equals 1 and it\u0027s only the denominator will cancel out,"},{"Start":"04:21.120 ","End":"04:24.325","Text":"and it might just work out nicely for us."},{"Start":"04:24.325 ","End":"04:28.115","Text":"I\u0027m going to have to scroll so I\u0027ll lose something."},{"Start":"04:28.115 ","End":"04:30.310","Text":"I can always go back and copy,"},{"Start":"04:30.310 ","End":"04:38.285","Text":"so I copied the original equation that\u0027s in black and integrating factor in blue here."},{"Start":"04:38.285 ","End":"04:40.610","Text":"If I multiply this out,"},{"Start":"04:40.610 ","End":"04:43.220","Text":"then let\u0027s see what we get with the dx."},{"Start":"04:43.220 ","End":"04:46.940","Text":"We get x over this x^2 plus y^2."},{"Start":"04:46.940 ","End":"04:49.610","Text":"Then these 2, it\u0027s like minus brackets,"},{"Start":"04:49.610 ","End":"04:51.470","Text":"x^2 plus y^2,"},{"Start":"04:51.470 ","End":"04:54.380","Text":"so that\u0027s minus 1."},{"Start":"04:54.380 ","End":"04:58.460","Text":"Here, just put the y in the numerator here,"},{"Start":"04:58.460 ","End":"05:00.200","Text":"so this is the equation I get,"},{"Start":"05:00.200 ","End":"05:01.430","Text":"this my new m,"},{"Start":"05:01.430 ","End":"05:05.570","Text":"this is my new n. To see that, we\u0027re okay,"},{"Start":"05:05.570 ","End":"05:09.995","Text":"we need to check what is M_y and what is N_x,"},{"Start":"05:09.995 ","End":"05:14.630","Text":"and if they\u0027re equal, we\u0027ve got an exact equation, so let\u0027s see."},{"Start":"05:14.630 ","End":"05:16.915","Text":"Turns out they are equal."},{"Start":"05:16.915 ","End":"05:19.910","Text":"I won\u0027t go into all the details of the calculation you know how to"},{"Start":"05:19.910 ","End":"05:24.140","Text":"differentiate this with respect to y and this with respect to x,"},{"Start":"05:24.140 ","End":"05:26.870","Text":"and check it, in both cases you get this."},{"Start":"05:26.870 ","End":"05:30.985","Text":"We now have an exact equation here,"},{"Start":"05:30.985 ","End":"05:33.950","Text":"and this time I will go ahead and solve it."},{"Start":"05:33.950 ","End":"05:35.840","Text":"You remember the technique,"},{"Start":"05:35.840 ","End":"05:38.195","Text":"but you know what, just to be safe, I\u0027ll remind you,"},{"Start":"05:38.195 ","End":"05:44.690","Text":"we\u0027re looking for a function f of x and y equals c as the solution,"},{"Start":"05:44.690 ","End":"05:49.970","Text":"and we know that this has to satisfy the 2 equations."},{"Start":"05:49.970 ","End":"05:52.670","Text":"This with respect to x is m,"},{"Start":"05:52.670 ","End":"05:58.700","Text":"and function with respect to y has got to equal n. I\u0027m going to start with this 1."},{"Start":"05:58.700 ","End":"06:00.950","Text":"We saw that sometimes this gets a bit difficult,"},{"Start":"06:00.950 ","End":"06:02.660","Text":"so we go for that 1 first,"},{"Start":"06:02.660 ","End":"06:04.760","Text":"but usually it just first come first serve."},{"Start":"06:04.760 ","End":"06:06.295","Text":"Let\u0027s try this one,"},{"Start":"06:06.295 ","End":"06:08.750","Text":"so f is the integral."},{"Start":"06:08.750 ","End":"06:15.410","Text":"Now this is what we have as m. You can just rewind and see that I copied it correctly."},{"Start":"06:15.410 ","End":"06:17.525","Text":"The integral of this dx,"},{"Start":"06:17.525 ","End":"06:19.970","Text":"I\u0027ll split it up with this minus into"},{"Start":"06:19.970 ","End":"06:23.375","Text":"2 separate integrals because this is an easy integral."},{"Start":"06:23.375 ","End":"06:26.180","Text":"Here it\u0027s also going to be easy because I roughly have"},{"Start":"06:26.180 ","End":"06:29.555","Text":"the derivative of the denominator on the numerator."},{"Start":"06:29.555 ","End":"06:32.450","Text":"I\u0027m sorry, roughly what I\u0027m missing is a factor of 2,"},{"Start":"06:32.450 ","End":"06:34.970","Text":"so if I put the 2 here and a 2 here,"},{"Start":"06:34.970 ","End":"06:36.395","Text":"no harm done,"},{"Start":"06:36.395 ","End":"06:38.900","Text":"and now I have the derivative of this here,"},{"Start":"06:38.900 ","End":"06:42.130","Text":"so this fits 1 of these patterns within natural log,"},{"Start":"06:42.130 ","End":"06:45.740","Text":"and the integral is the 1/2 from here."},{"Start":"06:45.740 ","End":"06:48.560","Text":"Here I have the natural log of the denominator."},{"Start":"06:48.560 ","End":"06:53.060","Text":"I don\u0027t need absolute value because x^2 plus y^2 can\u0027t be negative."},{"Start":"06:53.060 ","End":"06:55.250","Text":"From this part, the minus 1,"},{"Start":"06:55.250 ","End":"06:56.900","Text":"I got minus x."},{"Start":"06:56.900 ","End":"07:01.175","Text":"Since the integral is dx I have to add a general function of y,"},{"Start":"07:01.175 ","End":"07:06.600","Text":"and the way we find y is by using the twin equation."},{"Start":"07:06.600 ","End":"07:08.820","Text":"Not only is fx equal to m,"},{"Start":"07:08.820 ","End":"07:10.430","Text":"here it is, I still have it written."},{"Start":"07:10.430 ","End":"07:14.330","Text":"I need to use the other one with respect to y is n,"},{"Start":"07:14.330 ","End":"07:18.680","Text":"so if I differentiate this with respect to y and then compare,"},{"Start":"07:18.680 ","End":"07:24.755","Text":"I can get g. The derivative of this with respect to y,"},{"Start":"07:24.755 ","End":"07:29.315","Text":"from the natural logarithm I get 1 over x^2 plus y^2."},{"Start":"07:29.315 ","End":"07:31.325","Text":"The half was there before,"},{"Start":"07:31.325 ","End":"07:34.460","Text":"and 2y is the inner derivative,"},{"Start":"07:34.460 ","End":"07:38.180","Text":"minus x gives me nothing with respect to y,"},{"Start":"07:38.180 ","End":"07:40.580","Text":"and g gives me g\u0027."},{"Start":"07:40.580 ","End":"07:44.795","Text":"All this has got to equal our function n,"},{"Start":"07:44.795 ","End":"07:48.280","Text":"which I\u0027m going to copy from above."},{"Start":"07:48.280 ","End":"07:50.840","Text":"At least I will in a moment because maybe simplifying"},{"Start":"07:50.840 ","End":"07:53.420","Text":"this will be better the 2 with the 2 cancels,"},{"Start":"07:53.420 ","End":"07:56.800","Text":"and just take the y in the numerator."},{"Start":"07:56.800 ","End":"07:59.925","Text":"If you go back and look what n was,"},{"Start":"07:59.925 ","End":"08:01.350","Text":"n was this,"},{"Start":"08:01.350 ","End":"08:02.970","Text":"so from these 2,"},{"Start":"08:02.970 ","End":"08:05.805","Text":"we see that g\u0027 has to be 0,"},{"Start":"08:05.805 ","End":"08:08.105","Text":"and if g\u0027 is 0,"},{"Start":"08:08.105 ","End":"08:11.300","Text":"then g is the integral of 0, which is also 0."},{"Start":"08:11.300 ","End":"08:15.760","Text":"We\u0027re not adding a constant because we\u0027re going to take care of the constant at the end."},{"Start":"08:15.760 ","End":"08:17.610","Text":"Now, what is this g good for?"},{"Start":"08:17.610 ","End":"08:19.320","Text":"Well, I\u0027ll just scroll back a bit,"},{"Start":"08:19.320 ","End":"08:21.855","Text":"see how far do I need to go?"},{"Start":"08:21.855 ","End":"08:23.225","Text":"There it is."},{"Start":"08:23.225 ","End":"08:24.980","Text":"This is what I was looking for."},{"Start":"08:24.980 ","End":"08:30.225","Text":"I\u0027ve got f, once I have g. Now I have that g is 0,"},{"Start":"08:30.225 ","End":"08:34.220","Text":"so all I have to do is take this bit 0,"},{"Start":"08:34.220 ","End":"08:35.445","Text":"just throw it out,"},{"Start":"08:35.445 ","End":"08:38.625","Text":"and that will give me what f is,"},{"Start":"08:38.625 ","End":"08:42.510","Text":"so here\u0027s f that we saw before,"},{"Start":"08:42.510 ","End":"08:49.905","Text":"and the last thing to do is to make this equal to c. This equals c,"},{"Start":"08:49.905 ","End":"08:51.630","Text":"and that is the answer,"},{"Start":"08:51.630 ","End":"08:54.160","Text":"so we are done."}],"Thumbnail":null,"ID":14636},{"Watched":false,"Name":"Exercise 6","Duration":"7m 3s","ChapterTopicVideoID":13838,"CourseChapterTopicPlaylistID":4222,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.790","Text":"We have here a differential equation to solve."},{"Start":"00:02.790 ","End":"00:05.775","Text":"First attempt let\u0027s see if it\u0027s exact."},{"Start":"00:05.775 ","End":"00:08.610","Text":"I copied it here and then we label"},{"Start":"00:08.610 ","End":"00:14.820","Text":"the function that goes with the dx is M and the one that goes with dy,"},{"Start":"00:14.820 ","End":"00:18.180","Text":"we call that N of x and y and we have to"},{"Start":"00:18.180 ","End":"00:21.690","Text":"check if the partial derivatives of this with respect to y,"},{"Start":"00:21.690 ","End":"00:24.975","Text":"and this with respect to x are the same or not."},{"Start":"00:24.975 ","End":"00:27.270","Text":"First, this with respect to y,"},{"Start":"00:27.270 ","End":"00:29.130","Text":"so x is a constant."},{"Start":"00:29.130 ","End":"00:35.400","Text":"We get 6xy^2 from here 4y^3 and from this one with respect to x,"},{"Start":"00:35.400 ","End":"00:37.940","Text":"the 2 minus 2 drops out."},{"Start":"00:37.940 ","End":"00:40.400","Text":"In this case, x is the constant,"},{"Start":"00:40.400 ","End":"00:42.275","Text":"so it\u0027s coefficient is y cubed,"},{"Start":"00:42.275 ","End":"00:47.910","Text":"that\u0027s what we get and these 2 are definitely not equal."},{"Start":"00:47.910 ","End":"00:49.370","Text":"They\u0027re are not the same function."},{"Start":"00:49.370 ","End":"00:52.760","Text":"They could be equal for some particular value like if y was 0,"},{"Start":"00:52.760 ","End":"00:59.400","Text":"but it\u0027s not the same function and so this is not an exact equation. Too bad."},{"Start":"00:59.400 ","End":"01:04.150","Text":"But, next best thing we could do is try and look for an integration factor."},{"Start":"01:04.150 ","End":"01:07.640","Text":"In the tutorial, I gave you some pointers as to how"},{"Start":"01:07.640 ","End":"01:11.450","Text":"to go about looking for 1 in the case that we\u0027re not exact."},{"Start":"01:11.450 ","End":"01:16.100","Text":"One of the tricks I showed you there was to figure out they\u0027re equals,"},{"Start":"01:16.100 ","End":"01:22.040","Text":"to take the difference and divide by either M or N. If we divide by M,"},{"Start":"01:22.040 ","End":"01:24.980","Text":"we\u0027re looking to get a function of y."},{"Start":"01:24.980 ","End":"01:28.565","Text":"If we divide by N, we\u0027re looking to get a function of just x."},{"Start":"01:28.565 ","End":"01:32.845","Text":"In this case, I tried them both and I don\u0027t know if this is the one that works."},{"Start":"01:32.845 ","End":"01:36.785","Text":"If I simplify the right-hand side,"},{"Start":"01:36.785 ","End":"01:38.900","Text":"first collecting these 2 I get this."},{"Start":"01:38.900 ","End":"01:43.490","Text":"But we can do a lot more because this numerator can be simplified by"},{"Start":"01:43.490 ","End":"01:48.395","Text":"y^2 and the denominator can be simplified by taking out y cubed."},{"Start":"01:48.395 ","End":"01:51.215","Text":"That\u0027s what I did if we take out y^2."},{"Start":"01:51.215 ","End":"01:56.505","Text":"In fact, here we can take out 3y^2 and we\u0027re left with 2x plus y."},{"Start":"01:56.505 ","End":"01:58.695","Text":"If I take y^3 out here,"},{"Start":"01:58.695 ","End":"02:02.430","Text":"I\u0027m also left with 2x plus y so this cancels."},{"Start":"02:02.430 ","End":"02:05.211","Text":"Y^2 doesn\u0027t completely cancel the y^3,"},{"Start":"02:05.211 ","End":"02:07.425","Text":"we\u0027re still left with a y here."},{"Start":"02:07.425 ","End":"02:11.575","Text":"Ultimately, we are left with 3 over y,"},{"Start":"02:11.575 ","End":"02:15.439","Text":"which is definitely a function of just y."},{"Start":"02:15.439 ","End":"02:17.825","Text":"Now in the case that happens,"},{"Start":"02:17.825 ","End":"02:22.130","Text":"then the rule says that we take e to the power of minus"},{"Start":"02:22.130 ","End":"02:26.750","Text":"the integral of this and that\u0027s going to be the integrating factor."},{"Start":"02:26.750 ","End":"02:31.220","Text":"We had something similar when we divide it by N and we got just a function of x."},{"Start":"02:31.220 ","End":"02:34.955","Text":"We add e to the power of only there we didn\u0027t have a minus."},{"Start":"02:34.955 ","End":"02:37.052","Text":"Make sure you get the correct formula,"},{"Start":"02:37.052 ","End":"02:38.900","Text":"this one has a minus in it."},{"Start":"02:38.900 ","End":"02:42.775","Text":"It\u0027s e to the minus and 3 over y here."},{"Start":"02:42.775 ","End":"02:49.445","Text":"This integral is minus 3 natural log(y) and this simplifies to 1 over y^3."},{"Start":"02:49.445 ","End":"02:55.310","Text":"You can in fact remember one of these rules with exponents that this is so."},{"Start":"02:55.310 ","End":"02:56.615","Text":"Just quickly show you the reason,"},{"Start":"02:56.615 ","End":"02:59.840","Text":"because I could always say this is e to the power of natural log of"},{"Start":"02:59.840 ","End":"03:04.730","Text":"y to the power of minus k and this is just y."},{"Start":"03:04.730 ","End":"03:07.580","Text":"This is y^minus k and then we just throw this"},{"Start":"03:07.580 ","End":"03:10.430","Text":"into the denominator and make the exponent positive,"},{"Start":"03:10.430 ","End":"03:13.270","Text":"so in case you have any doubts about this."},{"Start":"03:13.270 ","End":"03:18.830","Text":"This is our integrating factor and I\u0027ve lost the original equation."},{"Start":"03:18.830 ","End":"03:21.185","Text":"Let\u0027s see where was that? This was this."},{"Start":"03:21.185 ","End":"03:27.060","Text":"Now I\u0027m going to take this and transform it by multiplying by 1 over y^3. Let\u0027s see."},{"Start":"03:27.800 ","End":"03:33.155","Text":"I copied, and I multiplied it by 1 over y^3 here and here,"},{"Start":"03:33.155 ","End":"03:36.140","Text":"and on the right-hand side but that\u0027s 0."},{"Start":"03:36.140 ","End":"03:42.200","Text":"I\u0027ve labeled them m and n. It\u0027s a different m and n than before but no confusion,"},{"Start":"03:42.200 ","End":"03:44.720","Text":"but it\u0027s not the same m and n that there was."},{"Start":"03:44.720 ","End":"03:46.280","Text":"This time we\u0027ll see,"},{"Start":"03:46.280 ","End":"03:49.730","Text":"hopefully if the theory works they should be exact."},{"Start":"03:49.730 ","End":"03:54.275","Text":"This with respect to y should be the partial derivative of this with respect to x."},{"Start":"03:54.275 ","End":"03:56.711","Text":"Just a bit of simplification first."},{"Start":"03:56.711 ","End":"04:03.350","Text":"Each of these is divisible by y^3 and we\u0027re left with this 2x here and y here."},{"Start":"04:03.350 ","End":"04:07.140","Text":"Here, the y^3 cancels and here we have minus 2 over y^3."},{"Start":"04:07.140 ","End":"04:09.705","Text":"Call this the new M,"},{"Start":"04:09.705 ","End":"04:11.820","Text":"this is the new N."},{"Start":"04:11.820 ","End":"04:15.815","Text":"Everything works out fine because the derivative of this with respect to y."},{"Start":"04:15.815 ","End":"04:18.040","Text":"We just look at this and it\u0027s 1,"},{"Start":"04:18.040 ","End":"04:19.960","Text":"this doesn\u0027t count it\u0027s only x."},{"Start":"04:19.960 ","End":"04:22.550","Text":"Here, this bit doesn\u0027t count it\u0027s only"},{"Start":"04:22.550 ","End":"04:25.910","Text":"y\u0027s or differentiate this with respect to x, which is also 1."},{"Start":"04:25.910 ","End":"04:28.220","Text":"I do have this equality."},{"Start":"04:28.220 ","End":"04:32.120","Text":"The equation really is exact."},{"Start":"04:32.120 ","End":"04:34.925","Text":"Now I could say this one,"},{"Start":"04:34.925 ","End":"04:37.310","Text":"this is our equation."},{"Start":"04:37.310 ","End":"04:39.450","Text":"Let\u0027s move to a new page."},{"Start":"04:39.450 ","End":"04:42.555","Text":"Here\u0027s our equation copied."},{"Start":"04:42.555 ","End":"04:46.805","Text":"You probably remember the theory that when we have an exact equation,"},{"Start":"04:46.805 ","End":"04:49.820","Text":"and we can expect to find the solution of this form,"},{"Start":"04:49.820 ","End":"04:53.345","Text":"with the conditions that f with respect to x is this,"},{"Start":"04:53.345 ","End":"04:58.775","Text":"with respect to y is this and we just pick one of them to start with."},{"Start":"04:58.775 ","End":"05:01.205","Text":"In each case we\u0027re going to get an integral."},{"Start":"05:01.205 ","End":"05:04.655","Text":"If one of them comes out difficult then we can try the other."},{"Start":"05:04.655 ","End":"05:07.990","Text":"I don\u0027t think this will be a difficult integral to do."},{"Start":"05:07.990 ","End":"05:10.530","Text":"Derivative with respect to x is M,"},{"Start":"05:10.530 ","End":"05:19.175","Text":"it means that we\u0027ve got the integral of M with respect to x and M is this 2x plus y."},{"Start":"05:19.175 ","End":"05:22.340","Text":"This integral with respect to x is from here,"},{"Start":"05:22.340 ","End":"05:24.390","Text":"x^2 from here xy."},{"Start":"05:24.390 ","End":"05:26.450","Text":"We don\u0027t just have a constant,"},{"Start":"05:26.450 ","End":"05:28.790","Text":"we have a whole function of y,"},{"Start":"05:28.790 ","End":"05:32.750","Text":"which is a constant when we\u0027re integrating with respect to x."},{"Start":"05:32.750 ","End":"05:39.260","Text":"Now, we have to find g(y) and to find this we use the equation we haven\u0027t used yet,"},{"Start":"05:39.260 ","End":"05:40.645","Text":"which is this one."},{"Start":"05:40.645 ","End":"05:43.550","Text":"I need f with respect to y,"},{"Start":"05:43.550 ","End":"05:45.170","Text":"which I can get from here."},{"Start":"05:45.170 ","End":"05:49.475","Text":"Which is this thing is nothing because it\u0027s just x\u0027s."},{"Start":"05:49.475 ","End":"05:51.140","Text":"From here with respect to y,"},{"Start":"05:51.140 ","End":"05:54.995","Text":"it\u0027s just the x and from here g\u0027(y)."},{"Start":"05:54.995 ","End":"05:58.115","Text":"Now here\u0027s N and I\u0027ll equate them."},{"Start":"05:58.115 ","End":"06:03.290","Text":"What I\u0027m saying is that this x plus g\u0027(y)=N from here,"},{"Start":"06:03.290 ","End":"06:08.475","Text":"which is x minus 2 over y^3 and of course the x will cancel."},{"Start":"06:08.475 ","End":"06:17.475","Text":"We\u0027ve got g\u0027(y) is this but I need g(y) because I want to substitute g(y) in here."},{"Start":"06:17.475 ","End":"06:26.973","Text":"I need to do an integration the integral of minus 2 over y^3 is just of 1 over y^2."},{"Start":"06:26.973 ","End":"06:29.210","Text":"You don\u0027t need to add a constant because the"},{"Start":"06:29.210 ","End":"06:32.345","Text":"constant in this case it\u0027s taken care of at the end."},{"Start":"06:32.345 ","End":"06:40.770","Text":"Now we put this 1 over y^2 instead g\u0027(y) and that will give us what F is."},{"Start":"06:40.770 ","End":"06:47.985","Text":"Just disappeared off screen but it was x plus y plus g\u0027(y), which is this."},{"Start":"06:47.985 ","End":"06:49.620","Text":"Now that we found f,"},{"Start":"06:49.620 ","End":"06:53.675","Text":"the last step is just to equate this to general constant."},{"Start":"06:53.675 ","End":"06:57.020","Text":"This equals c and that here is"},{"Start":"06:57.020 ","End":"07:03.840","Text":"the general solution to the differential equation. That\u0027s it."}],"Thumbnail":null,"ID":14637},{"Watched":false,"Name":"Exercise 7","Duration":"4m 54s","ChapterTopicVideoID":13839,"CourseChapterTopicPlaylistID":4222,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.315","Text":"In this exercise, we have to solve a differential equation,"},{"Start":"00:03.315 ","End":"00:07.260","Text":"and the first thing I\u0027ll do is see maybe it\u0027s exact."},{"Start":"00:07.260 ","End":"00:11.490","Text":"I\u0027ll label this one M and this part N. I want to see if"},{"Start":"00:11.490 ","End":"00:16.145","Text":"the derivatives of this with respect to y and this with respect to x are equal."},{"Start":"00:16.145 ","End":"00:18.130","Text":"The first one with respect y,"},{"Start":"00:18.130 ","End":"00:19.830","Text":"gives me 2y minus 1."},{"Start":"00:19.830 ","End":"00:26.225","Text":"Second one, this with respect to x is 1 and this functions are not the same."},{"Start":"00:26.225 ","End":"00:32.645","Text":"It\u0027s not exact but there\u0027s still hope yet to finding an integration factor."},{"Start":"00:32.645 ","End":"00:36.379","Text":"Something we can multiply everything by and make it exact."},{"Start":"00:36.379 ","End":"00:40.070","Text":"If you remember in the tutorial I showed you a few tricks."},{"Start":"00:40.070 ","End":"00:44.435","Text":"One of them is to subtract M minus N and then divide that"},{"Start":"00:44.435 ","End":"00:49.635","Text":"by M and get a function of y or divide by N and get a function of x."},{"Start":"00:49.635 ","End":"00:51.545","Text":"In those cases will know what to do."},{"Start":"00:51.545 ","End":"00:54.380","Text":"Let\u0027s try. Well, I tried it out before."},{"Start":"00:54.380 ","End":"00:55.910","Text":"I know that one of them works."},{"Start":"00:55.910 ","End":"00:57.650","Text":"Divided by M,"},{"Start":"00:57.650 ","End":"01:02.330","Text":"I\u0027ve got 2y minus 1 minus 1 over M,"},{"Start":"01:02.330 ","End":"01:05.210","Text":"which is y^2 minus y."},{"Start":"01:05.210 ","End":"01:07.765","Text":"If I simplify this,"},{"Start":"01:07.765 ","End":"01:10.575","Text":"this is 2y minus 2 in the numerator."},{"Start":"01:10.575 ","End":"01:13.110","Text":"It\u0027s 2 (y minus 1) here, y minus 1,"},{"Start":"01:13.110 ","End":"01:19.665","Text":"the y minus 1 cancels and I\u0027m left with 2 over y and I\u0027II call this g(y)."},{"Start":"01:19.665 ","End":"01:21.080","Text":"This minus this over,"},{"Start":"01:21.080 ","End":"01:24.095","Text":"this is a pure function of y, no x\u0027s in it."},{"Start":"01:24.095 ","End":"01:26.525","Text":"We can get an integration factor."},{"Start":"01:26.525 ","End":"01:34.230","Text":"The formula is e to the power of minus the integral of this,"},{"Start":"01:35.120 ","End":"01:39.250","Text":"one of them has plus and one of them has a minus."},{"Start":"01:40.940 ","End":"01:45.830","Text":"Here we are e to the minus integral of this is e to"},{"Start":"01:45.830 ","End":"01:49.280","Text":"the minus integral of 2 over y is twice"},{"Start":"01:49.280 ","End":"01:52.954","Text":"natural log of y and we\u0027ve seen this kind of thing before."},{"Start":"01:52.954 ","End":"01:54.910","Text":"E to the natural log of y is y,"},{"Start":"01:54.910 ","End":"01:57.935","Text":"y^minus 2, it\u0027s 1 over y squared."},{"Start":"01:57.935 ","End":"02:00.890","Text":"This is going to be our integrating factor,"},{"Start":"02:00.890 ","End":"02:04.820","Text":"and we\u0027ll multiply the original equation by this."},{"Start":"02:04.820 ","End":"02:08.450","Text":"After we do that, we\u0027ll get an exact equation."},{"Start":"02:08.450 ","End":"02:11.675","Text":"That\u0027s what the theory guarantees for this trick."},{"Start":"02:11.675 ","End":"02:16.160","Text":"I took our original equation that\u0027s in black and multiply it by the 1 over y"},{"Start":"02:16.160 ","End":"02:21.145","Text":"squared in blue everywhere and a little bit of simplification."},{"Start":"02:21.145 ","End":"02:26.120","Text":"It\u0027s just algebra multiplying out this times this is 1 and so on and so on."},{"Start":"02:26.120 ","End":"02:31.430","Text":"Now this is the new M and N not the same M and N as before."},{"Start":"02:31.430 ","End":"02:34.010","Text":"But we\u0027re going to check this one."},{"Start":"02:34.010 ","End":"02:36.350","Text":"If everything\u0027s gone right should be exact."},{"Start":"02:36.350 ","End":"02:38.855","Text":"Take M with respect to y,"},{"Start":"02:38.855 ","End":"02:41.660","Text":"and N with respect to x this time they should come out equal."},{"Start":"02:41.660 ","End":"02:46.250","Text":"Let\u0027s see that this derivative with respect to y is plus 1 over y squared"},{"Start":"02:46.250 ","End":"02:50.870","Text":"because there\u0027s a minus minus and this with respect to x is also 1 over y^2."},{"Start":"02:50.870 ","End":"02:52.610","Text":"So these are equal."},{"Start":"02:52.610 ","End":"02:55.910","Text":"We really do have an exact equation with"},{"Start":"02:55.910 ","End":"03:00.325","Text":"this new M and N. Let me continue on another page."},{"Start":"03:00.325 ","End":"03:06.170","Text":"We\u0027re looking for a function of the form F(x,y)=c."},{"Start":"03:06.170 ","End":"03:08.630","Text":"We know these 2 conditions."},{"Start":"03:08.630 ","End":"03:12.020","Text":"Derivative of F with respect to x is M, with respect to y,"},{"Start":"03:12.020 ","End":"03:14.905","Text":"it\u0027s n. We don\u0027t see M and N anymore,"},{"Start":"03:14.905 ","End":"03:16.980","Text":"I\u0027II just bring them in as needed."},{"Start":"03:16.980 ","End":"03:21.585","Text":"M, if you look back is 1 minus 1 over y,"},{"Start":"03:21.585 ","End":"03:24.980","Text":"the new M and so I left to do is integrate this with respect"},{"Start":"03:24.980 ","End":"03:30.010","Text":"to x in order to get F. F is x minus x over y."},{"Start":"03:30.010 ","End":"03:31.780","Text":"When we integrate with respect to x,"},{"Start":"03:31.780 ","End":"03:36.160","Text":"we get a general function of y to find what g(y) is,"},{"Start":"03:36.160 ","End":"03:37.510","Text":"we use this equation,"},{"Start":"03:37.510 ","End":"03:42.970","Text":"so the other one will help us to find g. If I write this in 2 different ways,"},{"Start":"03:42.970 ","End":"03:47.350","Text":"on the one hand, F with respect to y from differentiating this and on the other hand,"},{"Start":"03:47.350 ","End":"03:50.275","Text":"they\u0027ll copy N from what it was before."},{"Start":"03:50.275 ","End":"03:57.100","Text":"From differentiating this, I get that F with respect to y is x gives me nothing here I"},{"Start":"03:57.100 ","End":"04:03.880","Text":"have minus and the derivative of this is minus x over y^2 and g prime of y,"},{"Start":"04:03.880 ","End":"04:06.685","Text":"which is just x over y^2 plus g prime of y."},{"Start":"04:06.685 ","End":"04:10.320","Text":"On the other hand, g is equal to n. If you look back,"},{"Start":"04:10.320 ","End":"04:14.055","Text":"you\u0027ll see that n is x over y^2."},{"Start":"04:14.055 ","End":"04:17.090","Text":"We have that this equals this,"},{"Start":"04:17.090 ","End":"04:18.960","Text":"and this cancels with this,"},{"Start":"04:18.960 ","End":"04:21.195","Text":"so g prime of y is 0."},{"Start":"04:21.195 ","End":"04:26.825","Text":"Then just an integration to find g. G is integral of 0, which is 0."},{"Start":"04:26.825 ","End":"04:28.670","Text":"If we want, then plus a constant,"},{"Start":"04:28.670 ","End":"04:32.150","Text":"we don\u0027t need the constant because that\u0027s taken care of at the end."},{"Start":"04:32.150 ","End":"04:36.735","Text":"Now I can put this 0 in place of g(y) here."},{"Start":"04:36.735 ","End":"04:40.505","Text":"We\u0027ve got F. F is x minus x over y."},{"Start":"04:40.505 ","End":"04:44.105","Text":"As I said, the F is just x minus x over y."},{"Start":"04:44.105 ","End":"04:47.300","Text":"Final step is just to let this equal a constant."},{"Start":"04:47.300 ","End":"04:50.330","Text":"This equals a constant and this here is"},{"Start":"04:50.330 ","End":"04:54.990","Text":"the general solution to the differential equation and we\u0027re done."}],"Thumbnail":null,"ID":14638},{"Watched":false,"Name":"Exercise 8","Duration":"6m 48s","ChapterTopicVideoID":13840,"CourseChapterTopicPlaylistID":4222,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"We have this differential equation here to solve."},{"Start":"00:03.570 ","End":"00:07.800","Text":"The first thing I\u0027d like to do is see maybe it\u0027s exact."},{"Start":"00:07.800 ","End":"00:11.250","Text":"I label this function M and this function"},{"Start":"00:11.250 ","End":"00:17.010","Text":"N. Then we check the derivative of M with respect to y and N with respect to x,"},{"Start":"00:17.010 ","End":"00:19.575","Text":"and we see that they are not equal."},{"Start":"00:19.575 ","End":"00:21.960","Text":"This is not the same function as this,"},{"Start":"00:21.960 ","End":"00:23.384","Text":"so it\u0027s not exact."},{"Start":"00:23.384 ","End":"00:26.820","Text":"I think may be an integrating factor."},{"Start":"00:26.820 ","End":"00:31.110","Text":"I see an integration factor wasn\u0027t provided with the exercise,"},{"Start":"00:31.110 ","End":"00:35.655","Text":"so there\u0027s 2 things that are standard tricks that might work."},{"Start":"00:35.655 ","End":"00:42.180","Text":"One of them is to compute this minus this over N and see if we get just a function of x."},{"Start":"00:42.180 ","End":"00:43.905","Text":"But when we do this,"},{"Start":"00:43.905 ","End":"00:47.925","Text":"this minus this over this function here,"},{"Start":"00:47.925 ","End":"00:49.410","Text":"simplified all I can,"},{"Start":"00:49.410 ","End":"00:50.630","Text":"it won\u0027t simplify anymore."},{"Start":"00:50.630 ","End":"00:52.920","Text":"It doesn\u0027t boil down to just a function of x."},{"Start":"00:52.920 ","End":"00:54.530","Text":"We try the other way round."},{"Start":"00:54.530 ","End":"00:57.220","Text":"If we divide this minus this over M,"},{"Start":"00:57.220 ","End":"01:03.050","Text":"then we get the same numerator as here but the denominator is the function M,"},{"Start":"01:03.050 ","End":"01:06.200","Text":"which is this which factorizes also."},{"Start":"01:06.200 ","End":"01:12.350","Text":"It doesn\u0027t come out to be a function of y so that\u0027s my 2 main hopes."},{"Start":"01:12.350 ","End":"01:14.480","Text":"Other than that in the tutorial,"},{"Start":"01:14.480 ","End":"01:21.785","Text":"we also mentioned this table that says that,"},{"Start":"01:21.785 ","End":"01:23.975","Text":"if the left-hand side,"},{"Start":"01:23.975 ","End":"01:26.570","Text":"this Mdx plus Ndy,"},{"Start":"01:26.570 ","End":"01:31.025","Text":"if it contains this expression,"},{"Start":"01:31.025 ","End":"01:36.890","Text":"then we try one of these as an integration factor."},{"Start":"01:36.890 ","End":"01:38.825","Text":"If it contains this,"},{"Start":"01:38.825 ","End":"01:43.415","Text":"then we can try this as an integration factor,"},{"Start":"01:43.415 ","End":"01:45.800","Text":"some value of N whole number,"},{"Start":"01:45.800 ","End":"01:48.650","Text":"and likewise third entry."},{"Start":"01:48.650 ","End":"01:51.680","Text":"Now, I\u0027ve lost the original function."},{"Start":"01:51.680 ","End":"01:54.335","Text":"I\u0027ve just copied it again down here."},{"Start":"01:54.335 ","End":"01:59.400","Text":"Let\u0027s see if we do have any one of these 3 forms."},{"Start":"02:00.530 ","End":"02:07.590","Text":"I can collect this ydx from here and this xdy from here,"},{"Start":"02:07.590 ","End":"02:13.430","Text":"and get ydx plus xdy and the rest of it will just be higher power terms."},{"Start":"02:13.430 ","End":"02:17.880","Text":"They\u0027re all with xy^2 and maybe this case with the, let\u0027s see,"},{"Start":"02:17.880 ","End":"02:20.350","Text":"ydx plus xdy,"},{"Start":"02:20.350 ","End":"02:24.289","Text":"that would be this one just in a backwards order."},{"Start":"02:24.289 ","End":"02:27.710","Text":"We would like to try 1 over xy^n."},{"Start":"02:27.710 ","End":"02:29.670","Text":"Now, which power of n should I choose?"},{"Start":"02:29.670 ","End":"02:34.367","Text":"Well, since I have everywhere y^2 and even I have x^2,"},{"Start":"02:34.367 ","End":"02:36.275","Text":"y^2 here, I\u0027d like it to cancel."},{"Start":"02:36.275 ","End":"02:38.720","Text":"Let\u0027s try this one with n=2."},{"Start":"02:38.720 ","End":"02:43.405","Text":"With n=2, it means 1 over xy^2."},{"Start":"02:43.405 ","End":"02:46.670","Text":"What I wrote in blue, that\u0027s going to be the integrating factor."},{"Start":"02:46.670 ","End":"02:51.180","Text":"Hopefully, this will make the equation exact now. Let\u0027s see."},{"Start":"02:51.620 ","End":"02:55.020","Text":"If I multiply out,"},{"Start":"02:55.020 ","End":"03:04.080","Text":"what I\u0027ll get is this times this is going to be 1 over x^2y that\u0027s this bit here,"},{"Start":"03:04.080 ","End":"03:05.340","Text":"it\u0027s the opposite order."},{"Start":"03:05.340 ","End":"03:08.400","Text":"This times this is x^2y^2,"},{"Start":"03:08.400 ","End":"03:11.810","Text":"so there\u0027s an x in the denominator and it\u0027s so minus is this,"},{"Start":"03:11.810 ","End":"03:17.660","Text":"and so on I\u0027ll let you check the algebra that from here we get this."},{"Start":"03:17.660 ","End":"03:24.000","Text":"This is our new M and N. Perhaps here, I mean,"},{"Start":"03:24.000 ","End":"03:28.210","Text":"if we did it right and if we\u0027re lucky we chose the right value of n,"},{"Start":"03:28.210 ","End":"03:31.315","Text":"I chose n to be 2 then maybe we\u0027ll get that"},{"Start":"03:31.315 ","End":"03:35.615","Text":"My=Nx for the new M and N so let\u0027s check that on a new page."},{"Start":"03:35.615 ","End":"03:37.840","Text":"I just copied the exercise first,"},{"Start":"03:37.840 ","End":"03:40.105","Text":"M with respect to y."},{"Start":"03:40.105 ","End":"03:44.530","Text":"This parts are constant and it\u0027s like 1 over y times something,"},{"Start":"03:44.530 ","End":"03:48.790","Text":"so it\u0027s minus 1 over y^2 in the same constant there."},{"Start":"03:48.790 ","End":"03:52.050","Text":"If I look at what N is with respect to x,"},{"Start":"03:52.050 ","End":"03:55.290","Text":"it\u0027s based on the 1 over x model,"},{"Start":"03:55.290 ","End":"03:59.700","Text":"the y^2 stays and 1 over x gives me minus 1 over x^2 so the same thing."},{"Start":"03:59.700 ","End":"04:02.450","Text":"This is equal to this and this is equal to this."},{"Start":"04:02.450 ","End":"04:08.270","Text":"My=Nx, so this is an exact equation."},{"Start":"04:08.270 ","End":"04:10.010","Text":"We solve it the usual way."},{"Start":"04:10.010 ","End":"04:12.710","Text":"We look for a solution F(x,y)=c,"},{"Start":"04:12.710 ","End":"04:15.125","Text":"where F satisfies 2 conditions."},{"Start":"04:15.125 ","End":"04:17.330","Text":"Derivative with respect to x is M,"},{"Start":"04:17.330 ","End":"04:20.900","Text":"with respect to y, it\u0027s N. We start with one of them."},{"Start":"04:20.900 ","End":"04:22.935","Text":"Let\u0027s say we start with this."},{"Start":"04:22.935 ","End":"04:25.955","Text":"If we know the derivative with respect to x is M,"},{"Start":"04:25.955 ","End":"04:27.395","Text":"this here is M,"},{"Start":"04:27.395 ","End":"04:31.110","Text":"I integrate it with respect to x to get F. The integral of"},{"Start":"04:31.110 ","End":"04:35.570","Text":"minus 1 over x is natural log of x. I didn\u0027t put the absolute value here."},{"Start":"04:35.570 ","End":"04:38.090","Text":"Let\u0027s assume that x is positive or something."},{"Start":"04:38.090 ","End":"04:43.535","Text":"Let\u0027s not worry about the small technical details and the integral of this,"},{"Start":"04:43.535 ","End":"04:46.475","Text":"it\u0027s based on 1 over x^2,"},{"Start":"04:46.475 ","End":"04:50.270","Text":"whose integral is minus 1 over x and the y just stays there."},{"Start":"04:50.270 ","End":"04:52.940","Text":"But we also have to add the constant,"},{"Start":"04:52.940 ","End":"04:54.050","Text":"which is not a constant,"},{"Start":"04:54.050 ","End":"04:57.680","Text":"it\u0027s a function of y when we take an integral with respect to x."},{"Start":"04:57.680 ","End":"05:01.100","Text":"Now, next task is to find g(y)."},{"Start":"05:01.100 ","End":"05:04.810","Text":"What we do is we use the other equation here that we didn\u0027t use."},{"Start":"05:04.810 ","End":"05:08.480","Text":"I\u0027m going to differentiate this with respect to y and then assign it"},{"Start":"05:08.480 ","End":"05:13.310","Text":"to N. The derivative of this with respect to y,"},{"Start":"05:13.310 ","End":"05:17.240","Text":"this bit gives me nothing because it\u0027s only an expression in x."},{"Start":"05:17.240 ","End":"05:19.880","Text":"Here I have minus 1 over x,"},{"Start":"05:19.880 ","End":"05:21.950","Text":"which is a constant times 1 over y,"},{"Start":"05:21.950 ","End":"05:23.540","Text":"so the minus 1 over x,"},{"Start":"05:23.540 ","End":"05:27.905","Text":"I take first and 1 over y gives me minus 1 over y^2,"},{"Start":"05:27.905 ","End":"05:32.730","Text":"and g(y) gives me g\u0027 (y)."},{"Start":"05:32.730 ","End":"05:36.860","Text":"Then simplify it, minus times minus is plus and x times y^2."},{"Start":"05:36.860 ","End":"05:39.365","Text":"As we said, F with respect to y has got to equal N,"},{"Start":"05:39.365 ","End":"05:41.150","Text":"that\u0027s this equation here."},{"Start":"05:41.150 ","End":"05:45.195","Text":"This, which is this has got to equal n,"},{"Start":"05:45.195 ","End":"05:47.780","Text":"its not on the page at the moment,"},{"Start":"05:47.780 ","End":"05:50.870","Text":"but n was 1 plus 1 over xy^2."},{"Start":"05:50.870 ","End":"06:01.075","Text":"This equals this. We can cancel this from both sides and that gives us that g\u0027(y)=1."},{"Start":"06:01.075 ","End":"06:04.755","Text":"G is just the integral of 1, which is y."},{"Start":"06:04.755 ","End":"06:06.450","Text":"We don\u0027t have the c at this point,"},{"Start":"06:06.450 ","End":"06:08.145","Text":"there\u0027s a c at the end."},{"Start":"06:08.145 ","End":"06:10.005","Text":"Now that we have g(y),"},{"Start":"06:10.005 ","End":"06:13.460","Text":"let\u0027s see where I did lose it. Here it is."},{"Start":"06:13.460 ","End":"06:19.145","Text":"F is equal to minus natural log of x minus 1 over xy plus g(y)."},{"Start":"06:19.145 ","End":"06:25.610","Text":"Now I know that this g(y) is just y. I\u0027m going to have to scroll off-screen,"},{"Start":"06:25.610 ","End":"06:27.250","Text":"but we\u0027ll remember that."},{"Start":"06:27.250 ","End":"06:31.820","Text":"Here\u0027s our F. It\u0027s what we had up there where instead of g(y),"},{"Start":"06:31.820 ","End":"06:35.270","Text":"we put y and there\u0027s only one more step to go,"},{"Start":"06:35.270 ","End":"06:38.725","Text":"is to assign this function to equal a constant,"},{"Start":"06:38.725 ","End":"06:40.000","Text":"and so finally,"},{"Start":"06:40.000 ","End":"06:41.180","Text":"this is the solution."},{"Start":"06:41.180 ","End":"06:42.540","Text":"This equals a constant,"},{"Start":"06:42.540 ","End":"06:48.150","Text":"this is the general solution to the differential equation. That\u0027s it."}],"Thumbnail":null,"ID":14639},{"Watched":false,"Name":"Exercise 9","Duration":"6m 58s","ChapterTopicVideoID":13841,"CourseChapterTopicPlaylistID":4222,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"Here we have a differential equation to solve,"},{"Start":"00:03.300 ","End":"00:06.255","Text":"it\u0027s a problem with an initial condition."},{"Start":"00:06.255 ","End":"00:07.740","Text":"When x is 1,"},{"Start":"00:07.740 ","End":"00:09.600","Text":"is equal to minus 1."},{"Start":"00:09.600 ","End":"00:11.010","Text":"We\u0027ll use this at the end."},{"Start":"00:11.010 ","End":"00:12.930","Text":"We\u0027ll first of all, see if we can solve a"},{"Start":"00:12.930 ","End":"00:15.585","Text":"generally then substitute the initial condition."},{"Start":"00:15.585 ","End":"00:17.820","Text":"This is in the chapter on exact equation."},{"Start":"00:17.820 ","End":"00:21.074","Text":"So we\u0027re going to expect that this might be an exact,"},{"Start":"00:21.074 ","End":"00:26.130","Text":"but it doesn\u0027t look in the form because we needed something dx plus something dy."},{"Start":"00:26.130 ","End":"00:28.185","Text":"So start off with a bit of algebra."},{"Start":"00:28.185 ","End":"00:30.870","Text":"Y prime is dy/dx,"},{"Start":"00:30.870 ","End":"00:34.590","Text":"and now we cross-multiply and we get this,"},{"Start":"00:34.590 ","End":"00:36.750","Text":"this dy equals this dx,"},{"Start":"00:36.750 ","End":"00:39.260","Text":"but we want everything on the left-hand side."},{"Start":"00:39.260 ","End":"00:41.090","Text":"So this over here becomes minus,"},{"Start":"00:41.090 ","End":"00:44.870","Text":"and this is what we have and we call this function M and this function"},{"Start":"00:44.870 ","End":"00:50.180","Text":"N. Let\u0027s see if it meets the conditions for an exact equation."},{"Start":"00:50.180 ","End":"00:57.260","Text":"Unfortunately, no, because the derivative of M with respect to y from here is -3x^2,"},{"Start":"00:57.260 ","End":"01:00.680","Text":"and the derivative of N with respect to x, this part gets nothing,"},{"Start":"01:00.680 ","End":"01:02.075","Text":"This part is 3x^2,"},{"Start":"01:02.075 ","End":"01:03.320","Text":"here, I got 3x^2,"},{"Start":"01:03.320 ","End":"01:06.335","Text":"here -3x^2 not the same function."},{"Start":"01:06.335 ","End":"01:08.720","Text":"We\u0027ll try some of tricks."},{"Start":"01:08.720 ","End":"01:14.720","Text":"One of them has to take this minus this and then divide it either by M or by N,"},{"Start":"01:14.720 ","End":"01:16.700","Text":"if it\u0027s by N,"},{"Start":"01:16.700 ","End":"01:19.190","Text":"we hope to get a function of x if it\u0027s by M,"},{"Start":"01:19.190 ","End":"01:20.645","Text":"we hope to get a function of y."},{"Start":"01:20.645 ","End":"01:21.820","Text":"This in fact is the case,"},{"Start":"01:21.820 ","End":"01:24.580","Text":"I\u0027m going to take this minus this over M,"},{"Start":"01:24.580 ","End":"01:26.485","Text":"like I said, this minus this over this,"},{"Start":"01:26.485 ","End":"01:32.140","Text":"which is this minus this over this and the moment,"},{"Start":"01:32.140 ","End":"01:36.810","Text":"it doesn\u0027t look like a function of just y but if"},{"Start":"01:36.810 ","End":"01:45.950","Text":"we just combine these two first and now we\u0027ll divide by the x squared."},{"Start":"01:45.950 ","End":"01:47.920","Text":"This is 2/y,"},{"Start":"01:47.920 ","End":"01:51.635","Text":"and we\u0027ll call that function g(y)."},{"Start":"01:51.635 ","End":"01:55.270","Text":"This is not the integration factor itself,"},{"Start":"01:55.270 ","End":"02:01.680","Text":"what we need is e to the power of minus the integral."},{"Start":"02:01.680 ","End":"02:08.870","Text":"So if we do that, the integral of 2/y is 2 natural log of y, there is a minus,"},{"Start":"02:08.870 ","End":"02:10.775","Text":"you\u0027ve seen this trick before,"},{"Start":"02:10.775 ","End":"02:15.410","Text":"e to the power of minus something can natural log y is 1/y^2,"},{"Start":"02:15.410 ","End":"02:19.670","Text":"just algebra and exponents."},{"Start":"02:19.670 ","End":"02:23.180","Text":"Now this is going to be our integration factor which will multiply by"},{"Start":"02:23.180 ","End":"02:27.900","Text":"this and this should come out to be exact now,"},{"Start":"02:27.900 ","End":"02:31.400","Text":"going to have to scroll and lose that."},{"Start":"02:31.400 ","End":"02:35.690","Text":"But this in black was just what we had before and now adding"},{"Start":"02:35.690 ","End":"02:40.495","Text":"the integration factor and want to simplify this a bit."},{"Start":"02:40.495 ","End":"02:43.190","Text":"The y^2 with this y means it\u0027s"},{"Start":"02:43.190 ","End":"02:47.720","Text":"just a single y in the denominator minus 3x^2 in the numerator."},{"Start":"02:47.720 ","End":"02:55.100","Text":"From here, we get x^3/y^2 + y^2 cancels it and we are just left with y^2 here."},{"Start":"02:55.100 ","End":"02:56.975","Text":"This is what we get now,"},{"Start":"02:56.975 ","End":"03:01.880","Text":"this is the new M and the new N. With this new M and N,"},{"Start":"03:01.880 ","End":"03:06.770","Text":"we now should get that M with respect to y equals N with respect to x not"},{"Start":"03:06.770 ","End":"03:12.620","Text":"the old M and N that we had before and let\u0027s just see that this was respect to y."},{"Start":"03:12.620 ","End":"03:14.045","Text":"Well, it\u0027s all a constant,"},{"Start":"03:14.045 ","End":"03:15.545","Text":"the 1/y gives me -1/y^2,"},{"Start":"03:15.545 ","End":"03:21.275","Text":"minus with minus cancels, so it\u0027s 3x^2/y^2."},{"Start":"03:21.275 ","End":"03:22.940","Text":"So this is equal to this."},{"Start":"03:22.940 ","End":"03:25.520","Text":"In the other hand, this with respect to x,"},{"Start":"03:25.520 ","End":"03:28.340","Text":"this part disappears because there\u0027s no x in it,"},{"Start":"03:28.340 ","End":"03:32.075","Text":"and here I get 3x^2/y^2."},{"Start":"03:32.075 ","End":"03:35.600","Text":"This equals this, these two are equal to each other."},{"Start":"03:35.600 ","End":"03:38.285","Text":"So this is an exact equation,"},{"Start":"03:38.285 ","End":"03:40.615","Text":"and I\u0027ll continue on the next page."},{"Start":"03:40.615 ","End":"03:45.035","Text":"Here we are, and we know it\u0027s exact, since it\u0027s exact,"},{"Start":"03:45.035 ","End":"03:48.275","Text":"we know to look for a solution of this form,"},{"Start":"03:48.275 ","End":"03:50.480","Text":"this sum function of x and y equals c,"},{"Start":"03:50.480 ","End":"03:53.315","Text":"where f has to satisfy two conditions."},{"Start":"03:53.315 ","End":"03:56.060","Text":"The partial derivatives have to equal M and N"},{"Start":"03:56.060 ","End":"04:00.065","Text":"respectively and what we do is we start with one of them,"},{"Start":"04:00.065 ","End":"04:02.180","Text":"whichever simplest to integrate."},{"Start":"04:02.180 ","End":"04:05.810","Text":"I would say this with respect to x is easier to integrate,"},{"Start":"04:05.810 ","End":"04:07.250","Text":"so we start with this."},{"Start":"04:07.250 ","End":"04:09.080","Text":"This means that to get F,"},{"Start":"04:09.080 ","End":"04:14.030","Text":"we just integrate M with respect to x,"},{"Start":"04:14.030 ","End":"04:20.615","Text":"y is a constant and the integral of 3x^2 is x^3."},{"Start":"04:20.615 ","End":"04:26.110","Text":"So it becomes the minus, because -x^3/y."},{"Start":"04:26.110 ","End":"04:29.640","Text":"We don\u0027t just put a constant in this case,"},{"Start":"04:29.640 ","End":"04:33.410","Text":"in this case we have a function of two variables and the integrate with respect to x,"},{"Start":"04:33.410 ","End":"04:35.720","Text":"we put an arbitrary function of y,"},{"Start":"04:35.720 ","End":"04:38.510","Text":"that\u0027s like a constant as far as X goes."},{"Start":"04:38.510 ","End":"04:40.775","Text":"Now we have to find what g is."},{"Start":"04:40.775 ","End":"04:44.030","Text":"To find this, we use the equation that we didn\u0027t use from these two,"},{"Start":"04:44.030 ","End":"04:45.500","Text":"we didn\u0027t use this one."},{"Start":"04:45.500 ","End":"04:48.350","Text":"So differentiate this with respect to y and then equate"},{"Start":"04:48.350 ","End":"04:51.290","Text":"it to n. The derivative of this with respect to y."},{"Start":"04:51.290 ","End":"04:53.390","Text":"1/y becomes -1/y^2,"},{"Start":"04:53.390 ","End":"04:58.145","Text":"the minus disappears and g becomes g prime."},{"Start":"04:58.145 ","End":"05:01.115","Text":"Now I have to equate this to N,"},{"Start":"05:01.115 ","End":"05:03.935","Text":"like we said here, derivative with respect to y is N,"},{"Start":"05:03.935 ","End":"05:07.970","Text":"which gives us this expression here, is equal to,"},{"Start":"05:07.970 ","End":"05:12.080","Text":"this is just N that I copied from above,"},{"Start":"05:12.080 ","End":"05:14.060","Text":"everything looks very similar."},{"Start":"05:14.060 ","End":"05:20.730","Text":"This part cancels, and that leaves us with g\u0027(y)= 2y^2,"},{"Start":"05:20.730 ","End":"05:24.110","Text":"is here because this cancels and at this point we just"},{"Start":"05:24.110 ","End":"05:28.505","Text":"have to do a little integration with respect to y."},{"Start":"05:28.505 ","End":"05:30.950","Text":"So g(y) is the integral of this,"},{"Start":"05:30.950 ","End":"05:34.340","Text":"which is just raise the power and divide by that."},{"Start":"05:34.340 ","End":"05:38.490","Text":"So it\u0027s 2y^3/3, don\u0027t put the constant in at this point."},{"Start":"05:38.490 ","End":"05:40.890","Text":"Now I have to substitute, where was it?"},{"Start":"05:40.890 ","End":"05:42.465","Text":"We had f up somewhere."},{"Start":"05:42.465 ","End":"05:48.350","Text":"Here it is. We have -x^3/y + g(y), which is this."},{"Start":"05:48.350 ","End":"05:50.580","Text":"Let\u0027s see if I remember that,"},{"Start":"05:50.830 ","End":"05:54.860","Text":"-x^3/y plus g(y) which is this."},{"Start":"05:54.860 ","End":"05:56.765","Text":"Once we have f,"},{"Start":"05:56.765 ","End":"05:59.090","Text":"then the general solution to"},{"Start":"05:59.090 ","End":"06:04.160","Text":"the differential equation is that F equals a constant and that is this."},{"Start":"06:04.160 ","End":"06:05.959","Text":"This is the general solution."},{"Start":"06:05.959 ","End":"06:10.385","Text":"But we don\u0027t want the general solution because we had initial conditions."},{"Start":"06:10.385 ","End":"06:14.825","Text":"If you remember, we had that y(1)= -1."},{"Start":"06:14.825 ","End":"06:17.590","Text":"Yes, here y(1) =-1,"},{"Start":"06:17.590 ","End":"06:23.020","Text":"which means basically they can substitute x = 1 and y =-1."},{"Start":"06:23.020 ","End":"06:25.500","Text":"So where I see x, I put 1,"},{"Start":"06:25.500 ","End":"06:27.655","Text":"where I see y I put -1."},{"Start":"06:27.655 ","End":"06:32.045","Text":"So we get this expression here equals c. If we compute this,"},{"Start":"06:32.045 ","End":"06:35.295","Text":"this is just 1/1 with a plus it\u0027s 1."},{"Start":"06:35.295 ","End":"06:44.615","Text":"And here I have -2/3,1 -2/3 is 1/3 that gives me C. So now I put C in here,"},{"Start":"06:44.615 ","End":"06:48.065","Text":"and now we have the solution without any constants."},{"Start":"06:48.065 ","End":"06:51.650","Text":"That\u0027s the solution to the problem with the initial condition."},{"Start":"06:51.650 ","End":"06:57.990","Text":"So this is our answer and we are finally done."}],"Thumbnail":null,"ID":14640},{"Watched":false,"Name":"Exercise 10","Duration":"6m 14s","ChapterTopicVideoID":7618,"CourseChapterTopicPlaylistID":4222,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this exercise, I\u0027m going to prove a result, well,"},{"Start":"00:04.320 ","End":"00:08.100","Text":"2 results that we\u0027ve already used in previous exercises,"},{"Start":"00:08.100 ","End":"00:10.455","Text":"but we use them without proof."},{"Start":"00:10.455 ","End":"00:14.700","Text":"In part a, we have to show that if we have a non-exact ODE,"},{"Start":"00:14.700 ","End":"00:17.580","Text":"ordinary differential equation of this form,"},{"Start":"00:17.580 ","End":"00:23.384","Text":"that if this expression turns out to be a function of x,"},{"Start":"00:23.384 ","End":"00:24.825","Text":"that\u0027s what I mean here,"},{"Start":"00:24.825 ","End":"00:26.250","Text":"the letter f is not important,"},{"Start":"00:26.250 ","End":"00:27.330","Text":"that f is not defined,"},{"Start":"00:27.330 ","End":"00:30.360","Text":"it just means that this is some function of x, which we\u0027ll call f,"},{"Start":"00:30.360 ","End":"00:34.560","Text":"then this is an integration factor e to"},{"Start":"00:34.560 ","End":"00:39.735","Text":"power of the integral of this function of x, and similar result,"},{"Start":"00:39.735 ","End":"00:46.190","Text":"this expression in this case it\u0027s the difference of the 2 partial derivatives over N,"},{"Start":"00:46.190 ","End":"00:48.710","Text":"in this case it\u0027s over M. If this happens to be"},{"Start":"00:48.710 ","End":"00:52.010","Text":"a function purely of y and call that function g,"},{"Start":"00:52.010 ","End":"00:54.710","Text":"then this is an integration factor."},{"Start":"00:54.710 ","End":"00:58.535","Text":"It\u0027s slightly different because here we don\u0027t have a minus and here we have a minus."},{"Start":"00:58.535 ","End":"01:00.995","Text":"Note that the numerator here,"},{"Start":"01:00.995 ","End":"01:08.085","Text":"this M_y minus N _x would be 0 if it were an exact equation,"},{"Start":"01:08.085 ","End":"01:10.610","Text":"and here we have it non-zero."},{"Start":"01:10.610 ","End":"01:13.280","Text":"If this thing was 0, then this over this would be 0,"},{"Start":"01:13.280 ","End":"01:16.610","Text":"this over this would be 0, so both these functions would be 0,"},{"Start":"01:16.610 ","End":"01:21.130","Text":"and in each case we\u0027d get e to the power of a constant,"},{"Start":"01:21.130 ","End":"01:24.320","Text":"so the integration factor will be a constant or 1."},{"Start":"01:24.320 ","End":"01:26.750","Text":"It means you don\u0027t really need an integrating factor."},{"Start":"01:26.750 ","End":"01:29.540","Text":"This numerator is not 0."},{"Start":"01:29.540 ","End":"01:33.560","Text":"Anyway, we\u0027re going to start from a different place where we take"},{"Start":"01:33.560 ","End":"01:38.360","Text":"the original equation and multiply it by some function Mu(x),"},{"Start":"01:38.360 ","End":"01:43.040","Text":"where we assume this Mu(x) is an integration factor that\u0027s purely a function of x."},{"Start":"01:43.040 ","End":"01:46.700","Text":"Then we\u0027ll see how it ties in to this."},{"Start":"01:46.700 ","End":"01:50.330","Text":"Let\u0027s assume that we have this Mu(x),"},{"Start":"01:50.330 ","End":"01:54.455","Text":"we\u0027ve multiplied it, and that now this is an exact equation."},{"Start":"01:54.455 ","End":"01:58.370","Text":"This exactness means that the derivative of this with"},{"Start":"01:58.370 ","End":"02:02.675","Text":"respect to y is equal to the derivative of this,"},{"Start":"02:02.675 ","End":"02:08.000","Text":"I\u0027m talking about this product and this product, with respect to x."},{"Start":"02:08.000 ","End":"02:11.060","Text":"Now I\u0027m going to do the partial differentiation."},{"Start":"02:11.060 ","End":"02:14.690","Text":"Notice that on the left it\u0027s easier because Mu(x) is a constant,"},{"Start":"02:14.690 ","End":"02:17.570","Text":"so it stays, we just have to differentiate this."},{"Start":"02:17.570 ","End":"02:19.130","Text":"But on the right-hand side,"},{"Start":"02:19.130 ","End":"02:23.865","Text":"we\u0027ll use the product rule because each of these has a derivative with respect to x."},{"Start":"02:23.865 ","End":"02:26.375","Text":"We\u0027ll end up as follows,"},{"Start":"02:26.375 ","End":"02:28.640","Text":"we already mentioned about the left-hand side,"},{"Start":"02:28.640 ","End":"02:31.370","Text":"the right-hand side is the derivative of this times"},{"Start":"02:31.370 ","End":"02:35.170","Text":"this plus the derivative of this times this,"},{"Start":"02:35.170 ","End":"02:37.085","Text":"and this is what we get."},{"Start":"02:37.085 ","End":"02:39.885","Text":"Let\u0027s just manipulate the equation a bit."},{"Start":"02:39.885 ","End":"02:45.545","Text":"Take this to the left-hand side and take Mu(x) out of the brackets."},{"Start":"02:45.545 ","End":"02:47.945","Text":"This is what we get on the left,"},{"Start":"02:47.945 ","End":"02:50.260","Text":"on the right this remains."},{"Start":"02:50.260 ","End":"02:58.280","Text":"Next, bring this Mu(x) to this side on the denominator and the N on this side,"},{"Start":"02:58.280 ","End":"02:59.975","Text":"and then flip sides."},{"Start":"02:59.975 ","End":"03:04.205","Text":"We\u0027ve got Mu\u0027 over Mu equals this expression,"},{"Start":"03:04.205 ","End":"03:08.005","Text":"which looks very familiar because that\u0027s the expression we had in the beginning."},{"Start":"03:08.005 ","End":"03:13.145","Text":"The assumption in part a was that this expression was purely a function of x,"},{"Start":"03:13.145 ","End":"03:15.005","Text":"we called it f(x)."},{"Start":"03:15.005 ","End":"03:18.920","Text":"That means that if I just plug this here instead of this,"},{"Start":"03:18.920 ","End":"03:25.880","Text":"that we get the equation Mu\u0027 over Mu is f. Now we\u0027ve seen this many times,"},{"Start":"03:25.880 ","End":"03:28.190","Text":"the derivative of a function over the function,"},{"Start":"03:28.190 ","End":"03:33.050","Text":"we know that the integral of this is just the natural log of the denominator."},{"Start":"03:33.050 ","End":"03:36.080","Text":"Taking the integral, as I said,"},{"Start":"03:36.080 ","End":"03:39.170","Text":"we get this natural log of the denominator."},{"Start":"03:39.170 ","End":"03:41.795","Text":"On the right, just the integral of the function."},{"Start":"03:41.795 ","End":"03:44.590","Text":"Now by the definition of the natural log,"},{"Start":"03:44.590 ","End":"03:47.010","Text":"if the natural log of this is this,"},{"Start":"03:47.010 ","End":"03:49.430","Text":"then each of the power of this is this."},{"Start":"03:49.430 ","End":"03:51.650","Text":"Our integrating factor,"},{"Start":"03:51.650 ","End":"03:54.185","Text":"I abbreviate the I. F for integrating factor,"},{"Start":"03:54.185 ","End":"03:58.260","Text":"Mu is e to the power of the integral of f(x)dx,"},{"Start":"03:58.260 ","End":"03:59.945","Text":"and that\u0027s what we have to show."},{"Start":"03:59.945 ","End":"04:01.975","Text":"We\u0027re done with part a."},{"Start":"04:01.975 ","End":"04:04.820","Text":"Now, part b was very similar,"},{"Start":"04:04.820 ","End":"04:11.870","Text":"except that we had the other expression with the M on the denominator instead of the N,"},{"Start":"04:11.870 ","End":"04:15.710","Text":"that, that was equal to some function of y."},{"Start":"04:15.710 ","End":"04:21.210","Text":"Start as before except that instead of Mu(x), we have Mu(y)."},{"Start":"04:21.430 ","End":"04:26.690","Text":"The exactness condition says that this derived with"},{"Start":"04:26.690 ","End":"04:31.885","Text":"respect to y is equal to this bit with respect to x."},{"Start":"04:31.885 ","End":"04:34.700","Text":"The computation is similar to before."},{"Start":"04:34.700 ","End":"04:39.656","Text":"The right-hand side is easier because Mu(y) is a constant as far as x goes,"},{"Start":"04:39.656 ","End":"04:42.845","Text":"so it just stays and here we use the product rule."},{"Start":"04:42.845 ","End":"04:47.210","Text":"Bring the expression on the right over to the left and take"},{"Start":"04:47.210 ","End":"04:51.755","Text":"out Mu(y) and throw this bit to the right,"},{"Start":"04:51.755 ","End":"04:53.840","Text":"that\u0027s where the minus comes in."},{"Start":"04:53.840 ","End":"04:59.120","Text":"As before, we divide both sides by Mu(y) and"},{"Start":"04:59.120 ","End":"05:04.280","Text":"then by M and so we\u0027ve got this expression,"},{"Start":"05:04.280 ","End":"05:05.750","Text":"we switch sides also."},{"Start":"05:05.750 ","End":"05:09.020","Text":"This was exactly the expression that we had at"},{"Start":"05:09.020 ","End":"05:13.174","Text":"the beginning that we said was a function of just y."},{"Start":"05:13.174 ","End":"05:16.720","Text":"We call this g(y)."},{"Start":"05:16.720 ","End":"05:18.620","Text":"From this and this,"},{"Start":"05:18.620 ","End":"05:21.200","Text":"just by cutting out the middle term,"},{"Start":"05:21.200 ","End":"05:25.940","Text":"we\u0027ve got that this is minus g(y) because notice there\u0027s a minus here,"},{"Start":"05:25.940 ","End":"05:27.620","Text":"but there\u0027s no minus here."},{"Start":"05:27.620 ","End":"05:30.245","Text":"Put a minus in front of both."},{"Start":"05:30.245 ","End":"05:32.915","Text":"Taking the integral of both sides,"},{"Start":"05:32.915 ","End":"05:39.260","Text":"we have the natural log of the denominator equals the integral of this."},{"Start":"05:39.260 ","End":"05:42.785","Text":"I could take the minus out of the integral sign."},{"Start":"05:42.785 ","End":"05:44.180","Text":"We don\u0027t need a constant,"},{"Start":"05:44.180 ","End":"05:48.064","Text":"we just want a particular integration factor."},{"Start":"05:48.064 ","End":"05:54.410","Text":"The constant is just going to change the answer by making it multiplied by a constant."},{"Start":"05:54.410 ","End":"05:55.985","Text":"We don\u0027t need the constant."},{"Start":"05:55.985 ","End":"06:00.391","Text":"We just use the definition of the law to say that if the log of this is this,"},{"Start":"06:00.391 ","End":"06:05.750","Text":"then e to the power of this is Mu(y)."},{"Start":"06:05.750 ","End":"06:07.370","Text":"The minus comes out in front,"},{"Start":"06:07.370 ","End":"06:09.443","Text":"like I said, and that\u0027s what we had to prove,"},{"Start":"06:09.443 ","End":"06:14.340","Text":"and so we\u0027re done with part b and then with the whole question."}],"Thumbnail":null,"ID":7701},{"Watched":false,"Name":"Exercise 11","Duration":"8m 55s","ChapterTopicVideoID":7619,"CourseChapterTopicPlaylistID":4222,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.667","Text":"In this exercise we\u0027re given an ordinary differential equation as follows,"},{"Start":"00:05.667 ","End":"00:07.725","Text":"and it\u0027s not exact."},{"Start":"00:07.725 ","End":"00:10.350","Text":"We have to find an integration factor,"},{"Start":"00:10.350 ","End":"00:13.605","Text":"but one which is just a function of xy."},{"Start":"00:13.605 ","End":"00:15.495","Text":"Now what is does it mean,"},{"Start":"00:15.495 ","End":"00:18.150","Text":"a function of just xy?"},{"Start":"00:18.150 ","End":"00:20.430","Text":"Well, let\u0027s jut finish the question."},{"Start":"00:20.430 ","End":"00:23.805","Text":"We have to find integration factor of the f form Mu(xy),"},{"Start":"00:23.805 ","End":"00:26.024","Text":"where Mu is a function one 1 variable."},{"Start":"00:26.024 ","End":"00:30.480","Text":"This means for example that it could be something"},{"Start":"00:30.480 ","End":"00:37.405","Text":"like (xy)^2 or sine(xy),"},{"Start":"00:37.405 ","End":"00:40.925","Text":"or any number of possibilities,"},{"Start":"00:40.925 ","End":"00:42.650","Text":"or 1 over xy."},{"Start":"00:42.650 ","End":"00:45.320","Text":"Xy have to appear together as a unit."},{"Start":"00:45.320 ","End":"00:49.445","Text":"You can\u0027t have just (xy)^2 then plus x, or something."},{"Start":"00:49.445 ","End":"00:52.670","Text":"For example in this case,"},{"Start":"00:52.670 ","End":"00:54.320","Text":"let\u0027s just take one of the examples here."},{"Start":"00:54.320 ","End":"00:57.965","Text":"If I took Mu as a function of a single variable t,"},{"Start":"00:57.965 ","End":"01:02.815","Text":"if Mu(t) was t^2 then Mu(xy) would be (xy)^2."},{"Start":"01:02.815 ","End":"01:05.990","Text":"In that case our function Mu would be as follows,"},{"Start":"01:05.990 ","End":"01:09.650","Text":"with the dummy variable t. I hope that\u0027s clear."},{"Start":"01:09.650 ","End":"01:11.525","Text":"Xy appear together."},{"Start":"01:11.525 ","End":"01:13.615","Text":"How do we do this?"},{"Start":"01:13.615 ","End":"01:16.333","Text":"Let\u0027s take this integration factor,"},{"Start":"01:16.333 ","End":"01:19.025","Text":"and multiply it by the equation,"},{"Start":"01:19.025 ","End":"01:20.845","Text":"and now it\u0027s supposed to be exact."},{"Start":"01:20.845 ","End":"01:22.400","Text":"Here I\u0027ve got the Mu(xy),"},{"Start":"01:22.400 ","End":"01:23.593","Text":"here I\u0027ve got the Mu(xy)."},{"Start":"01:23.593 ","End":"01:25.565","Text":"Just multiply the original."},{"Start":"01:25.565 ","End":"01:29.660","Text":"This becomes my function m,"},{"Start":"01:29.660 ","End":"01:31.940","Text":"and this becomes the function n,"},{"Start":"01:31.940 ","End":"01:36.455","Text":"and this with respect to y has to equal this with respect to x,"},{"Start":"01:36.455 ","End":"01:38.060","Text":"partial derivatives I mean."},{"Start":"01:38.060 ","End":"01:39.740","Text":"I wrote that with this notation,"},{"Start":"01:39.740 ","End":"01:45.115","Text":"derivative with respect to y of this equals derivative with respect to x of this."},{"Start":"01:45.115 ","End":"01:48.290","Text":"These partial derivatives could be a little tricky"},{"Start":"01:48.290 ","End":"01:52.010","Text":"especially when we get to the Mu(xy) part,"},{"Start":"01:52.010 ","End":"01:58.445","Text":"so I want to bring semi general results as part of the chain rule."},{"Start":"01:58.445 ","End":"02:02.015","Text":"I\u0027ll go over this, and I\u0027ll show you the relevance of this to our case."},{"Start":"02:02.015 ","End":"02:04.370","Text":"In this case, we suppose we have two functions."},{"Start":"02:04.370 ","End":"02:07.513","Text":"We have a function h of two variables; x and y,"},{"Start":"02:07.513 ","End":"02:11.660","Text":"and we have a function f of a single variable and I have the composition of"},{"Start":"02:11.660 ","End":"02:16.415","Text":"f with h. This f(h(x and y)),"},{"Start":"02:16.415 ","End":"02:19.580","Text":"it can have a derivative with respect to x and the derivative with"},{"Start":"02:19.580 ","End":"02:23.680","Text":"respect to y because it also depends on x and y."},{"Start":"02:23.680 ","End":"02:27.845","Text":"The chain rule says as follows that with respect to x,"},{"Start":"02:27.845 ","End":"02:31.820","Text":"we take the derivative of f. F is a function of one variable,"},{"Start":"02:31.820 ","End":"02:35.190","Text":"so it\u0027s just f\u0027(h( x,"},{"Start":"02:35.190 ","End":"02:42.175","Text":"y)) and then we take the inner derivative which is h with respect to x also at x, y."},{"Start":"02:42.175 ","End":"02:44.630","Text":"Also for y,"},{"Start":"02:44.630 ","End":"02:48.229","Text":"same thing just with x replaced by y."},{"Start":"02:48.229 ","End":"02:49.970","Text":"What is this to do with us?"},{"Start":"02:49.970 ","End":"02:54.660","Text":"In our case the f is exactly the Mu;"},{"Start":"02:54.660 ","End":"02:58.460","Text":"that\u0027s the outer function of one variable,"},{"Start":"02:58.460 ","End":"03:01.760","Text":"and the inner function in our case is just the xy."},{"Start":"03:01.760 ","End":"03:07.075","Text":"So h(x, y) is just xy."},{"Start":"03:07.075 ","End":"03:09.030","Text":"When I say Mu(xy),"},{"Start":"03:09.030 ","End":"03:12.510","Text":"that will be f(h( x, y)."},{"Start":"03:12.510 ","End":"03:14.550","Text":"F is Mu, h(x,"},{"Start":"03:14.550 ","End":"03:16.365","Text":"y) is xy Mu(xy)."},{"Start":"03:16.365 ","End":"03:20.510","Text":"I\u0027m going to use this as part of the derivative."},{"Start":"03:20.510 ","End":"03:23.435","Text":"This is wrapped up with a product,"},{"Start":"03:23.435 ","End":"03:25.400","Text":"so we\u0027re going to use the product rule together with"},{"Start":"03:25.400 ","End":"03:28.400","Text":"this chain rule and I\u0027ll show you what we get."},{"Start":"03:28.400 ","End":"03:29.900","Text":"Well, we get this mess,"},{"Start":"03:29.900 ","End":"03:31.610","Text":"and that\u0027s going in more detail."},{"Start":"03:31.610 ","End":"03:34.405","Text":"On the left-hand side I\u0027m using the product rule."},{"Start":"03:34.405 ","End":"03:38.690","Text":"With the product rule, I take each time the derivative of one factor,"},{"Start":"03:38.690 ","End":"03:40.595","Text":"and the other factor as is."},{"Start":"03:40.595 ","End":"03:47.565","Text":"First of all, the derivative of this with respect to y is this and then Mu(xy) as is."},{"Start":"03:47.565 ","End":"03:49.275","Text":"For the second bit,"},{"Start":"03:49.275 ","End":"03:50.566","Text":"I\u0027ve got this;"},{"Start":"03:50.566 ","End":"03:57.600","Text":"(y^4)-(4xy) as is, but I need to differentiate this with respect to y."},{"Start":"03:57.600 ","End":"04:06.305","Text":"This is where I\u0027m going to use this equality here to say that we get f\u0027 is just Mu\u0027,"},{"Start":"04:06.305 ","End":"04:11.690","Text":"h(x, y) is just xy and h with respect to y;"},{"Start":"04:11.690 ","End":"04:14.750","Text":"this xy, the derivative with respect to y,"},{"Start":"04:14.750 ","End":"04:17.330","Text":"is just x as a constant times y."},{"Start":"04:17.330 ","End":"04:19.350","Text":"That gives us this x."},{"Start":"04:19.350 ","End":"04:22.820","Text":"This is what we get from applying this rule."},{"Start":"04:22.820 ","End":"04:25.291","Text":"Similarly with the other bit;"},{"Start":"04:25.291 ","End":"04:27.485","Text":"exactly the same idea,"},{"Start":"04:27.485 ","End":"04:34.430","Text":"but when we take the derivative of Mu(xy) with respect to x then"},{"Start":"04:34.430 ","End":"04:37.490","Text":"again this time on the right-hand side we\u0027ll use"},{"Start":"04:37.490 ","End":"04:41.930","Text":"this equality to say that we get this bit here."},{"Start":"04:41.930 ","End":"04:50.955","Text":"It\u0027s the outer derivative f\u0027 and then h with respect to x is just y."},{"Start":"04:50.955 ","End":"04:55.235","Text":"That\u0027s what we get from the derivative of this with respect to x."},{"Start":"04:55.235 ","End":"04:58.050","Text":"We need to tidy up this mess."},{"Start":"04:58.050 ","End":"05:01.700","Text":"I\u0027m going to remove the highlight because I want to use a different highlighting."},{"Start":"05:01.700 ","End":"05:07.475","Text":"I want to collect together the terms with Mu\u0027(xy) which I have here,"},{"Start":"05:07.475 ","End":"05:11.630","Text":"and here, and I also want to collect together"},{"Start":"05:11.630 ","End":"05:16.680","Text":"the terms with Mu(xy) which I have here, and here."},{"Start":"05:16.680 ","End":"05:21.962","Text":"The bits with the Mu I put on the left and then take outside the brackets,"},{"Start":"05:21.962 ","End":"05:26.490","Text":"notice that I get a minus here because I brought this term over to the left."},{"Start":"05:26.490 ","End":"05:30.410","Text":"Similarly for the Mu\u0027 terms I put everything to"},{"Start":"05:30.410 ","End":"05:36.135","Text":"the right and I get a minus in front of the term from here,"},{"Start":"05:36.135 ","End":"05:38.480","Text":"and that\u0027s starting to take shape."},{"Start":"05:38.480 ","End":"05:41.930","Text":"I need to simplify still more because I have to get everything as"},{"Start":"05:41.930 ","End":"05:46.055","Text":"a function of just xy which is not the case at the moment."},{"Start":"05:46.055 ","End":"05:47.885","Text":"Just opening brackets;"},{"Start":"05:47.885 ","End":"05:49.430","Text":"this is simplification,"},{"Start":"05:49.430 ","End":"05:53.225","Text":"we get (4y^3)-(2y^3) is 2y^3 and so on,"},{"Start":"05:53.225 ","End":"05:56.735","Text":"minus 4x minus, minus 6x, and so on."},{"Start":"05:56.735 ","End":"05:59.330","Text":"On the left I took 2 out."},{"Start":"05:59.330 ","End":"06:04.625","Text":"On the right I just combined like terms, (2xy^4)-(xy^4)."},{"Start":"06:04.625 ","End":"06:06.574","Text":"Similarly this and this combine,"},{"Start":"06:06.574 ","End":"06:08.880","Text":"and now we\u0027re up to here."},{"Start":"06:08.880 ","End":"06:10.860","Text":"It\u0027s still not in the form of something of xy,"},{"Start":"06:10.860 ","End":"06:13.560","Text":"the (y^3)+x is not good for me,"},{"Start":"06:13.560 ","End":"06:21.628","Text":"but note that if you took xy out of this expression what you\u0027re left with is (y^3)+x,"},{"Start":"06:21.628 ","End":"06:23.150","Text":"and now I have this on the right,"},{"Start":"06:23.150 ","End":"06:25.576","Text":"and on the left, so I can get rid of it."},{"Start":"06:25.576 ","End":"06:28.100","Text":"This cancels with this,"},{"Start":"06:28.100 ","End":"06:31.820","Text":"and then we just as usual change sides a bit."},{"Start":"06:31.820 ","End":"06:35.255","Text":"This goes into the denominator,"},{"Start":"06:35.255 ","End":"06:37.330","Text":"and xy goes in the denominator,"},{"Start":"06:37.330 ","End":"06:39.470","Text":"or the other side, and then we switch sides."},{"Start":"06:39.470 ","End":"06:44.940","Text":"We get to this point here where we have Mu\u0027 over Mu."},{"Start":"06:44.940 ","End":"06:47.465","Text":"Xy is like a single unit."},{"Start":"06:47.465 ","End":"06:57.338","Text":"You could think of it like a t. Just think of it like Mu\u0027(t) over Mu(t) is 2 over t,"},{"Start":"06:57.338 ","End":"07:00.180","Text":"and if we had this then we would say,"},{"Start":"07:00.180 ","End":"07:02.780","Text":"\"Yes, we can take the integral of both sides."},{"Start":"07:02.780 ","End":"07:08.750","Text":"We\u0027ve seen this form before and get the natural log of the denominator.\""},{"Start":"07:08.750 ","End":"07:14.135","Text":"If we do that; this is just the way to think about it xy is a single variable,"},{"Start":"07:14.135 ","End":"07:19.099","Text":"we get the integral of this as natural log of the denominator."},{"Start":"07:19.099 ","End":"07:26.015","Text":"If it was 2 over t with 2 natural log(t), this is what we get."},{"Start":"07:26.015 ","End":"07:28.310","Text":"We could write a plus constant,"},{"Start":"07:28.310 ","End":"07:30.845","Text":"but we\u0027re just looking for a specific solution."},{"Start":"07:30.845 ","End":"07:33.500","Text":"We don\u0027t need to do a general case."},{"Start":"07:33.500 ","End":"07:36.200","Text":"In previous exercises I haven\u0027t"},{"Start":"07:36.200 ","End":"07:39.440","Text":"bothered with absolute values just make life a bit easier."},{"Start":"07:39.440 ","End":"07:42.980","Text":"I don\u0027t want to separate it into positive and negative cases."},{"Start":"07:42.980 ","End":"07:47.657","Text":"In the end you\u0027ll see that what we get is an integrating factor,"},{"Start":"07:47.657 ","End":"07:50.690","Text":"so not worrying too much about the little technical details."},{"Start":"07:50.690 ","End":"07:52.115","Text":"So the absolute value."},{"Start":"07:52.115 ","End":"07:53.210","Text":"Others are the constant,"},{"Start":"07:53.210 ","End":"07:57.240","Text":"it doesn\u0027t matter because we only want one particular answer."},{"Start":"07:57.240 ","End":"08:01.040","Text":"From here before I throw out the natural logarithms,"},{"Start":"08:01.040 ","End":"08:06.140","Text":"we\u0027ll just use the property of logarithm to put this 2 into the exponent here."},{"Start":"08:06.140 ","End":"08:08.510","Text":"Now throughout the logs because the logarithms"},{"Start":"08:08.510 ","End":"08:10.870","Text":"are equal the quantities themselves are equal,"},{"Start":"08:10.870 ","End":"08:14.890","Text":"and I get this Mu(xy) is (xy)^2."},{"Start":"08:14.890 ","End":"08:23.450","Text":"This is the integrating factor."},{"Start":"08:23.450 ","End":"08:28.085","Text":"As an extra exercise I suggest you take the original equation,"},{"Start":"08:28.085 ","End":"08:34.135","Text":"multiply it by (xy)^2 and see that you really do get an exact equation."},{"Start":"08:34.135 ","End":"08:38.375","Text":"Just to cement the idea of this function of one variable Mu,"},{"Start":"08:38.375 ","End":"08:41.990","Text":"in our case the function Mu if I describe it in terms of"},{"Start":"08:41.990 ","End":"08:46.890","Text":"a dummy variable t Mu(t) is t^2 but that\u0027s not important."},{"Start":"08:46.890 ","End":"08:49.530","Text":"But then Mu(xy) would be (xy)^2,"},{"Start":"08:49.530 ","End":"08:53.070","Text":"so it is of the form Mu of something."},{"Start":"08:53.070 ","End":"08:55.630","Text":"That\u0027s all then, I\u0027m done."}],"Thumbnail":null,"ID":7702},{"Watched":false,"Name":"Exercise 12","Duration":"7m 29s","ChapterTopicVideoID":7620,"CourseChapterTopicPlaylistID":4222,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.250","Text":"In this exercise we\u0027re given an ordinary differential equation as follows,"},{"Start":"00:05.250 ","End":"00:09.760","Text":"and it happens not to be exact."},{"Start":"00:10.100 ","End":"00:14.340","Text":"You can check that the derivative of this with respect"},{"Start":"00:14.340 ","End":"00:18.165","Text":"to y is not equal to the derivative of this with respect to x."},{"Start":"00:18.165 ","End":"00:21.150","Text":"What we want to do is find an integration factor"},{"Start":"00:21.150 ","End":"00:23.580","Text":"that if we multiply it will make it exact."},{"Start":"00:23.580 ","End":"00:26.700","Text":"But, we want a specific form of an integration factor,"},{"Start":"00:26.700 ","End":"00:27.960","Text":"one that is a function,"},{"Start":"00:27.960 ","End":"00:31.500","Text":"Mu of x plus y. I have done something like this before,"},{"Start":"00:31.500 ","End":"00:33.600","Text":"but I\u0027ll just explain again what this means."},{"Start":"00:33.600 ","End":"00:35.850","Text":"Mu is a function of a single variable."},{"Start":"00:35.850 ","End":"00:39.675","Text":"For example, if Mu of t was"},{"Start":"00:39.675 ","End":"00:45.675","Text":"1/t then Mu of x"},{"Start":"00:45.675 ","End":"00:50.035","Text":"plus y would mean 1 over x plus y."},{"Start":"00:50.035 ","End":"00:54.380","Text":"It just means that we have some expression which is only an x plus y."},{"Start":"00:54.380 ","End":"00:56.870","Text":"You don\u0027t get x separately and y separately,"},{"Start":"00:56.870 ","End":"00:59.915","Text":"they appear together as x plus y."},{"Start":"00:59.915 ","End":"01:02.320","Text":"That\u0027s just for instance."},{"Start":"01:02.320 ","End":"01:07.565","Text":"If this is an integration factor then I\u0027m going to multiply by the equation."},{"Start":"01:07.565 ","End":"01:10.895","Text":"I\u0027ve put it in here, and here."},{"Start":"01:10.895 ","End":"01:13.880","Text":"That\u0027s supposed to now make the equation exact,"},{"Start":"01:13.880 ","End":"01:18.650","Text":"meaning if I take this bit and differentiate with respect to y,"},{"Start":"01:18.650 ","End":"01:23.795","Text":"and this bit with respect to x then they should be equal."},{"Start":"01:23.795 ","End":"01:30.740","Text":"I\u0027ll just write this with respect to y equals this with respect to x."},{"Start":"01:30.740 ","End":"01:31.813","Text":"This is like our m,"},{"Start":"01:31.813 ","End":"01:34.025","Text":"and n in definition."},{"Start":"01:34.025 ","End":"01:36.530","Text":"Now to differentiate this,"},{"Start":"01:36.530 ","End":"01:37.970","Text":"it could be a little bit tricky."},{"Start":"01:37.970 ","End":"01:40.640","Text":"What do we do with this Mu of x plus y?"},{"Start":"01:40.640 ","End":"01:43.288","Text":"How do we differentiate that with respect to y,"},{"Start":"01:43.288 ","End":"01:46.865","Text":"and how do we differentiate this with respect to x?"},{"Start":"01:46.865 ","End":"01:49.010","Text":"We\u0027re going to need that as part of"},{"Start":"01:49.010 ","End":"01:53.390","Text":"the general differentiation involving chain rule and so on."},{"Start":"01:53.390 ","End":"01:57.380","Text":"I brought in these 2 formulas and 1,"},{"Start":"01:57.380 ","End":"02:02.810","Text":"which I\u0027ve used before when we have a function of 1 variable on the outside,"},{"Start":"02:02.810 ","End":"02:05.270","Text":"and inside a function of 2 variables and we want the"},{"Start":"02:05.270 ","End":"02:09.110","Text":"partial derivative with respect to x, and with respect to y."},{"Start":"02:09.110 ","End":"02:12.500","Text":"We\u0027ve used this before. Let me just say that the key for using this"},{"Start":"02:12.500 ","End":"02:15.915","Text":"in our case is that f is exactly Mu,"},{"Start":"02:15.915 ","End":"02:18.840","Text":"and the function of 2 variables,"},{"Start":"02:18.840 ","End":"02:24.115","Text":"h( x, y) is just the expression x plus y."},{"Start":"02:24.115 ","End":"02:26.453","Text":"If we consider that here,"},{"Start":"02:26.453 ","End":"02:28.498","Text":"now let\u0027s do the derivative,"},{"Start":"02:28.498 ","End":"02:31.955","Text":"and pay special attention when we differentiate the Mu part."},{"Start":"02:31.955 ","End":"02:35.945","Text":"Now for the left-hand side I see a product."},{"Start":"02:35.945 ","End":"02:40.460","Text":"With a product use the product rule, differentiate the first,"},{"Start":"02:40.460 ","End":"02:41.982","Text":"the second as is,"},{"Start":"02:41.982 ","End":"02:43.685","Text":"and then vice versa,"},{"Start":"02:43.685 ","End":"02:46.310","Text":"here\u0027s this expression as is."},{"Start":"02:46.310 ","End":"02:48.500","Text":"Now I need the derivative of this."},{"Start":"02:48.500 ","End":"02:56.600","Text":"Using the formula here what I get is f\u0027 is Mu\u0027 of x plus y."},{"Start":"02:56.600 ","End":"02:59.450","Text":"Then I need this h_x,"},{"Start":"02:59.450 ","End":"03:03.095","Text":"which is the derivative of this with respect to x."},{"Start":"03:03.095 ","End":"03:04.910","Text":"It\u0027s x plus a constant,"},{"Start":"03:04.910 ","End":"03:05.990","Text":"like x plus 5,"},{"Start":"03:05.990 ","End":"03:08.255","Text":"the derivative is 1."},{"Start":"03:08.255 ","End":"03:14.000","Text":"This bit I underlined it\u0027s with respect to y,"},{"Start":"03:14.000 ","End":"03:15.650","Text":"but same thing applies."},{"Start":"03:15.650 ","End":"03:17.825","Text":"The derivative with respect to y,"},{"Start":"03:17.825 ","End":"03:20.825","Text":"like 3 plus y is also 1."},{"Start":"03:20.825 ","End":"03:22.925","Text":"This is with respect to y."},{"Start":"03:22.925 ","End":"03:24.650","Text":"Similarly on the other side,"},{"Start":"03:24.650 ","End":"03:27.140","Text":"and this is the bit that we get when we"},{"Start":"03:27.140 ","End":"03:31.170","Text":"differentiate this Mu of x plus y with respect to x."},{"Start":"03:31.170 ","End":"03:34.245","Text":"Put a prime on the f which is Mu,"},{"Start":"03:34.245 ","End":"03:37.670","Text":"and on the outside we need this with respect to y,"},{"Start":"03:37.670 ","End":"03:40.745","Text":"which as I said, is also 1 in both cases."},{"Start":"03:40.745 ","End":"03:43.150","Text":"Now we want to simplify this mess,"},{"Start":"03:43.150 ","End":"03:46.610","Text":"and we\u0027ll collect together some similar stuff."},{"Start":"03:46.610 ","End":"03:50.630","Text":"For example, here I see Mu\u0027 of x plus y,"},{"Start":"03:50.630 ","End":"03:53.950","Text":"and here I see Mu\u0027 of x plus y."},{"Start":"03:53.950 ","End":"03:56.220","Text":"Here I see Mu of x plus y,"},{"Start":"03:56.220 ","End":"03:59.550","Text":"and I\u0027ll collect that with the other Mu of x plus y,"},{"Start":"03:59.550 ","End":"04:00.590","Text":"one of them on the left,"},{"Start":"04:00.590 ","End":"04:01.865","Text":"one of them on the right."},{"Start":"04:01.865 ","End":"04:06.590","Text":"Let\u0027s say we take the Mu of x plus y on the left what we\u0027re left with is this,"},{"Start":"04:06.590 ","End":"04:12.415","Text":"which is here, but minus this here because we brought it over to the other side."},{"Start":"04:12.415 ","End":"04:19.325","Text":"Similarly, the Mu\u0027 on the right we get this as is here,"},{"Start":"04:19.325 ","End":"04:23.899","Text":"minus one that came from the left over to the right."},{"Start":"04:23.899 ","End":"04:29.869","Text":"This line is what I get when I start opening brackets and combining like terms."},{"Start":"04:29.869 ","End":"04:36.095","Text":"For example, the 9 minus the 3 gives me 6y^2 and the 2 minus 6 is minus 4x."},{"Start":"04:36.095 ","End":"04:39.235","Text":"Similarly, here combining x^2,"},{"Start":"04:39.235 ","End":"04:41.160","Text":"this is what we\u0027re left with."},{"Start":"04:41.160 ","End":"04:45.425","Text":"I need some more space here. Let\u0027s see."},{"Start":"04:45.425 ","End":"04:48.920","Text":"I\u0027m starting to do some algebra because I want to get everything in terms of"},{"Start":"04:48.920 ","End":"04:54.680","Text":"just x plus y. I want to try and see if I can factorize this,"},{"Start":"04:54.680 ","End":"04:57.965","Text":"and maybe this also, and get rid of a common factor."},{"Start":"04:57.965 ","End":"05:01.175","Text":"From here I can factorize in groups."},{"Start":"05:01.175 ","End":"05:05.025","Text":"If I take the 3y^2 out of"},{"Start":"05:05.025 ","End":"05:10.515","Text":"this and this then this is what I get, 3y^2, I\u0027m left with y plus x."},{"Start":"05:10.515 ","End":"05:12.268","Text":"From the first,"},{"Start":"05:12.268 ","End":"05:16.070","Text":"and the last I can get minus 2x,"},{"Start":"05:16.070 ","End":"05:19.255","Text":"and then I also get x plus y."},{"Start":"05:19.255 ","End":"05:24.760","Text":"But I wrote it as y plus x because I wanted it to look the same as the other factor."},{"Start":"05:24.760 ","End":"05:29.190","Text":"Now I can take y plus x out of the brackets here,"},{"Start":"05:29.190 ","End":"05:32.295","Text":"and we\u0027re left with the 3y^2 minus the 2x."},{"Start":"05:32.295 ","End":"05:38.355","Text":"On the left, if I just take 2 out of this I\u0027ve also got 3y^2 minus 2x."},{"Start":"05:38.355 ","End":"05:42.500","Text":"This term cancels with this factor,"},{"Start":"05:42.500 ","End":"05:43.758","Text":"not a term,"},{"Start":"05:43.758 ","End":"05:46.235","Text":"and now it\u0027s already a lot simpler."},{"Start":"05:46.235 ","End":"05:49.790","Text":"What we still have to do now is to separate."},{"Start":"05:49.790 ","End":"05:52.100","Text":"I\u0027m going to divide by Mu,"},{"Start":"05:52.100 ","End":"05:55.085","Text":"take the Mu and put it under the Mu\u0027,"},{"Start":"05:55.085 ","End":"05:59.375","Text":"take the y plus x and put it over here,"},{"Start":"05:59.375 ","End":"06:01.735","Text":"and then switch sides also."},{"Start":"06:01.735 ","End":"06:03.807","Text":"What we\u0027re left, not hard to see,"},{"Start":"06:03.807 ","End":"06:06.830","Text":"this Mu\u0027 over Mu is 2 over y plus x,"},{"Start":"06:06.830 ","End":"06:09.935","Text":"which I write now as x plus y."},{"Start":"06:09.935 ","End":"06:13.280","Text":"Now just think of x plus y as some dummy variable"},{"Start":"06:13.280 ","End":"06:17.299","Text":"t. It\u0027s like x plus y could be anything,"},{"Start":"06:17.299 ","End":"06:20.900","Text":"like Mu\u0027 of t over Mu of t"},{"Start":"06:20.900 ","End":"06:25.550","Text":"equals 2 over t. If you think of it this way then here you would take the"},{"Start":"06:25.550 ","End":"06:29.465","Text":"integral of both sides and get natural log of Mu"},{"Start":"06:29.465 ","End":"06:33.950","Text":"and here twice natural log of t. This is what I\u0027m doing."},{"Start":"06:33.950 ","End":"06:37.475","Text":"I\u0027m just thinking of x plus y as a single variable."},{"Start":"06:37.475 ","End":"06:38.900","Text":"This is what we get."},{"Start":"06:38.900 ","End":"06:41.800","Text":"Now the 2 can go inside."},{"Start":"06:41.800 ","End":"06:43.550","Text":"By the rules of logarithms,"},{"Start":"06:43.550 ","End":"06:46.474","Text":"I can take this, put it inside the exponent."},{"Start":"06:46.474 ","End":"06:49.378","Text":"Now if the logarithms of 2 things are equal,"},{"Start":"06:49.378 ","End":"06:56.745","Text":"and the things themselves are equal now this expression here is a function of x plus y."},{"Start":"06:56.745 ","End":"06:58.880","Text":"In fact, not that it\u0027s important,"},{"Start":"06:58.880 ","End":"07:02.950","Text":"but you could define Mu of t is equal to t^2."},{"Start":"07:02.950 ","End":"07:04.280","Text":"That\u0027s the function of t,"},{"Start":"07:04.280 ","End":"07:05.825","Text":"but we apply it to x plus y."},{"Start":"07:05.825 ","End":"07:12.410","Text":"This is a function of x plus y. I recommend if you have the time to"},{"Start":"07:12.410 ","End":"07:15.500","Text":"see that this really is an integration factor go back to"},{"Start":"07:15.500 ","End":"07:19.685","Text":"the original differential equation and multiply by x plus y^2,"},{"Start":"07:19.685 ","End":"07:21.380","Text":"and check that it\u0027s exact."},{"Start":"07:21.380 ","End":"07:23.000","Text":"I do not ask you to solve it,"},{"Start":"07:23.000 ","End":"07:26.270","Text":"just check that we get an exact equation."},{"Start":"07:26.270 ","End":"07:30.420","Text":"That\u0027s the integrating factor and we are done."}],"Thumbnail":null,"ID":7703},{"Watched":false,"Name":"Exercise 13","Duration":"9m 55s","ChapterTopicVideoID":7621,"CourseChapterTopicPlaylistID":4222,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.460","Text":"In this exercise, we\u0027re given an ordinary differential equation, this 1 here."},{"Start":"00:05.460 ","End":"00:08.955","Text":"If you check, you will see that it\u0027s not exact."},{"Start":"00:08.955 ","End":"00:13.125","Text":"That\u0027s why we want to look for an integration factor."},{"Start":"00:13.125 ","End":"00:16.500","Text":"We don\u0027t know exactly what the integration factor,"},{"Start":"00:16.500 ","End":"00:17.850","Text":"what form it will be."},{"Start":"00:17.850 ","End":"00:23.040","Text":"But here it suggested that we look for something of the form x to the power of something,"},{"Start":"00:23.040 ","End":"00:24.749","Text":"y to the power of something."},{"Start":"00:24.749 ","End":"00:27.300","Text":"We\u0027ve used Greek letters Alpha and Beta."},{"Start":"00:27.300 ","End":"00:28.950","Text":"But if you don\u0027t like Greek letters,"},{"Start":"00:28.950 ","End":"00:31.830","Text":"you could put m and n here or something."},{"Start":"00:31.830 ","End":"00:35.309","Text":"Ensure we\u0027re looking for integration factor of this form."},{"Start":"00:35.309 ","End":"00:37.395","Text":"I want to make some remarks."},{"Start":"00:37.395 ","End":"00:40.820","Text":"The first remark is that if I was just given this equation to solve,"},{"Start":"00:40.820 ","End":"00:44.150","Text":"I wouldn\u0027t even be looking for an integration factor."},{"Start":"00:44.150 ","End":"00:47.390","Text":"I would just take x out here and you can see then that I\u0027ve got"},{"Start":"00:47.390 ","End":"00:50.930","Text":"separation of variables that bring the x\u0027s to 1 side,"},{"Start":"00:50.930 ","End":"00:52.340","Text":"the y\u0027s to the other side,"},{"Start":"00:52.340 ","End":"00:54.380","Text":"and it would be more straightforward."},{"Start":"00:54.380 ","End":"00:58.310","Text":"We\u0027re doing this for the practice and for learning the concept."},{"Start":"00:58.310 ","End":"00:59.810","Text":"Now in previous exercise,"},{"Start":"00:59.810 ","End":"01:01.640","Text":"we were given something more specific,"},{"Start":"01:01.640 ","End":"01:04.115","Text":"not with unknowns Alpha and Beta."},{"Start":"01:04.115 ","End":"01:09.635","Text":"We were given of the form μ(x, y) or μ(x+y)."},{"Start":"01:09.635 ","End":"01:12.140","Text":"This, although it\u0027s more work,"},{"Start":"01:12.140 ","End":"01:14.375","Text":"is more flexible, more general,"},{"Start":"01:14.375 ","End":"01:17.200","Text":"because sometimes we have an idea it\u0027s going to be x to the something,"},{"Start":"01:17.200 ","End":"01:18.565","Text":"y to the something,"},{"Start":"01:18.565 ","End":"01:21.770","Text":"but we\u0027re not quite sure what those somethings are."},{"Start":"01:21.770 ","End":"01:24.035","Text":"This is more general."},{"Start":"01:24.035 ","End":"01:26.590","Text":"Let\u0027s start working."},{"Start":"01:26.590 ","End":"01:28.740","Text":"On the other comment I think I already made that yes,"},{"Start":"01:28.740 ","End":"01:30.250","Text":"some people don\u0027t like Greek letters,"},{"Start":"01:30.250 ","End":"01:33.455","Text":"then just use m and n instead of Alpha and Beta."},{"Start":"01:33.455 ","End":"01:35.675","Text":"Here I jump 2 steps in 1."},{"Start":"01:35.675 ","End":"01:40.310","Text":"First of all, I multiplied out by the integration factor,"},{"Start":"01:40.310 ","End":"01:43.070","Text":"this on both sides of the equation."},{"Start":"01:43.070 ","End":"01:45.140","Text":"Well, on the right it\u0027s just 0."},{"Start":"01:45.140 ","End":"01:46.910","Text":"But if we do that,"},{"Start":"01:46.910 ","End":"01:48.985","Text":"after we multiplied out,"},{"Start":"01:48.985 ","End":"01:51.180","Text":"but this is maybe even 3 steps in 1,"},{"Start":"01:51.180 ","End":"01:54.630","Text":"we multiply everything by μ(x^Alpha, y^Beta)."},{"Start":"01:54.630 ","End":"01:58.520","Text":"We get something here plus something else here equals 0."},{"Start":"01:58.520 ","End":"02:02.940","Text":"What we get here is this times the integration factor, similarly here."},{"Start":"02:02.940 ","End":"02:06.590","Text":"Then we say that the partial derivative with respect to y of"},{"Start":"02:06.590 ","End":"02:12.620","Text":"the new m equals derivative with respect to x of the new n. This is what we get."},{"Start":"02:12.620 ","End":"02:16.250","Text":"Just take a few shortcuts because it\u0027s not the first exercise."},{"Start":"02:16.250 ","End":"02:20.870","Text":"Now, I remind you of particular form of the chain rule that we\u0027ve used"},{"Start":"02:20.870 ","End":"02:22.730","Text":"before when we have a function of"},{"Start":"02:22.730 ","End":"02:25.970","Text":"1 variable on the outside and inside a function of 2 variables,"},{"Start":"02:25.970 ","End":"02:29.270","Text":"x and y, the partial derivative with respect to x,"},{"Start":"02:29.270 ","End":"02:31.310","Text":"the partial derivative with respect to y."},{"Start":"02:31.310 ","End":"02:32.945","Text":"We\u0027ve seen this before."},{"Start":"02:32.945 ","End":"02:39.395","Text":"We\u0027re going to apply this where f is exactly μ and h(x,"},{"Start":"02:39.395 ","End":"02:44.850","Text":"y) is this expression, x^Alpha y^Beta."},{"Start":"02:45.770 ","End":"02:50.990","Text":"Using the product rule and this variant of the chain rule,"},{"Start":"02:50.990 ","End":"02:52.790","Text":"we get the following mess."},{"Start":"02:52.790 ","End":"02:54.680","Text":"I\u0027ll just briefly go over it."},{"Start":"02:54.680 ","End":"02:58.520","Text":"Product rule, so derivative of x^2 y^3,"},{"Start":"02:58.520 ","End":"03:00.330","Text":"μ of this as is,"},{"Start":"03:00.330 ","End":"03:01.875","Text":"and then vice versa."},{"Start":"03:01.875 ","End":"03:04.410","Text":"The x^2 y^3 here,"},{"Start":"03:04.410 ","End":"03:06.720","Text":"that\u0027s from here as is."},{"Start":"03:06.720 ","End":"03:12.660","Text":"All this bit here is what we get from this here."},{"Start":"03:12.660 ","End":"03:15.664","Text":"If I differentiate this with respect to y,"},{"Start":"03:15.664 ","End":"03:19.370","Text":"first of all, I get μ\u0027 of the same thing."},{"Start":"03:19.370 ","End":"03:25.260","Text":"Then I need the derivative of this with respect to y."},{"Start":"03:25.640 ","End":"03:27.660","Text":"Since x is a constant,"},{"Start":"03:27.660 ","End":"03:29.500","Text":"the x^Alpha stays,"},{"Start":"03:29.500 ","End":"03:34.115","Text":"and the derivative of this with respect to y is Beta y^Beta minus 1."},{"Start":"03:34.115 ","End":"03:38.025","Text":"The Beta is here and the exponent of y is here,"},{"Start":"03:38.025 ","End":"03:39.945","Text":"so this is what we get."},{"Start":"03:39.945 ","End":"03:47.615","Text":"Similarly on the right-hand side where we differentiate this bit using this rule."},{"Start":"03:47.615 ","End":"03:52.189","Text":"This time the derivative of h with respect to x,"},{"Start":"03:52.189 ","End":"03:55.205","Text":"we get Alpha x^Alpha minus 1,"},{"Start":"03:55.205 ","End":"03:59.030","Text":"and y^Beta, and all the rest of it is just product rule."},{"Start":"03:59.030 ","End":"04:01.145","Text":"Now I want to tidy this up a bit."},{"Start":"04:01.145 ","End":"04:07.790","Text":"I know I must indicate that this bit is what we got from the right-hand side here,"},{"Start":"04:07.790 ","End":"04:13.670","Text":"where f is μ and h with respect to x was this bit here."},{"Start":"04:13.670 ","End":"04:20.420","Text":"1 thing that bothers me is that here I have x^Alpha minus 1,"},{"Start":"04:20.420 ","End":"04:23.255","Text":"well, also except here Beta minus 1."},{"Start":"04:23.255 ","End":"04:27.035","Text":"I mostly want things in terms of x^Alpha, y^Beta."},{"Start":"04:27.035 ","End":"04:32.450","Text":"What I\u0027m going to do is if I get rid of this minus 1,"},{"Start":"04:32.450 ","End":"04:35.480","Text":"this is like multiplying by x to the minus 1,"},{"Start":"04:35.480 ","End":"04:42.130","Text":"I could divide this thing by x and say 1 plus y^2,"},{"Start":"04:42.130 ","End":"04:44.435","Text":"and then I wouldn\u0027t have to have the minus 1."},{"Start":"04:44.435 ","End":"04:48.905","Text":"Similarly here, I have a Beta to the minus 1,"},{"Start":"04:48.905 ","End":"04:51.770","Text":"which is like y^Beta, y to the minus 1."},{"Start":"04:51.770 ","End":"04:57.475","Text":"I could get rid of the minus 1 if I reduce the power of y by 1."},{"Start":"04:57.475 ","End":"05:01.070","Text":"Then I get here by the same trick as I did before."},{"Start":"05:01.070 ","End":"05:06.305","Text":"I took the places where I have μ of this and μ of this,"},{"Start":"05:06.305 ","End":"05:10.945","Text":"and I gathered those up on the left."},{"Start":"05:10.945 ","End":"05:15.440","Text":"Of course, I get a minus here because I\u0027ve got this minus this."},{"Start":"05:15.440 ","End":"05:18.395","Text":"Then the same thing with wherever I have μ\u0027"},{"Start":"05:18.395 ","End":"05:22.430","Text":"of this expression here and here μ\u0027 of this expression."},{"Start":"05:22.430 ","End":"05:24.620","Text":"I put those on the right."},{"Start":"05:24.620 ","End":"05:29.520","Text":"What was on the right is this after I corrected it."},{"Start":"05:29.520 ","End":"05:32.060","Text":"On the left, this, again,"},{"Start":"05:32.060 ","End":"05:34.860","Text":"after the correction with a minus."},{"Start":"05:34.860 ","End":"05:36.840","Text":"Now I\u0027m in a good place,"},{"Start":"05:36.840 ","End":"05:38.370","Text":"so we\u0027ve seen this thing before."},{"Start":"05:38.370 ","End":"05:43.593","Text":"What we want to do now is put the μ\u0027 over the μ in the denominator,"},{"Start":"05:43.593 ","End":"05:45.410","Text":"the rest of it in the denominator there,"},{"Start":"05:45.410 ","End":"05:46.915","Text":"and maybe switch sides,"},{"Start":"05:46.915 ","End":"05:48.625","Text":"but not quite yet soon."},{"Start":"05:48.625 ","End":"05:51.635","Text":"I just want to do a bit of tidying up first."},{"Start":"05:51.635 ","End":"05:54.245","Text":"What I did here was take the x^Alpha,"},{"Start":"05:54.245 ","End":"05:56.795","Text":"y^Beta from here and here,"},{"Start":"05:56.795 ","End":"05:59.730","Text":"and bring it out of the brackets."},{"Start":"05:59.730 ","End":"06:01.905","Text":"Then this is what we\u0027re left with."},{"Start":"06:01.905 ","End":"06:04.140","Text":"They haven\u0027t touched the other side."},{"Start":"06:04.140 ","End":"06:08.030","Text":"This might be a good point to select Alpha and Beta."},{"Start":"06:08.030 ","End":"06:13.525","Text":"Now, I see I have 1 plus y^2 and here I have a 1 plus y^2."},{"Start":"06:13.525 ","End":"06:18.130","Text":"It looks like I could take Alpha to be minus 1,"},{"Start":"06:18.130 ","End":"06:20.900","Text":"but we could also take it to be 1."},{"Start":"06:20.900 ","End":"06:23.705","Text":"If we\u0027ve take both of them as being minus,"},{"Start":"06:23.705 ","End":"06:24.800","Text":"it will work out."},{"Start":"06:24.800 ","End":"06:26.465","Text":"There will be a common minus."},{"Start":"06:26.465 ","End":"06:29.855","Text":"Maybe I\u0027ll want to switch sides and minus will become a plus."},{"Start":"06:29.855 ","End":"06:34.255","Text":"Here also, if I have a 3 here and a minus Beta here,"},{"Start":"06:34.255 ","End":"06:37.400","Text":"I could take Beta equals 3. I\u0027ll just make a remark."},{"Start":"06:37.400 ","End":"06:44.150","Text":"You could also solve it by taking Beta equals minus 3 and Alpha equals minus 1,"},{"Start":"06:44.150 ","End":"06:48.305","Text":"and that you would get actually the same answer in the end."},{"Start":"06:48.305 ","End":"06:51.810","Text":"But anyway, we\u0027ll take it as not to work with negatives."},{"Start":"06:51.810 ","End":"06:53.085","Text":"We\u0027ll take this,"},{"Start":"06:53.085 ","End":"06:54.965","Text":"and you\u0027ll see how it works out."},{"Start":"06:54.965 ","End":"06:57.350","Text":"If I substitute these values of Alpha and Beta,"},{"Start":"06:57.350 ","End":"06:58.820","Text":"this is what I get."},{"Start":"06:58.820 ","End":"07:04.210","Text":"Really, this square bracket is the same as this square brackets except for a minus."},{"Start":"07:04.210 ","End":"07:09.890","Text":"Pull the minus out of here and then we\u0027ve got exactly the same as square brackets here."},{"Start":"07:09.890 ","End":"07:13.670","Text":"I think we\u0027re just about ready to do what I said I was going to do before,"},{"Start":"07:13.670 ","End":"07:17.995","Text":"but kept postponing is to do this μ\u0027/μ,"},{"Start":"07:17.995 ","End":"07:20.354","Text":"and this over this."},{"Start":"07:20.354 ","End":"07:21.700","Text":"Well, should\u0027ve indicated."},{"Start":"07:21.700 ","End":"07:26.525","Text":"I\u0027m canceling this square bracket because it\u0027s common to both."},{"Start":"07:26.525 ","End":"07:30.710","Text":"Then if I bring the minus to the other side,"},{"Start":"07:30.710 ","End":"07:40.040","Text":"I\u0027ve got minus μ\u0027/μ and this minus xy^3 on the other side is minus 1/xy^3."},{"Start":"07:40.040 ","End":"07:45.440","Text":"Now we\u0027re in a good situation of having μ\u0027/μ."},{"Start":"07:45.440 ","End":"07:49.265","Text":"Notice that everywhere we have xy^3 cubed."},{"Start":"07:49.265 ","End":"07:52.415","Text":"If I said t is xy^3,"},{"Start":"07:52.415 ","End":"07:54.740","Text":"then this pretty much says"},{"Start":"07:54.740 ","End":"08:02.635","Text":"μ\u0027(t)/μ(t) equals minus 1/t."},{"Start":"08:02.635 ","End":"08:04.080","Text":"If I think of it that way,"},{"Start":"08:04.080 ","End":"08:07.930","Text":"as xy^3 being the single letter t,"},{"Start":"08:07.930 ","End":"08:10.040","Text":"then we can take the integral."},{"Start":"08:10.040 ","End":"08:15.075","Text":"Here, we\u0027ve got natural log of μ(t), t is this."},{"Start":"08:15.075 ","End":"08:16.770","Text":"As I\u0027ve said in previous exercises,"},{"Start":"08:16.770 ","End":"08:20.465","Text":"we don\u0027t make life more difficult than we need to with absolute values,"},{"Start":"08:20.465 ","End":"08:25.990","Text":"just restrict the conditions so that we don\u0027t know this thing is going to be plus."},{"Start":"08:25.990 ","End":"08:28.110","Text":"Here we have minus 1/t,"},{"Start":"08:28.110 ","End":"08:30.790","Text":"so minus natural log."},{"Start":"08:31.670 ","End":"08:33.690","Text":"This is a minus 1."},{"Start":"08:33.690 ","End":"08:35.430","Text":"We can bring it into the exponent."},{"Start":"08:35.430 ","End":"08:37.710","Text":"Just the rules of logarithms."},{"Start":"08:37.710 ","End":"08:40.070","Text":"This really should have a bracket just to avoid"},{"Start":"08:40.070 ","End":"08:43.865","Text":"ambiguity that it\u0027s the log of this whole thing."},{"Start":"08:43.865 ","End":"08:48.200","Text":"The logarithms are equal and the quantities are equal."},{"Start":"08:48.200 ","End":"08:53.560","Text":"Our μ ends up being 1/xy^3."},{"Start":"08:53.560 ","End":"08:58.175","Text":"This is the integrating factor that we were looking for."},{"Start":"08:58.175 ","End":"09:01.445","Text":"As I\u0027ve said in previous exercises, if you have time,"},{"Start":"09:01.445 ","End":"09:03.500","Text":"go back and check that this really works,"},{"Start":"09:03.500 ","End":"09:07.535","Text":"that this makes the equation exact if you multiply by it."},{"Start":"09:07.535 ","End":"09:08.830","Text":"Just a small remark."},{"Start":"09:08.830 ","End":"09:12.530","Text":"In case you\u0027re saying what about the x^Alpha, y^Beta?"},{"Start":"09:12.860 ","End":"09:21.695","Text":"Well, it came out in the end that Alpha is minus 1 and Beta is minus 3."},{"Start":"09:21.695 ","End":"09:26.765","Text":"I remember I mentioned above that the Alpha and Beta we chose,"},{"Start":"09:26.765 ","End":"09:28.820","Text":"they worked as far as that went,"},{"Start":"09:28.820 ","End":"09:32.140","Text":"but we could have started off with minus 3 and minus 1."},{"Start":"09:32.140 ","End":"09:36.665","Text":"Then the difference is that we wouldn\u0027t have got a minus here,"},{"Start":"09:36.665 ","End":"09:38.855","Text":"wouldn\u0027t have got a minus here,"},{"Start":"09:38.855 ","End":"09:41.750","Text":"and we would have just got the x to the minus 1,"},{"Start":"09:41.750 ","End":"09:44.030","Text":"y to the minus 3, but without the minus here,"},{"Start":"09:44.030 ","End":"09:46.580","Text":"would have come out the same thing in the end."},{"Start":"09:46.580 ","End":"09:47.915","Text":"This is what we would have got,"},{"Start":"09:47.915 ","End":"09:49.265","Text":"x to the minus 1,"},{"Start":"09:49.265 ","End":"09:50.675","Text":"y to the minus 3."},{"Start":"09:50.675 ","End":"09:55.890","Text":"In any event, this is the integration factor and we are done."}],"Thumbnail":null,"ID":7704},{"Watched":false,"Name":"Exercise 14","Duration":"10m ","ChapterTopicVideoID":7622,"CourseChapterTopicPlaylistID":4222,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.295","Text":"In this exercise, we\u0027re given"},{"Start":"00:02.295 ","End":"00:07.005","Text":"an ordinary differential equation but in general of this form,"},{"Start":"00:07.005 ","End":"00:09.600","Text":"we want to find the condition on the equation,"},{"Start":"00:09.600 ","End":"00:13.068","Text":"some condition on M and N in order that it have"},{"Start":"00:13.068 ","End":"00:17.790","Text":"an integration factor for this equation which is a function of just x,"},{"Start":"00:17.790 ","End":"00:22.210","Text":"y, some function, some Mu(xy)."},{"Start":"00:22.340 ","End":"00:25.710","Text":"Then once we\u0027ve solved part a,"},{"Start":"00:25.710 ","End":"00:31.155","Text":"we\u0027re going to use the results to find an integration factor for this equation."},{"Start":"00:31.155 ","End":"00:33.780","Text":"Presumably this will satisfy the conditions that we"},{"Start":"00:33.780 ","End":"00:36.615","Text":"found here and then we\u0027ll use it to solve it."},{"Start":"00:36.615 ","End":"00:41.150","Text":"What I want to remark is that of course we could have just started from Part b,"},{"Start":"00:41.150 ","End":"00:43.490","Text":"assuming we have an integration factor,"},{"Start":"00:43.490 ","End":"00:45.290","Text":"some function Mu(xy),"},{"Start":"00:45.290 ","End":"00:50.120","Text":"and then multiply that and then seeing if it works out or not"},{"Start":"00:50.120 ","End":"00:55.265","Text":"and find out what that is but we want to do this in more general."},{"Start":"00:55.265 ","End":"00:57.290","Text":"This is like a generalization,"},{"Start":"00:57.290 ","End":"01:01.040","Text":"then we can check right away the condition here that it\u0027s met."},{"Start":"01:01.040 ","End":"01:03.125","Text":"In fact, in Part a,"},{"Start":"01:03.125 ","End":"01:04.700","Text":"not only will we find the condition,"},{"Start":"01:04.700 ","End":"01:09.080","Text":"but it will also give us a formula for what is that integration factor."},{"Start":"01:09.080 ","End":"01:11.045","Text":"This is just a generalization."},{"Start":"01:11.045 ","End":"01:16.220","Text":"We\u0027ve even done something like this in the past to look for integration factor of xy,"},{"Start":"01:16.220 ","End":"01:18.095","Text":"we had a previous exercise."},{"Start":"01:18.095 ","End":"01:20.290","Text":"What we\u0027re doing is more general."},{"Start":"01:20.290 ","End":"01:23.085","Text":"We\u0027re calling this function of xy,"},{"Start":"01:23.085 ","End":"01:27.865","Text":"Mu(xy), and multiplying it by both sides of the equation."},{"Start":"01:27.865 ","End":"01:32.000","Text":"Now I\u0027m going to apply the condition that this is an exact equation."},{"Start":"01:32.000 ","End":"01:38.210","Text":"This with respect to y is going to be the derivative of this with respect to x."},{"Start":"01:38.210 ","End":"01:41.140","Text":"That\u0027s what I\u0027ve written over here."},{"Start":"01:41.140 ","End":"01:44.900","Text":"Here\u0027s the case of the chain rule that we need."},{"Start":"01:44.900 ","End":"01:46.775","Text":"We\u0027ve used this in the past,"},{"Start":"01:46.775 ","End":"01:48.260","Text":"where in our case,"},{"Start":"01:48.260 ","End":"01:53.620","Text":"f is exactly the function Mu and our h(x,"},{"Start":"01:53.620 ","End":"01:57.860","Text":"y) in this case is the expression in x and y,"},{"Start":"01:57.860 ","End":"02:02.020","Text":"namely xy that we want to be a function of and"},{"Start":"02:02.020 ","End":"02:06.470","Text":"applying that and the product rule, this is what we get."},{"Start":"02:06.470 ","End":"02:09.150","Text":"We\u0027ve done this thing before."},{"Start":"02:09.220 ","End":"02:13.655","Text":"What we have is the chain rule M with respect to y,"},{"Start":"02:13.655 ","End":"02:16.265","Text":"this Mu as is and then vice versa."},{"Start":"02:16.265 ","End":"02:17.540","Text":"We have M as is,"},{"Start":"02:17.540 ","End":"02:20.915","Text":"and then the derivative of this bit,"},{"Start":"02:20.915 ","End":"02:24.980","Text":"we use this formula here so we get Mu prime,"},{"Start":"02:24.980 ","End":"02:28.160","Text":"which is the f prime that\u0027s here."},{"Start":"02:28.160 ","End":"02:35.235","Text":"Then the derivative h with respect to y is just the constant x as far as y goes."},{"Start":"02:35.235 ","End":"02:39.970","Text":"That\u0027s this bit here is from this formula, the second one."},{"Start":"02:39.970 ","End":"02:41.960","Text":"Similarly in the other bit,"},{"Start":"02:41.960 ","End":"02:44.250","Text":"if we use this formula here,"},{"Start":"02:44.250 ","End":"02:46.835","Text":"h with respect to x is y,"},{"Start":"02:46.835 ","End":"02:51.630","Text":"and that\u0027s what we get and the rest of it is just use of the product rule."},{"Start":"02:52.310 ","End":"02:55.220","Text":"What I did is what we\u0027ve done before,"},{"Start":"02:55.220 ","End":"02:57.980","Text":"separate the Mu from the Mu prime,"},{"Start":"02:57.980 ","End":"03:02.065","Text":"Mu(xy) on the left and what we\u0027re left with is M_y,"},{"Start":"03:02.065 ","End":"03:04.965","Text":"the N_x from here becomes a minus."},{"Start":"03:04.965 ","End":"03:06.960","Text":"Likewise on the right Mu(xy),"},{"Start":"03:06.960 ","End":"03:09.195","Text":"we have Mu prime."},{"Start":"03:09.195 ","End":"03:12.080","Text":"Here we have y times n, which is here,"},{"Start":"03:12.080 ","End":"03:14.420","Text":"and this x times M,"},{"Start":"03:14.420 ","End":"03:16.175","Text":"it comes out minus."},{"Start":"03:16.175 ","End":"03:17.630","Text":"This is what we get."},{"Start":"03:17.630 ","End":"03:19.520","Text":"Now a bit of manipulation."},{"Start":"03:19.520 ","End":"03:22.970","Text":"Mu prime over Mu equals this over this,"},{"Start":"03:22.970 ","End":"03:24.890","Text":"but switch sides also,"},{"Start":"03:24.890 ","End":"03:26.530","Text":"so this is what we get."},{"Start":"03:26.530 ","End":"03:31.535","Text":"Now, we think of xy like a single quantity like t,"},{"Start":"03:31.535 ","End":"03:33.440","Text":"like Mu is a function of one variable,"},{"Start":"03:33.440 ","End":"03:38.690","Text":"call it t. What we get is like Mu prime of t over Mu(t)."},{"Start":"03:38.690 ","End":"03:41.645","Text":"In order to integrate this and say,"},{"Start":"03:41.645 ","End":"03:43.805","Text":"we got natural log of Mu(t),"},{"Start":"03:43.805 ","End":"03:47.395","Text":"we want the right-hand side also to be a function of just"},{"Start":"03:47.395 ","End":"03:52.220","Text":"t. This might be like g of t and then we can say that"},{"Start":"03:52.220 ","End":"03:57.620","Text":"the integral of this is the integral of g. Our condition is that"},{"Start":"03:57.620 ","End":"04:04.610","Text":"this expression here in x and y is going to be a function of xy alone."},{"Start":"04:04.610 ","End":"04:05.885","Text":"It\u0027s going to be some g,"},{"Start":"04:05.885 ","End":"04:08.065","Text":"some function of xy."},{"Start":"04:08.065 ","End":"04:15.065","Text":"Then the integration factor Mu is going to be e to the power of the integral of this."},{"Start":"04:15.065 ","End":"04:17.030","Text":"This is not quite precise."},{"Start":"04:17.030 ","End":"04:20.090","Text":"I don\u0027t say integral with respect to what."},{"Start":"04:20.090 ","End":"04:22.430","Text":"Let\u0027s continue here."},{"Start":"04:22.430 ","End":"04:32.960","Text":"What this would mean that the natural log of Mu(t) would equal the integral of g(t)dt."},{"Start":"04:32.960 ","End":"04:35.840","Text":"Which is some function G(t)."},{"Start":"04:35.840 ","End":"04:37.490","Text":"Forget about the constant."},{"Start":"04:37.490 ","End":"04:45.275","Text":"What we say from here is that Mu in general would be e to the power of this,"},{"Start":"04:45.275 ","End":"04:48.135","Text":"e^g or if you like,"},{"Start":"04:48.135 ","End":"04:54.995","Text":"Mu is e to the power of integral of g but this g is a function of one variable."},{"Start":"04:54.995 ","End":"04:58.580","Text":"But at the end, what we want is not Mu(t),"},{"Start":"04:58.580 ","End":"05:02.240","Text":"but Mu(xy), which means plug in xy."},{"Start":"05:02.240 ","End":"05:04.745","Text":"Take this function of xy,"},{"Start":"05:04.745 ","End":"05:08.615","Text":"integrate it as a general function not of xy,"},{"Start":"05:08.615 ","End":"05:11.405","Text":"t or whatever, then plug in xy."},{"Start":"05:11.405 ","End":"05:13.159","Text":"Well, we\u0027ll see this in the example."},{"Start":"05:13.159 ","End":"05:17.450","Text":"I\u0027m just saying that this beware, it\u0027s shorthand notation."},{"Start":"05:17.450 ","End":"05:18.965","Text":"Take the integral of this,"},{"Start":"05:18.965 ","End":"05:20.510","Text":"the anti-derivative of this function,"},{"Start":"05:20.510 ","End":"05:21.800","Text":"little g, call it G,"},{"Start":"05:21.800 ","End":"05:23.930","Text":"and then plug back the xy."},{"Start":"05:23.930 ","End":"05:26.500","Text":"Let\u0027s go and do Part b now."},{"Start":"05:26.500 ","End":"05:28.340","Text":"Here we are in Part b,"},{"Start":"05:28.340 ","End":"05:32.135","Text":"I just copied the equation and labeled it M and"},{"Start":"05:32.135 ","End":"05:36.680","Text":"N. Now the condition we got in Part a is this,"},{"Start":"05:36.680 ","End":"05:43.514","Text":"which says that if we evaluate this expression and it comes out to be a function g(xy),"},{"Start":"05:43.514 ","End":"05:46.504","Text":"then the integration factor is given by this formula,"},{"Start":"05:46.504 ","End":"05:57.630","Text":"which is an imprecise way of what it really means is that Mu is equal to e^G(xy),"},{"Start":"05:58.000 ","End":"06:02.490","Text":"where G is the anti-derivative of g. Or"},{"Start":"06:02.490 ","End":"06:06.320","Text":"you could say that G is the integral of g, just briefly."},{"Start":"06:06.320 ","End":"06:08.480","Text":"Let\u0027s see then this is M,"},{"Start":"06:08.480 ","End":"06:10.360","Text":"this is N, let\u0027s compute this."},{"Start":"06:10.360 ","End":"06:13.425","Text":"M with respect to y,"},{"Start":"06:13.425 ","End":"06:15.450","Text":"you can just see it."},{"Start":"06:15.450 ","End":"06:16.740","Text":"We get this y^2,"},{"Start":"06:16.740 ","End":"06:18.165","Text":"gives us 2y,"},{"Start":"06:18.165 ","End":"06:19.905","Text":"the y gives us 1, so on."},{"Start":"06:19.905 ","End":"06:22.530","Text":"N with respect to x is just 1."},{"Start":"06:22.530 ","End":"06:24.600","Text":"Then we need this expression,"},{"Start":"06:24.600 ","End":"06:27.360","Text":"y times n is just yx,"},{"Start":"06:27.360 ","End":"06:32.305","Text":"x times M is x times this expression, which is this."},{"Start":"06:32.305 ","End":"06:35.180","Text":"This whole expression here becomes this,"},{"Start":"06:35.180 ","End":"06:39.650","Text":"which doesn\u0027t look like a function of xy in this mode at first,"},{"Start":"06:39.650 ","End":"06:41.270","Text":"but can be simplified."},{"Start":"06:41.270 ","End":"06:47.315","Text":"What I did was I canceled the 1 with the minus 1 and yx with minus xy."},{"Start":"06:47.315 ","End":"06:48.650","Text":"That gives us this,"},{"Start":"06:48.650 ","End":"06:51.200","Text":"but we can keep going because here,"},{"Start":"06:51.200 ","End":"06:54.385","Text":"natural log(x) cancel the natural log(x)."},{"Start":"06:54.385 ","End":"06:57.795","Text":"Here we have xy, and x^2 y^2."},{"Start":"06:57.795 ","End":"06:59.350","Text":"I could cancel the xy,"},{"Start":"06:59.350 ","End":"07:01.070","Text":"take one of the x\u0027s away,"},{"Start":"07:01.070 ","End":"07:03.335","Text":"and take one of the y\u0027s the way."},{"Start":"07:03.335 ","End":"07:05.120","Text":"If we do this,"},{"Start":"07:05.120 ","End":"07:06.980","Text":"let\u0027s see what we\u0027re left with."},{"Start":"07:06.980 ","End":"07:09.130","Text":"This not bad at all."},{"Start":"07:09.130 ","End":"07:11.570","Text":"This is definitely a function of xy,"},{"Start":"07:11.570 ","End":"07:14.870","Text":"I mean I could say this is g(xy),"},{"Start":"07:14.870 ","End":"07:20.840","Text":"where g is defined by g(t) is minus 2 over t so g(xy) is this,"},{"Start":"07:20.840 ","End":"07:23.065","Text":"this is just to be more formal."},{"Start":"07:23.065 ","End":"07:30.185","Text":"Then the integration factor we said was e to the power of the integral but really,"},{"Start":"07:30.185 ","End":"07:34.520","Text":"what we mean is by the integral is not saying"},{"Start":"07:34.520 ","End":"07:39.020","Text":"it\u0027s the integral of minus 2 over xy with respect to what."},{"Start":"07:39.020 ","End":"07:42.500","Text":"What it means is that we take the integral of g and we\u0027ve"},{"Start":"07:42.500 ","End":"07:46.600","Text":"got g(t) is minus 2 natural log(t)."},{"Start":"07:46.600 ","End":"07:50.235","Text":"Then instead of t, we put xy,"},{"Start":"07:50.235 ","End":"07:56.115","Text":"so that\u0027s where we get this e to the minus t is replaced by xy."},{"Start":"07:56.115 ","End":"08:00.795","Text":"We get minus 2 natural log of xy, this part here."},{"Start":"08:00.795 ","End":"08:02.435","Text":"How did I get from here to here,"},{"Start":"08:02.435 ","End":"08:04.159","Text":"just using rules of exponents,"},{"Start":"08:04.159 ","End":"08:05.280","Text":"I\u0027ll show you at the side."},{"Start":"08:05.280 ","End":"08:07.330","Text":"I could pull a formula out of the hat,"},{"Start":"08:07.330 ","End":"08:09.010","Text":"let us do it more directly."},{"Start":"08:09.010 ","End":"08:13.160","Text":"This is e to the power of natural log(xy),"},{"Start":"08:13.520 ","End":"08:17.220","Text":"then times minus 2."},{"Start":"08:17.220 ","End":"08:24.175","Text":"I\u0027ll just take the minus 2 and that gives me e to the power of natural log(xy),"},{"Start":"08:24.175 ","End":"08:29.545","Text":"all to the power minus 2 using exponents of exponents, we have a product."},{"Start":"08:29.545 ","End":"08:33.790","Text":"Now, e to the natural log of something is the thing itself."},{"Start":"08:33.790 ","End":"08:37.240","Text":"Take natural log and e to the power of it it\u0027s just xy."},{"Start":"08:37.240 ","End":"08:43.315","Text":"Then this to the minus 2 is 1 over x^2 y^2,"},{"Start":"08:43.315 ","End":"08:46.165","Text":"it\u0027s (xy)^2 and then open it up."},{"Start":"08:46.165 ","End":"08:48.450","Text":"This is what we get."},{"Start":"08:48.450 ","End":"08:54.380","Text":"I could stop here and say this is the integration factor but let\u0027s take"},{"Start":"08:54.380 ","End":"08:59.929","Text":"it a little bit further and just verify that it makes the equation exact."},{"Start":"08:59.929 ","End":"09:03.694","Text":"This here is the equation to both sides,"},{"Start":"09:03.694 ","End":"09:08.585","Text":"multiply by the integration factor 1 over x^2 y^2."},{"Start":"09:08.585 ","End":"09:12.670","Text":"I just put x^2 y^2 on the denominator here and here."},{"Start":"09:12.670 ","End":"09:14.580","Text":"Then we get this equation."},{"Start":"09:14.580 ","End":"09:16.820","Text":"I still haven\u0027t shown that this is exact,"},{"Start":"09:16.820 ","End":"09:21.575","Text":"but the partial derivative of this with respect to y."},{"Start":"09:21.575 ","End":"09:25.755","Text":"Let me just label this M and this N different M and N than before."},{"Start":"09:25.755 ","End":"09:28.130","Text":"Then M with respect to y,"},{"Start":"09:28.130 ","End":"09:30.935","Text":"this is all with just x doesn\u0027t count."},{"Start":"09:30.935 ","End":"09:34.370","Text":"This is like 1 over y with a constant,"},{"Start":"09:34.370 ","End":"09:40.900","Text":"so it\u0027s minus 1 over y^2 and the same constant sticks."},{"Start":"09:40.900 ","End":"09:43.870","Text":"N with respect to x, again,"},{"Start":"09:43.870 ","End":"09:47.090","Text":"I have 1 over x with an extra constant,"},{"Start":"09:47.090 ","End":"09:52.010","Text":"so it\u0027s minus 1 over x^2 with that constant y^2."},{"Start":"09:52.010 ","End":"09:54.695","Text":"These are indeed equal."},{"Start":"09:54.695 ","End":"10:00.900","Text":"This is exact and this really was an integration factor. I\u0027m done."}],"Thumbnail":null,"ID":7705},{"Watched":false,"Name":"Exercise 15","Duration":"3m 38s","ChapterTopicVideoID":7623,"CourseChapterTopicPlaylistID":4222,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.290","Text":"In this exercise, we consider an ordinary differential equation in"},{"Start":"00:04.290 ","End":"00:08.100","Text":"this form and we want to find the condition on the equation,"},{"Start":"00:08.100 ","End":"00:09.795","Text":"condition on M and N,"},{"Start":"00:09.795 ","End":"00:16.500","Text":"such that it should have an integration factor which is a function of x plus y,"},{"Start":"00:16.500 ","End":"00:21.025","Text":"not anything else, like some Mu of x plus y."},{"Start":"00:21.025 ","End":"00:26.545","Text":"What I\u0027ll do is multiply both sides by the integration factor, like so."},{"Start":"00:26.545 ","End":"00:28.170","Text":"Mu of x plus y here,"},{"Start":"00:28.170 ","End":"00:29.870","Text":"Mu of x plus y here."},{"Start":"00:29.870 ","End":"00:34.910","Text":"Now the condition that it should be exact is that this differentiated with"},{"Start":"00:34.910 ","End":"00:41.045","Text":"respect to y should equal this differentiated with respect to x, i.e."},{"Start":"00:41.045 ","End":"00:43.100","Text":"what\u0027s written here."},{"Start":"00:43.100 ","End":"00:47.270","Text":"I\u0027m going to bring the formula I keep bringing, this one here."},{"Start":"00:47.270 ","End":"00:51.200","Text":"One form of the chain rule that suits our purposes,"},{"Start":"00:51.200 ","End":"00:56.066","Text":"where f is Mu and h(x,"},{"Start":"00:56.066 ","End":"00:58.820","Text":"y) is just x plus y."},{"Start":"00:58.820 ","End":"01:02.150","Text":"We\u0027re going to use the product rule, but for this bit,"},{"Start":"01:02.150 ","End":"01:04.280","Text":"here with respect to x and y,"},{"Start":"01:04.280 ","End":"01:05.705","Text":"we\u0027re going use this."},{"Start":"01:05.705 ","End":"01:10.100","Text":"Basically, it just means put a prime in front of the f or Mu,"},{"Start":"01:10.100 ","End":"01:15.005","Text":"and then the inner derivative here and here in both cases will be 1,"},{"Start":"01:15.005 ","End":"01:16.400","Text":"because with respect to x or y,"},{"Start":"01:16.400 ","End":"01:19.760","Text":"it\u0027ll be 1, so we end up with this."},{"Start":"01:19.760 ","End":"01:22.100","Text":"Like I said, it\u0027s just the product derivative of M,"},{"Start":"01:22.100 ","End":"01:23.780","Text":"the other bit as is,"},{"Start":"01:23.780 ","End":"01:25.400","Text":"then M as is,"},{"Start":"01:25.400 ","End":"01:29.075","Text":"and this differentiated, which we said using this"},{"Start":"01:29.075 ","End":"01:33.410","Text":"will be Mu\u0027 and the inner derivative is 1."},{"Start":"01:33.410 ","End":"01:39.110","Text":"Likewise, here we have Mu\u0027 of this and inner derivative is 1 and again,"},{"Start":"01:39.110 ","End":"01:41.240","Text":"the product, we\u0027ve done this thing before,"},{"Start":"01:41.240 ","End":"01:43.985","Text":"I\u0027m not going into too much detail."},{"Start":"01:43.985 ","End":"01:47.720","Text":"Then as usual, we bring the stuff with Mu onto the left and"},{"Start":"01:47.720 ","End":"01:51.920","Text":"the Mu\u0027 onto the right and collect."},{"Start":"01:51.920 ","End":"01:54.530","Text":"What we get is this."},{"Start":"01:54.530 ","End":"01:57.875","Text":"We\u0027re going to divide by Mu"},{"Start":"01:57.875 ","End":"02:03.125","Text":"here and divide by the N minus M over there and this is what we get."},{"Start":"02:03.125 ","End":"02:06.080","Text":"I think of x plus y like a variable t,"},{"Start":"02:06.080 ","End":"02:10.520","Text":"so we have Mu\u0027(t) over Mu(t)."},{"Start":"02:10.520 ","End":"02:15.890","Text":"We want it to also be an expression in just t, which is x plus y."},{"Start":"02:15.890 ","End":"02:25.708","Text":"This has got to be some g(t)."},{"Start":"02:25.708 ","End":"02:27.470","Text":"So our condition is that this minus this over this is a function of x plus y. I need"},{"Start":"02:27.470 ","End":"02:29.015","Text":"more space here."},{"Start":"02:29.015 ","End":"02:30.440","Text":"This is the condition."},{"Start":"02:30.440 ","End":"02:33.365","Text":"If this is a function of x plus y,"},{"Start":"02:33.365 ","End":"02:35.480","Text":"then well, looking at this,"},{"Start":"02:35.480 ","End":"02:43.010","Text":"we could take the integral of both sides and get natural log of Mu equals G,"},{"Start":"02:43.010 ","End":"02:48.320","Text":"where G is the integral of g. Then we can say"},{"Start":"02:48.320 ","End":"02:53.720","Text":"that Mu is e^G and we want Mu of not t,"},{"Start":"02:53.720 ","End":"02:54.920","Text":"well, in this case,"},{"Start":"02:54.920 ","End":"02:57.095","Text":"it would be t, but here,"},{"Start":"02:57.095 ","End":"03:02.270","Text":"we would want x plus y. I\u0027m writing Mu as the integral"},{"Start":"03:02.270 ","End":"03:07.550","Text":"of this but this is just a bit of a shorthand because really,"},{"Start":"03:07.550 ","End":"03:09.845","Text":"we look at it as a function of a single variable,"},{"Start":"03:09.845 ","End":"03:12.410","Text":"we take the integral of g,"},{"Start":"03:12.410 ","End":"03:18.395","Text":"which is G, what this really is is e^G."},{"Start":"03:18.395 ","End":"03:20.660","Text":"Then here we put the t,"},{"Start":"03:20.660 ","End":"03:22.340","Text":"which is x plus y,"},{"Start":"03:22.340 ","End":"03:30.020","Text":"where G is the integral of g. This is just a shorthand way of writing that."},{"Start":"03:30.020 ","End":"03:31.760","Text":"That\u0027s in general. We don\u0027t have"},{"Start":"03:31.760 ","End":"03:35.824","Text":"a specific example to try it on like we did in the previous exercise."},{"Start":"03:35.824 ","End":"03:39.030","Text":"I\u0027ll just leave it at that."}],"Thumbnail":null,"ID":7706},{"Watched":false,"Name":"Exercise 16","Duration":"4m 51s","ChapterTopicVideoID":7624,"CourseChapterTopicPlaylistID":4222,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"This exercise is similar to some previous ones."},{"Start":"00:02.910 ","End":"00:08.085","Text":"We have a general first order differential equation in this form."},{"Start":"00:08.085 ","End":"00:10.080","Text":"We want to find a condition on the equation,"},{"Start":"00:10.080 ","End":"00:14.970","Text":"now here conditioned on M and N in order for it to have an integration factor,"},{"Start":"00:14.970 ","End":"00:17.535","Text":"which is a function of just x over y,"},{"Start":"00:17.535 ","End":"00:20.760","Text":"no separate x\u0027s and y\u0027s."},{"Start":"00:20.760 ","End":"00:27.180","Text":"What we mean is some function Mu of x over y."},{"Start":"00:27.180 ","End":"00:33.025","Text":"Let\u0027s multiply this equation by the integration factor,"},{"Start":"00:33.025 ","End":"00:35.120","Text":"that will give us this."},{"Start":"00:35.120 ","End":"00:39.560","Text":"Now we apply the condition that the derivative of this with respect to y"},{"Start":"00:39.560 ","End":"00:45.085","Text":"should equal the derivative of this with respect to x like so."},{"Start":"00:45.085 ","End":"00:51.320","Text":"I produced the couple of formulas that we\u0027ve used all along,"},{"Start":"00:51.320 ","End":"00:54.230","Text":"when we have a function of 1 variable,"},{"Start":"00:54.230 ","End":"00:59.225","Text":"f is like Mu and h(x,"},{"Start":"00:59.225 ","End":"01:02.329","Text":"y) is a function of 2 variables,"},{"Start":"01:02.329 ","End":"01:07.490","Text":"which is what we have the expression x over y and if we apply this basically"},{"Start":"01:07.490 ","End":"01:13.740","Text":"this tells us when we do the differentiation how to derive this bit."},{"Start":"01:18.860 ","End":"01:23.615","Text":"This should not be here, just erase it."},{"Start":"01:23.615 ","End":"01:25.385","Text":"Yes, sorry."},{"Start":"01:25.385 ","End":"01:28.250","Text":"Anyway, let\u0027s continue."},{"Start":"01:28.250 ","End":"01:36.300","Text":"We use the product rule on each of these pieces derivative of M with respect to y,"},{"Start":"01:36.300 ","End":"01:43.370","Text":"M as is, and the derivative of this with respect to y."},{"Start":"01:43.370 ","End":"01:46.460","Text":"We use this part of the formula here we get"},{"Start":"01:46.460 ","End":"01:52.879","Text":"mu prime and then the integer derivative of x over y with respect to y."},{"Start":"01:52.879 ","End":"02:00.980","Text":"Let\u0027s just write that here maybe that hy is minus x over y^2,"},{"Start":"02:00.980 ","End":"02:05.265","Text":"also hx is just 1 over y,"},{"Start":"02:05.265 ","End":"02:06.660","Text":"that\u0027s the 1 over y here,"},{"Start":"02:06.660 ","End":"02:11.025","Text":"that\u0027s the minus x over y squared here all the rest of it is standard."},{"Start":"02:11.025 ","End":"02:15.650","Text":"I mean as usual, we\u0027re going to separate the mu from the mu prime."},{"Start":"02:15.650 ","End":"02:22.670","Text":"Stuff with mu on the left that gives us the my from here and then minus nx"},{"Start":"02:22.670 ","End":"02:26.120","Text":"from here because it\u0027s switched sides and similarly the mu prime stuff on"},{"Start":"02:26.120 ","End":"02:29.600","Text":"the right and we had a minus here,"},{"Start":"02:29.600 ","End":"02:32.350","Text":"so we bring it to the right, it becomes a plus."},{"Start":"02:32.350 ","End":"02:35.470","Text":"Need more space,"},{"Start":"02:37.130 ","End":"02:39.510","Text":"I didn\u0027t like the fraction here,"},{"Start":"02:39.510 ","End":"02:41.865","Text":"so let\u0027s make a common denominator,"},{"Start":"02:41.865 ","End":"02:43.675","Text":"pull the y out."},{"Start":"02:43.675 ","End":"02:49.645","Text":"I\u0027ll have to multiply this by y then and I have a 1 over y squared here."},{"Start":"02:49.645 ","End":"02:56.610","Text":"Then as usually want mu prime over mu has got to equal this over this."},{"Start":"02:57.070 ","End":"03:00.170","Text":"But the y squared goes into"},{"Start":"03:00.170 ","End":"03:03.930","Text":"the numerator from here when"},{"Start":"03:03.930 ","End":"03:08.705","Text":"we divide by it and here we have this expression here, goes down here."},{"Start":"03:08.705 ","End":"03:14.120","Text":"Could cancel bit y goes with 1 of the y\u0027s here and now"},{"Start":"03:14.120 ","End":"03:19.190","Text":"if the expression that we have on the right-hand side is a function of x over y,"},{"Start":"03:19.190 ","End":"03:22.370","Text":"then everything\u0027s in terms of x over y,"},{"Start":"03:22.370 ","End":"03:29.910","Text":"then this thing can help us find the integration factor."},{"Start":"03:30.130 ","End":"03:34.070","Text":"We just get e to the power of this function of x over y."},{"Start":"03:34.070 ","End":"03:36.335","Text":"But this is really a bit of a shorthand."},{"Start":"03:36.335 ","End":"03:41.010","Text":"What it really means is that when you come to actually do it,"},{"Start":"03:41.010 ","End":"03:45.030","Text":"is you just identify what this g is."},{"Start":"03:49.610 ","End":"03:54.840","Text":"This thing is like a t Mu prime(t) over Mu(t)"},{"Start":"03:54.840 ","End":"04:01.880","Text":"is some function g(t) so integrating both sides,"},{"Start":"04:01.880 ","End":"04:07.445","Text":"we get natural log of Mu(t) and the integral of this,"},{"Start":"04:07.445 ","End":"04:09.470","Text":"let\u0027s call it g(t),"},{"Start":"04:09.470 ","End":"04:17.090","Text":"where big G is the primitive anti-derivative integral, whatever."},{"Start":"04:17.090 ","End":"04:21.350","Text":"Then we get the mu is e^g."},{"Start":"04:21.350 ","End":"04:29.690","Text":"Our integration factor is not with t,"},{"Start":"04:29.690 ","End":"04:37.590","Text":"but with x over y so we need e^G of x"},{"Start":"04:37.590 ","End":"04:47.045","Text":"over y where G is the integral of the function we identified here."},{"Start":"04:47.045 ","End":"04:51.480","Text":"We\u0027ll just leave it at that and we are done."}],"Thumbnail":null,"ID":7707}],"ID":4222},{"Name":"Linear Equations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Linear Equations","Duration":"4m 48s","ChapterTopicVideoID":7611,"CourseChapterTopicPlaylistID":4223,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.915","Text":"Continuing with the ODEs, ordinary differential equations,"},{"Start":"00:03.915 ","End":"00:09.675","Text":"of the first order and the next variety will be linear ordinary differential equations."},{"Start":"00:09.675 ","End":"00:13.920","Text":"Linear differential equation of the first order is something that"},{"Start":"00:13.920 ","End":"00:18.125","Text":"either isn\u0027t in the form or you can bring it to the form y\u0027,"},{"Start":"00:18.125 ","End":"00:22.890","Text":"derivative of y, plus some function of x times y equals some other function of x."},{"Start":"00:22.890 ","End":"00:24.630","Text":"Let\u0027s start with an example."},{"Start":"00:24.630 ","End":"00:30.760","Text":"We have this equation which is not quite in this form because we want y\u0027 isolate."},{"Start":"00:30.760 ","End":"00:32.200","Text":"But a little bit of algebra,"},{"Start":"00:32.200 ","End":"00:33.440","Text":"bring this to the left,"},{"Start":"00:33.440 ","End":"00:37.293","Text":"bring this to the right, and divide by x and we\u0027ll get this."},{"Start":"00:37.293 ","End":"00:44.540","Text":"This is of this form because if we let the x be a(x) here and the e^x/x,"},{"Start":"00:44.540 ","End":"00:45.920","Text":"we call that b(x),"},{"Start":"00:45.920 ","End":"00:47.555","Text":"then it fits this form."},{"Start":"00:47.555 ","End":"00:50.509","Text":"The next thing we want to know is how to solve such equations."},{"Start":"00:50.509 ","End":"00:52.130","Text":"Suppose I have this equation."},{"Start":"00:52.130 ","End":"00:57.410","Text":"Turns out that there is just a closed formula that we can find the solution to it."},{"Start":"00:57.410 ","End":"00:59.000","Text":"It goes as follows,"},{"Start":"00:59.000 ","End":"01:01.730","Text":"y equals e to the power of minus."},{"Start":"01:01.730 ","End":"01:03.460","Text":"Now what\u0027s capital A(x) well,"},{"Start":"01:03.460 ","End":"01:04.835","Text":"I also have to tell you that."},{"Start":"01:04.835 ","End":"01:06.080","Text":"You could start here,"},{"Start":"01:06.080 ","End":"01:07.975","Text":"say, let capital A,"},{"Start":"01:07.975 ","End":"01:12.650","Text":"be a primitive of little a and an indefinite integral,"},{"Start":"01:12.650 ","End":"01:14.225","Text":"then just pick one."},{"Start":"01:14.225 ","End":"01:17.090","Text":"You don\u0027t have to bother with a constant because it turns out that"},{"Start":"01:17.090 ","End":"01:20.510","Text":"this single constant takes care of this constant also."},{"Start":"01:20.510 ","End":"01:23.225","Text":"We take this and we plug it in here,"},{"Start":"01:23.225 ","End":"01:28.610","Text":"e to the minus this function and then we have another integral here of b(x) from"},{"Start":"01:28.610 ","End":"01:34.625","Text":"here times e^A(x) from there dx and at the end is a single constant here."},{"Start":"01:34.625 ","End":"01:37.205","Text":"Notice that we have 2 integrations to do."},{"Start":"01:37.205 ","End":"01:39.560","Text":"First, we compute this integral."},{"Start":"01:39.560 ","End":"01:41.795","Text":"We have this other integral and the rest of it is"},{"Start":"01:41.795 ","End":"01:45.725","Text":"just algebra so 2 integrations and substitution."},{"Start":"01:45.725 ","End":"01:48.950","Text":"We have some formulas that are worth remembering when solving."},{"Start":"01:48.950 ","End":"01:52.745","Text":"We\u0027re going to have to do some integrations and what I\u0027d like to give you"},{"Start":"01:52.745 ","End":"01:57.170","Text":"these which are particularly useful when I have a little square box."},{"Start":"01:57.170 ","End":"02:00.860","Text":"It\u0027s just some function of x, but e^k,"},{"Start":"02:00.860 ","End":"02:03.290","Text":"natural log of something is the same as"},{"Start":"02:03.290 ","End":"02:05.915","Text":"that something to the power of k. That\u0027s easy to show,"},{"Start":"02:05.915 ","End":"02:07.445","Text":"we won\u0027t go into the proofs here."},{"Start":"02:07.445 ","End":"02:09.880","Text":"If I have a minus, it\u0027s one over."},{"Start":"02:09.880 ","End":"02:13.790","Text":"Also we have a template integral that if I have e to the power of something,"},{"Start":"02:13.790 ","End":"02:16.340","Text":"but I have that something\u0027s derivative then the integral is"},{"Start":"02:16.340 ","End":"02:18.964","Text":"just e to that something plus the constant of integration."},{"Start":"02:18.964 ","End":"02:20.270","Text":"These are just useful,"},{"Start":"02:20.270 ","End":"02:21.455","Text":"will come in handy."},{"Start":"02:21.455 ","End":"02:27.110","Text":"Not every teacher or professor will accept plugging into a formula as an answer."},{"Start":"02:27.110 ","End":"02:28.190","Text":"If that\u0027s the case,"},{"Start":"02:28.190 ","End":"02:33.670","Text":"you might have to prove this formula and so I\u0027m now going to prove the formula."},{"Start":"02:33.670 ","End":"02:36.680","Text":"By the way, the proof is not very difficult and it is possible it\u0027s quite"},{"Start":"02:36.680 ","End":"02:40.505","Text":"feasible to memorize it in case you\u0027re required to do so."},{"Start":"02:40.505 ","End":"02:43.295","Text":"We start with the original linear equation."},{"Start":"02:43.295 ","End":"02:50.000","Text":"Multiply both sides by e^A(x) in case you forgot what A(x) is,"},{"Start":"02:50.000 ","End":"02:54.620","Text":"A(x) was the integral of a(x) dx."},{"Start":"02:54.620 ","End":"02:59.105","Text":"Something I\u0027m going to use later because this function is an indefinite integral of this,"},{"Start":"02:59.105 ","End":"03:02.990","Text":"then certainly A\u0027(x) will be a(x)."},{"Start":"03:02.990 ","End":"03:05.690","Text":"Opposite of integration is differentiation."},{"Start":"03:05.690 ","End":"03:08.330","Text":"Multiplying leaves me with this expression."},{"Start":"03:08.330 ","End":"03:11.810","Text":"I just put it in all 3 terms and now what I claim is"},{"Start":"03:11.810 ","End":"03:17.930","Text":"the left-hand side is the derivative using the product rule of this expression."},{"Start":"03:17.930 ","End":"03:19.820","Text":"In other words, the derivative of what\u0027s in brackets"},{"Start":"03:19.820 ","End":"03:22.040","Text":"is this and this follows from the product rule."},{"Start":"03:22.040 ","End":"03:31.275","Text":"I\u0027ll remind you that if we have u times v derivative is u\u0027v plus uv\u0027,"},{"Start":"03:31.275 ","End":"03:34.440","Text":"this is u and this is v,"},{"Start":"03:34.440 ","End":"03:38.775","Text":"Then this is u\u0027, and this is v. I want u,"},{"Start":"03:38.775 ","End":"03:46.875","Text":"which is y, and v\u0027 is e^A(x) times the inner derivative A\u0027(x)."},{"Start":"03:46.875 ","End":"03:50.760","Text":"This A\u0027(x) is big A is little a."},{"Start":"03:50.760 ","End":"03:53.480","Text":"This derivative is this on the right-hand side,"},{"Start":"03:53.480 ","End":"03:55.775","Text":"I\u0027m just copying as is."},{"Start":"03:55.775 ","End":"03:58.160","Text":"Put an integral sign in front of each and"},{"Start":"03:58.160 ","End":"04:01.235","Text":"then the integral of the derivative is the thing itself."},{"Start":"04:01.235 ","End":"04:03.215","Text":"It\u0027s going to equal the integral of this."},{"Start":"04:03.215 ","End":"04:08.285","Text":"But we need constant because when I go from the derivative back to the function,"},{"Start":"04:08.285 ","End":"04:11.255","Text":"I could pick up some random constant."},{"Start":"04:11.255 ","End":"04:19.460","Text":"Next step is to divide both sides by e^A(x), I should say."},{"Start":"04:19.460 ","End":"04:25.280","Text":"Then when I say one over it can be to the negative power and this is,"},{"Start":"04:25.280 ","End":"04:26.555","Text":"if you check above,"},{"Start":"04:26.555 ","End":"04:28.955","Text":"it was the formula we had before."},{"Start":"04:28.955 ","End":"04:32.330","Text":"Basically that\u0027s the proof in case your are asked."},{"Start":"04:32.330 ","End":"04:36.215","Text":"Let me just emphasize that the check with your teacher,"},{"Start":"04:36.215 ","End":"04:38.420","Text":"if you are allowed to use the formula,"},{"Start":"04:38.420 ","End":"04:42.470","Text":"sample exercises are after this tutorial and there\u0027ll be"},{"Start":"04:42.470 ","End":"04:46.730","Text":"quite a few of them so will help you understand this theory better."},{"Start":"04:46.730 ","End":"04:48.960","Text":"That\u0027s it for now."}],"Thumbnail":null,"ID":7676},{"Watched":false,"Name":"Exercise 1","Duration":"2m 58s","ChapterTopicVideoID":7604,"CourseChapterTopicPlaylistID":4223,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"Here we have to solve linear differential equation."},{"Start":"00:03.600 ","End":"00:07.980","Text":"It\u0027s linear because it fits the template which is that we have"},{"Start":"00:07.980 ","End":"00:13.665","Text":"y\u0027 plus some function of x times y equals another function of x, a, and b here."},{"Start":"00:13.665 ","End":"00:15.120","Text":"Here, a(x) is 2,"},{"Start":"00:15.120 ","End":"00:17.340","Text":"b(x) is 4 and we have"},{"Start":"00:17.340 ","End":"00:21.585","Text":"a standard solution for this case and it\u0027s given in this formula here."},{"Start":"00:21.585 ","End":"00:24.230","Text":"In our particular case, as I say,"},{"Start":"00:24.230 ","End":"00:32.885","Text":"a(x) is 2x and b(x) is 4x and what we want to do is figure out this formula here."},{"Start":"00:32.885 ","End":"00:35.060","Text":"I didn\u0027t say what A(x) is,"},{"Start":"00:35.060 ","End":"00:41.060","Text":"but it\u0027s standard thing that A(x) is the indefinite integral of little a(x),"},{"Start":"00:41.060 ","End":"00:43.790","Text":"so let\u0027s do that as a little side calculation."},{"Start":"00:43.790 ","End":"00:46.115","Text":"We\u0027ll have a couple of side calculations."},{"Start":"00:46.115 ","End":"00:49.340","Text":"a(x) is the integral of a(x)dx,"},{"Start":"00:49.340 ","End":"00:52.890","Text":"a(x) is 2xdx, so it\u0027s just x^2."},{"Start":"00:52.890 ","End":"00:54.960","Text":"When we compute capital A(x),"},{"Start":"00:54.960 ","End":"00:56.870","Text":"we don\u0027t need to add a constant."},{"Start":"00:56.870 ","End":"00:59.315","Text":"This is because if I add a constant here"},{"Start":"00:59.315 ","End":"01:02.290","Text":"and here because it\u0027s opposite signs of the exponents,"},{"Start":"01:02.290 ","End":"01:05.725","Text":"they cancel each other out so it doesn\u0027t make any difference."},{"Start":"01:05.725 ","End":"01:09.130","Text":"We don\u0027t need a constant in computing A(x)."},{"Start":"01:09.130 ","End":"01:12.440","Text":"The second thing we want to compute is the integral,"},{"Start":"01:12.440 ","End":"01:15.530","Text":"this bit here, which I\u0027ll call double asterisk."},{"Start":"01:15.530 ","End":"01:17.000","Text":"What we have computed, as I said,"},{"Start":"01:17.000 ","End":"01:20.825","Text":"is this bit I marked b(x)e to A(x)."},{"Start":"01:20.825 ","End":"01:22.735","Text":"Now b(x) is 4x,"},{"Start":"01:22.735 ","End":"01:27.675","Text":"A(x) we already computed as x^2."},{"Start":"01:27.675 ","End":"01:29.119","Text":"To compute this integral,"},{"Start":"01:29.119 ","End":"01:33.995","Text":"I\u0027d like to use the following, which is well-known."},{"Start":"01:33.995 ","End":"01:39.905","Text":"If we have the integral of e to some function of x and we have the derivative alongside,"},{"Start":"01:39.905 ","End":"01:43.585","Text":"then that integral is just e^f(x)."},{"Start":"01:43.585 ","End":"01:45.570","Text":"Now if f(x) x is x^2."},{"Start":"01:45.570 ","End":"01:47.990","Text":"We don\u0027t exactly have f\u0027, f\u0027 is 2x,"},{"Start":"01:47.990 ","End":"01:49.580","Text":"but we know the usual routine."},{"Start":"01:49.580 ","End":"01:52.910","Text":"We break it up in such a way that it is convenient."},{"Start":"01:52.910 ","End":"01:54.380","Text":"If I put a 2 outside,"},{"Start":"01:54.380 ","End":"01:58.175","Text":"now I do have the derivative 2x(x^2)."},{"Start":"01:58.175 ","End":"02:00.680","Text":"Using this particular formula,"},{"Start":"02:00.680 ","End":"02:04.035","Text":"I get that the answer would just be 2e to the x^2,"},{"Start":"02:04.035 ","End":"02:07.635","Text":"the 2 from here, and the x^2 from the formula."},{"Start":"02:07.635 ","End":"02:12.320","Text":"Now I come back to the formula above,"},{"Start":"02:12.320 ","End":"02:14.405","Text":"that what we need is this."},{"Start":"02:14.405 ","End":"02:17.240","Text":"Now we\u0027ve already computed all the bits we need."},{"Start":"02:17.240 ","End":"02:19.775","Text":"The a(x), which I found here,"},{"Start":"02:19.775 ","End":"02:22.645","Text":"I\u0027m going to substitute here."},{"Start":"02:22.645 ","End":"02:26.000","Text":"This expression, the double-asterisk,"},{"Start":"02:26.000 ","End":"02:27.695","Text":"I\u0027m going to substitute."},{"Start":"02:27.695 ","End":"02:30.265","Text":"That\u0027s all this bit here."},{"Start":"02:30.265 ","End":"02:40.935","Text":"What we get is that y=e to the minus a(x) is x^2 and this bit here is 2e to the x^2 + c."},{"Start":"02:40.935 ","End":"02:46.770","Text":"Then we tidy up a bit because the e(-x^2) will cancel with e(x^2) and"},{"Start":"02:46.770 ","End":"02:52.930","Text":"so we\u0027ll get that the answer is 2 + ce to the -x^2."},{"Start":"02:52.930 ","End":"02:55.340","Text":"If we had an initial condition, we could find c,"},{"Start":"02:55.340 ","End":"02:59.610","Text":"but we don\u0027t so we leave it with the constant and we\u0027re done."}],"Thumbnail":null,"ID":7678},{"Watched":false,"Name":"Exercise 2","Duration":"4m 35s","ChapterTopicVideoID":7605,"CourseChapterTopicPlaylistID":4223,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.330","Text":"Here we have a differential equation to solve,"},{"Start":"00:03.330 ","End":"00:06.570","Text":"and we\u0027re told that x is not equal to 0."},{"Start":"00:06.570 ","End":"00:09.525","Text":"In fact, in anticipation of what\u0027s going to come,"},{"Start":"00:09.525 ","End":"00:10.575","Text":"allow me to just,"},{"Start":"00:10.575 ","End":"00:11.690","Text":"instead of this condition,"},{"Start":"00:11.690 ","End":"00:14.520","Text":"write x bigger than 0."},{"Start":"00:14.520 ","End":"00:16.590","Text":"That would be more convenient for us."},{"Start":"00:16.590 ","End":"00:18.810","Text":"Now, I anticipate that this is going to be"},{"Start":"00:18.810 ","End":"00:23.220","Text":"a linear differential equation after we do a bit of algebra."},{"Start":"00:23.220 ","End":"00:24.840","Text":"For linear differential equations,"},{"Start":"00:24.840 ","End":"00:26.130","Text":"we\u0027re going to need this formula,"},{"Start":"00:26.130 ","End":"00:27.555","Text":"and let me keep it up here."},{"Start":"00:27.555 ","End":"00:30.510","Text":"Meanwhile, let me just copy this,"},{"Start":"00:30.510 ","End":"00:35.450","Text":"and then we can manipulate it algebraically to get it into this form."},{"Start":"00:35.450 ","End":"00:40.580","Text":"First thing I can do is to bring just the y over to the other side,"},{"Start":"00:40.580 ","End":"00:44.870","Text":"and now I\u0027m going to divide by x."},{"Start":"00:44.870 ","End":"00:50.700","Text":"What I\u0027ll get if I divide by x is y prime minus 1"},{"Start":"00:50.700 ","End":"00:56.420","Text":"over x times y equals x squared plus 3x minus 2."},{"Start":"00:56.420 ","End":"00:58.370","Text":"I just lowered the powers by one,"},{"Start":"00:58.370 ","End":"01:05.570","Text":"and now I do have it in the form y prime plus a of x times y equals b of x."},{"Start":"01:05.570 ","End":"01:08.930","Text":"Because if a of x is this and b of x is this,"},{"Start":"01:08.930 ","End":"01:11.075","Text":"then it exactly fits the template."},{"Start":"01:11.075 ","End":"01:13.490","Text":"I want to compute this thing, but before that,"},{"Start":"01:13.490 ","End":"01:17.000","Text":"I\u0027d like to do two sub-computations, if you will."},{"Start":"01:17.000 ","End":"01:20.645","Text":"I want to compute what is a of x,"},{"Start":"01:20.645 ","End":"01:22.550","Text":"which I didn\u0027t write here,"},{"Start":"01:22.550 ","End":"01:26.030","Text":"but we all know that it\u0027s the indefinite integral of little a of"},{"Start":"01:26.030 ","End":"01:30.080","Text":"x. I also wanted to compute this whole thing here."},{"Start":"01:30.080 ","End":"01:31.335","Text":"This I call asterisk,"},{"Start":"01:31.335 ","End":"01:34.565","Text":"I call double asterisk. Here we are."},{"Start":"01:34.565 ","End":"01:38.235","Text":"Capital A of x is the integral of little a of x,"},{"Start":"01:38.235 ","End":"01:40.670","Text":"integral of minus 1 over x, dx,"},{"Start":"01:40.670 ","End":"01:44.420","Text":"the minus comes in front and we get the natural log of"},{"Start":"01:44.420 ","End":"01:48.350","Text":"x because we assume that x was bigger than 0 upfront."},{"Start":"01:48.350 ","End":"01:49.820","Text":"If I didn\u0027t give you this condition,"},{"Start":"01:49.820 ","End":"01:52.010","Text":"you\u0027d have to put the absolute value,"},{"Start":"01:52.010 ","End":"01:53.854","Text":"or do it in two cases,"},{"Start":"01:53.854 ","End":"01:57.290","Text":"it\u0027s a nuisance, or you could just go ahead and make this assumption."},{"Start":"01:57.290 ","End":"02:02.570","Text":"Most professors will allow for x bigger than 0 or even phrase it that way."},{"Start":"02:02.570 ","End":"02:04.670","Text":"The absolute value is just a bit of a nuisance,"},{"Start":"02:04.670 ","End":"02:06.980","Text":"but it doesn\u0027t make it really difficult."},{"Start":"02:06.980 ","End":"02:10.369","Text":"Now, the next thing we wanted was the other integral,"},{"Start":"02:10.369 ","End":"02:13.145","Text":"which was the double-asterisk, was this bit."},{"Start":"02:13.145 ","End":"02:17.170","Text":"Now, b of x is x squared plus 3x minus 2,"},{"Start":"02:17.170 ","End":"02:18.995","Text":"and now we have capital A of x,"},{"Start":"02:18.995 ","End":"02:24.560","Text":"which is minus natural log of x. I want to simplify this to show you what"},{"Start":"02:24.560 ","End":"02:31.525","Text":"the side minus natural log of x is natural log of x to the minus 1,"},{"Start":"02:31.525 ","End":"02:35.250","Text":"which is natural log of 1 over x."},{"Start":"02:35.250 ","End":"02:38.360","Text":"When I take e to the power of both sides,"},{"Start":"02:38.360 ","End":"02:42.980","Text":"I get e to the power of natural log of 1 over x."},{"Start":"02:42.980 ","End":"02:46.610","Text":"Now, the e and the natural log will"},{"Start":"02:46.610 ","End":"02:50.810","Text":"cancel each other because if I take the logarithm and then take the exponent,"},{"Start":"02:50.810 ","End":"02:52.805","Text":"I\u0027m back where I started."},{"Start":"02:52.805 ","End":"02:56.285","Text":"This just becomes 1 over x."},{"Start":"02:56.285 ","End":"03:01.610","Text":"Now I can go back here and instead of e to the minus natural log x,"},{"Start":"03:01.610 ","End":"03:05.255","Text":"I can write 1 over x here instead,"},{"Start":"03:05.255 ","End":"03:07.745","Text":"and then I can divide it out."},{"Start":"03:07.745 ","End":"03:14.430","Text":"We get x plus 3 minus 2 times 1 over x."},{"Start":"03:14.430 ","End":"03:16.340","Text":"When we take the integral,"},{"Start":"03:16.340 ","End":"03:17.840","Text":"this is x squared over 2,"},{"Start":"03:17.840 ","End":"03:19.190","Text":"this is 3x,"},{"Start":"03:19.190 ","End":"03:22.345","Text":"and this is minus 2 natural log of x."},{"Start":"03:22.345 ","End":"03:26.734","Text":"Now we can substitute in the other formula that we had above."},{"Start":"03:26.734 ","End":"03:30.305","Text":"Minus a of x is natural log of x,"},{"Start":"03:30.305 ","End":"03:34.010","Text":"because a of x was minus natural log of x,"},{"Start":"03:34.010 ","End":"03:35.900","Text":"and this whole thing,"},{"Start":"03:35.900 ","End":"03:40.250","Text":"it\u0027s what we wrote here plus the constant of integration."},{"Start":"03:40.250 ","End":"03:42.860","Text":"I want to add that for big A of x,"},{"Start":"03:42.860 ","End":"03:45.290","Text":"we don\u0027t need a constant of integration."},{"Start":"03:45.290 ","End":"03:48.065","Text":"If we do add a constant of integration,"},{"Start":"03:48.065 ","End":"03:53.390","Text":"it just cancels out because in this formula we have a minus a of x and a plus a of x."},{"Start":"03:53.390 ","End":"03:54.845","Text":"If we add a constant,"},{"Start":"03:54.845 ","End":"03:56.870","Text":"the constant parts will cancel out,"},{"Start":"03:56.870 ","End":"04:00.290","Text":"and we just have to remember that you don\u0027t need to see in this part,"},{"Start":"04:00.290 ","End":"04:02.450","Text":"but you do in this, which is that."},{"Start":"04:02.450 ","End":"04:08.380","Text":"What\u0027s left is simplify a bit the same logic as before,"},{"Start":"04:08.380 ","End":"04:13.460","Text":"we have e to the power of natural log of x, and again,"},{"Start":"04:13.460 ","End":"04:17.330","Text":"the e to the power of cancels with the natural log,"},{"Start":"04:17.330 ","End":"04:21.150","Text":"so we\u0027re just left with x."},{"Start":"04:21.150 ","End":"04:24.155","Text":"If I put x in front of here,"},{"Start":"04:24.155 ","End":"04:27.260","Text":"then I will get this,"},{"Start":"04:27.260 ","End":"04:35.340","Text":"and this is the answer with the limitation that x bigger than 0, we\u0027re done."}],"Thumbnail":null,"ID":7679},{"Watched":false,"Name":"Exercise 3","Duration":"3m 18s","ChapterTopicVideoID":7606,"CourseChapterTopicPlaylistID":4223,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"Here we have a differential equation to solve,"},{"Start":"00:02.640 ","End":"00:06.990","Text":"and I believe we can bring this into the form of a linear differential equation."},{"Start":"00:06.990 ","End":"00:10.710","Text":"The linear differential equation looks like this and with a bit of algebra,"},{"Start":"00:10.710 ","End":"00:12.615","Text":"we\u0027ll get this to look like that."},{"Start":"00:12.615 ","End":"00:15.870","Text":"For example, if I put the y over to the other side,"},{"Start":"00:15.870 ","End":"00:17.355","Text":"that\u0027s already closer,"},{"Start":"00:17.355 ","End":"00:20.820","Text":"and then if I divide by x minus 2,"},{"Start":"00:20.820 ","End":"00:29.025","Text":"then what I\u0027ll get is that y\u0027 plus some function of x times y= another function of x."},{"Start":"00:29.025 ","End":"00:30.960","Text":"That\u0027s what we call a(x)."},{"Start":"00:30.960 ","End":"00:35.400","Text":"And this is what we call b(x) and that\u0027s what we apply to"},{"Start":"00:35.400 ","End":"00:41.285","Text":"this formula but the A(x) is the indefinite integral of a(x)."},{"Start":"00:41.285 ","End":"00:43.520","Text":"Let\u0027s do that. In fact,"},{"Start":"00:43.520 ","End":"00:46.040","Text":"there\u0027s gonna be two side exercises;"},{"Start":"00:46.040 ","End":"00:49.685","Text":"One to compute A(x) and the other,"},{"Start":"00:49.685 ","End":"00:52.355","Text":"we\u0027re going to compute this bit over here."},{"Start":"00:52.355 ","End":"00:53.900","Text":"This bit, the A(x),"},{"Start":"00:53.900 ","End":"00:55.550","Text":"I\u0027ll call that the asterisk,"},{"Start":"00:55.550 ","End":"00:57.770","Text":"and this bit here will be the double asterisk."},{"Start":"00:57.770 ","End":"00:59.584","Text":"Here we start with the asterisk,"},{"Start":"00:59.584 ","End":"01:01.460","Text":"and that is the integral of a(x),"},{"Start":"01:01.460 ","End":"01:04.190","Text":"which is the integral of minus 1 over x minus 2."},{"Start":"01:04.190 ","End":"01:05.570","Text":"The minus comes out in front."},{"Start":"01:05.570 ","End":"01:06.995","Text":"This is a natural log."},{"Start":"01:06.995 ","End":"01:08.630","Text":"Remember that x is bigger than 2,"},{"Start":"01:08.630 ","End":"01:10.880","Text":"so we\u0027re okay with the domain,"},{"Start":"01:10.880 ","End":"01:14.900","Text":"I want to add though, we don\u0027t need to put a plus constant when we\u0027re computing A(x)."},{"Start":"01:14.900 ","End":"01:16.430","Text":"We\u0027ve already discussed this before."},{"Start":"01:16.430 ","End":"01:17.495","Text":"In the other bit,"},{"Start":"01:17.495 ","End":"01:19.580","Text":"we have to compute b(x) e^ A(x),"},{"Start":"01:19.580 ","End":"01:24.080","Text":"the integral of that we already have A(x), which is this."},{"Start":"01:24.080 ","End":"01:28.310","Text":"Which I copy from here and we can simplify a bit."},{"Start":"01:28.310 ","End":"01:32.030","Text":"What we can do is say that this is 1 over x minus 2."},{"Start":"01:32.030 ","End":"01:36.590","Text":"That\u0027s just because the exponent on the logarithm cancel each other out"},{"Start":"01:36.590 ","End":"01:41.540","Text":"basically minus the logarithm of x minus 2,"},{"Start":"01:41.540 ","End":"01:43.160","Text":"but the laws of logarithm,"},{"Start":"01:43.160 ","End":"01:48.005","Text":"it\u0027s x minus 2^-1 It\u0027s as if there was a minus 1 there,"},{"Start":"01:48.005 ","End":"01:50.975","Text":"so it\u0027s x minus 2^-1,"},{"Start":"01:50.975 ","End":"01:55.310","Text":"which is a natural logarithm of 1 over x minus"},{"Start":"01:55.310 ","End":"02:00.845","Text":"2 and when we take e to the power of natural log of something,"},{"Start":"02:00.845 ","End":"02:03.760","Text":"it\u0027s just that something itself because"},{"Start":"02:03.760 ","End":"02:08.810","Text":"the exponent on the natural logarithm are opposite functions of each other."},{"Start":"02:08.810 ","End":"02:11.690","Text":"Take the log then the exponent that gets back to the original."},{"Start":"02:11.690 ","End":"02:18.125","Text":"The next thing we can do is to say that if I cancel the x minus 2 here,"},{"Start":"02:18.125 ","End":"02:20.350","Text":"I cancel with one of the x minus 2,"},{"Start":"02:20.350 ","End":"02:22.330","Text":"it\u0027s as if I just cancel the 2 here,"},{"Start":"02:22.330 ","End":"02:25.420","Text":"and then 2x minus 2 is 2x minus 4."},{"Start":"02:25.420 ","End":"02:27.649","Text":"That\u0027s a pretty straightforward integral,"},{"Start":"02:27.649 ","End":"02:30.580","Text":"and let\u0027s just x squared minus 4x."},{"Start":"02:30.580 ","End":"02:32.515","Text":"Now we have to plug in."},{"Start":"02:32.515 ","End":"02:36.620","Text":"I\u0027m bringing back the original formula that we have to plug into,"},{"Start":"02:36.620 ","End":"02:38.900","Text":"so A(x) in here,"},{"Start":"02:38.900 ","End":"02:42.155","Text":"and I put this in there,"},{"Start":"02:42.155 ","End":"02:45.095","Text":"and the other one, this thing,"},{"Start":"02:45.095 ","End":"02:47.900","Text":"I just plug in this thing,"},{"Start":"02:47.900 ","End":"02:54.125","Text":"and so we end up getting e^- and there is a minus already there."},{"Start":"02:54.125 ","End":"03:00.285","Text":"Here we have x squared minus 4x plus the constant this time but like I said before,"},{"Start":"03:00.285 ","End":"03:05.135","Text":"e and natural log are inverse of each other just like I did here."},{"Start":"03:05.135 ","End":"03:07.400","Text":"Again, I can do that same trick here."},{"Start":"03:07.400 ","End":"03:09.980","Text":"Notice that the minus and minus disappear,"},{"Start":"03:09.980 ","End":"03:11.315","Text":"so that gives me a plus."},{"Start":"03:11.315 ","End":"03:12.425","Text":"I\u0027m using this formula,"},{"Start":"03:12.425 ","End":"03:18.420","Text":"this is just x minus 2 and the rest of it stays as it is and we are done."}],"Thumbnail":null,"ID":7680},{"Watched":false,"Name":"Exercise 4","Duration":"3m 47s","ChapterTopicVideoID":7603,"CourseChapterTopicPlaylistID":4223,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"Here we have another differential equation to solve."},{"Start":"00:02.730 ","End":"00:05.460","Text":"It turns out we can bring this to linear form."},{"Start":"00:05.460 ","End":"00:09.870","Text":"I\u0027ll remind you, a linear differential equation is something that looks like this."},{"Start":"00:09.870 ","End":"00:13.500","Text":"y\u0027 plus some function of x times y equals another function of x."},{"Start":"00:13.500 ","End":"00:15.600","Text":"Then we have the formula for solving it."},{"Start":"00:15.600 ","End":"00:19.340","Text":"Here I see I have to divide by x^3 to get y\u0027 on its own."},{"Start":"00:19.340 ","End":"00:22.410","Text":"If we divide by x^3 here we have y\u0027."},{"Start":"00:22.410 ","End":"00:24.090","Text":"On the right-hand side, we have 1."},{"Start":"00:24.090 ","End":"00:27.090","Text":"This bit, if we divide by x^3, we get this."},{"Start":"00:27.090 ","End":"00:29.325","Text":"This is what I call little a(x),"},{"Start":"00:29.325 ","End":"00:31.260","Text":"and this is b(x)."},{"Start":"00:31.260 ","End":"00:34.980","Text":"What we want to do is to compute what\u0027s in here,"},{"Start":"00:34.980 ","End":"00:37.485","Text":"but we\u0027ll do it in steps."},{"Start":"00:37.485 ","End":"00:40.080","Text":"The first thing we\u0027ll do is compute A(x),"},{"Start":"00:40.080 ","End":"00:44.645","Text":"which is our convention for the integral of little a(x)."},{"Start":"00:44.645 ","End":"00:47.555","Text":"This bit is what I call double asterisk."},{"Start":"00:47.555 ","End":"00:51.455","Text":"Just the a(x) is what I call asterisk."},{"Start":"00:51.455 ","End":"00:54.020","Text":"Asterisk is the integral of a(x),"},{"Start":"00:54.020 ","End":"00:58.759","Text":"which is the integral of 2 minus 3x^2 over x^3 dx."},{"Start":"00:58.759 ","End":"01:01.145","Text":"I break it up into 2 pieces."},{"Start":"01:01.145 ","End":"01:04.670","Text":"I get this over this minus this over this,"},{"Start":"01:04.670 ","End":"01:07.385","Text":"and then I put it in the exponent notation."},{"Start":"01:07.385 ","End":"01:09.860","Text":"This is 2x to the minus 3."},{"Start":"01:09.860 ","End":"01:13.700","Text":"Now we can easily take the integral using the power rule."},{"Start":"01:13.700 ","End":"01:17.680","Text":"We raise the power by 1 and we get x to the minus 2,"},{"Start":"01:17.680 ","End":"01:20.035","Text":"and then we divide by minus 2."},{"Start":"01:20.035 ","End":"01:22.125","Text":"We get a minus just."},{"Start":"01:22.125 ","End":"01:27.345","Text":"The x to the minus 2 is 1 over x^2 and the integral of 1 over x is natural log of x."},{"Start":"01:27.345 ","End":"01:29.250","Text":"When computing capital a(x),"},{"Start":"01:29.250 ","End":"01:30.600","Text":"we don\u0027t need to add a constant,"},{"Start":"01:30.600 ","End":"01:31.920","Text":"we\u0027ve discussed this before."},{"Start":"01:31.920 ","End":"01:35.825","Text":"Now we come to the second bit which I indicated as double asterisk."},{"Start":"01:35.825 ","End":"01:37.790","Text":"It\u0027s the integral of this."},{"Start":"01:37.790 ","End":"01:40.495","Text":"Now, b(x) is 1,"},{"Start":"01:40.495 ","End":"01:44.030","Text":"we saw that above and a(x) we computed,"},{"Start":"01:44.030 ","End":"01:46.730","Text":"it\u0027s just here, I just copied that there."},{"Start":"01:46.730 ","End":"01:51.395","Text":"We just got this integral which we can manipulate now,"},{"Start":"01:51.395 ","End":"01:55.295","Text":"the exponent can be written as a product of exponents."},{"Start":"01:55.295 ","End":"01:59.480","Text":"It\u0027s e to the power of this times e to the power of that."},{"Start":"01:59.480 ","End":"02:04.280","Text":"Now, we have e to the -3 natural log of x and there\u0027s a formula,"},{"Start":"02:04.280 ","End":"02:05.780","Text":"we don\u0027t need to use the formula,"},{"Start":"02:05.780 ","End":"02:07.280","Text":"we could do it directly,"},{"Start":"02:07.280 ","End":"02:13.055","Text":"but I\u0027ll use this formula e^ -k times natural log of x is 1 over x^k,"},{"Start":"02:13.055 ","End":"02:16.695","Text":"since our k is 3 here,"},{"Start":"02:16.695 ","End":"02:18.350","Text":"we get 1 over x^3."},{"Start":"02:18.350 ","End":"02:20.180","Text":"Now how are we going to solve this integral?"},{"Start":"02:20.180 ","End":"02:23.600","Text":"There\u0027s another useful tool that if we have e to"},{"Start":"02:23.600 ","End":"02:27.740","Text":"the power of something and then we have its derivative alongside,"},{"Start":"02:27.740 ","End":"02:29.240","Text":"then we know it\u0027s integral."},{"Start":"02:29.240 ","End":"02:33.950","Text":"The thing is that if we want to take f as -1 over x^2,"},{"Start":"02:33.950 ","End":"02:37.460","Text":"it\u0027s derivative is 2 over x^3. You can check."},{"Start":"02:37.460 ","End":"02:38.825","Text":"If I put a 2 here,"},{"Start":"02:38.825 ","End":"02:41.810","Text":"no problem as long as I compensate by writing a 2 in"},{"Start":"02:41.810 ","End":"02:44.810","Text":"the denominator also which I can do in front of the integral sign."},{"Start":"02:44.810 ","End":"02:50.010","Text":"Now I\u0027m using this formula here with f being -1 over x^2."},{"Start":"02:50.010 ","End":"02:55.200","Text":"What I get is the half from here, e^-1 over x^2."},{"Start":"02:55.200 ","End":"03:02.625","Text":"Here we are, here\u0027s the formula again and a(x) was equal to this mess here."},{"Start":"03:02.625 ","End":"03:06.480","Text":"There\u0027s a minus in front and b(x) was just equal to this."},{"Start":"03:06.480 ","End":"03:08.085","Text":"When we have the plus c,"},{"Start":"03:08.085 ","End":"03:09.920","Text":"what we can do is take the minus,"},{"Start":"03:09.920 ","End":"03:13.905","Text":"minus to be a plus both here and here."},{"Start":"03:13.905 ","End":"03:17.510","Text":"I\u0027ve got rid of the minuses and the next thing is to write"},{"Start":"03:17.510 ","End":"03:22.085","Text":"the exponent of a sum as the product of the exponents."},{"Start":"03:22.085 ","End":"03:26.630","Text":"We can use that formula we had before for this to simplify"},{"Start":"03:26.630 ","End":"03:31.285","Text":"it and this comes out to be just x^3."},{"Start":"03:31.285 ","End":"03:37.580","Text":"That\u0027s pretty much it except that we can now cancel e^1 over x^2."},{"Start":"03:37.580 ","End":"03:40.320","Text":"The x^3 becomes just x^3,"},{"Start":"03:40.320 ","End":"03:43.695","Text":"and here also x^3, e^1 over x^2."},{"Start":"03:43.695 ","End":"03:44.840","Text":"After opening the brackets,"},{"Start":"03:44.840 ","End":"03:47.910","Text":"this is what we get and we are done."}],"Thumbnail":null,"ID":7677},{"Watched":false,"Name":"Exercise 5","Duration":"3m 48s","ChapterTopicVideoID":7607,"CourseChapterTopicPlaylistID":4223,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.355","Text":"Here we have a linear differential equation with a couple of differences."},{"Start":"00:05.355 ","End":"00:08.280","Text":"The independent variable is t and not x,"},{"Start":"00:08.280 ","End":"00:10.050","Text":"but that\u0027s no big deal."},{"Start":"00:10.050 ","End":"00:11.790","Text":"We also have an initial condition,"},{"Start":"00:11.790 ","End":"00:15.210","Text":"which means it will be able to get rid of the constant at the end."},{"Start":"00:15.210 ","End":"00:19.500","Text":"Let me remind you of the formula for linear differential equation."},{"Start":"00:19.500 ","End":"00:22.080","Text":"This is the form of a linear differential equation."},{"Start":"00:22.080 ","End":"00:27.795","Text":"This time we put a t instead of x and we have it in this form,"},{"Start":"00:27.795 ","End":"00:32.960","Text":"because if we express it as a(t) is 1,"},{"Start":"00:32.960 ","End":"00:36.065","Text":"we have 1y and b(t) is 2+2t,"},{"Start":"00:36.065 ","End":"00:38.330","Text":"then we\u0027ve exactly got it in this form with the"},{"Start":"00:38.330 ","End":"00:40.760","Text":"a and the b and let me just remind you that"},{"Start":"00:40.760 ","End":"00:45.875","Text":"capital A(t) is the integral of little a(t)."},{"Start":"00:45.875 ","End":"00:48.485","Text":"I like to use asterisks,"},{"Start":"00:48.485 ","End":"00:53.810","Text":"so this part here is going to be asterisk and I\u0027ll do that separately,"},{"Start":"00:53.810 ","End":"00:55.675","Text":"and this integral here,"},{"Start":"00:55.675 ","End":"00:57.420","Text":"I\u0027ll call double asterisk,"},{"Start":"00:57.420 ","End":"00:59.520","Text":"are 2 partial computations."},{"Start":"00:59.520 ","End":"01:02.570","Text":"The first 1 is the asterisk."},{"Start":"01:02.570 ","End":"01:04.565","Text":"This is an easy integration."},{"Start":"01:04.565 ","End":"01:06.485","Text":"Little a(t) is 1,"},{"Start":"01:06.485 ","End":"01:09.365","Text":"so the integral of 1dt is just t,"},{"Start":"01:09.365 ","End":"01:13.979","Text":"the other bit, the double asterisk is this expression."},{"Start":"01:13.979 ","End":"01:16.845","Text":"Let\u0027s see, go back up and see what b was,"},{"Start":"01:16.845 ","End":"01:19.350","Text":"b was equal to 2+2t,"},{"Start":"01:19.350 ","End":"01:23.250","Text":"a(t) was just t so that\u0027s this t here,"},{"Start":"01:23.250 ","End":"01:27.880","Text":"it\u0027s this 1, and what we get is 2te^t."},{"Start":"01:27.880 ","End":"01:31.385","Text":"Now how did we get from here to here?"},{"Start":"01:31.385 ","End":"01:33.005","Text":"I\u0027ll leave that at the end."},{"Start":"01:33.005 ","End":"01:34.445","Text":"I\u0027ll just make a note,"},{"Start":"01:34.445 ","End":"01:37.100","Text":"still have to show that from here to here."},{"Start":"01:37.100 ","End":"01:38.525","Text":"This is what we get."},{"Start":"01:38.525 ","End":"01:40.205","Text":"I\u0027ll do that at the end."},{"Start":"01:40.205 ","End":"01:42.710","Text":"Of course you could also just check that this is true"},{"Start":"01:42.710 ","End":"01:45.185","Text":"by differentiating this and getting this."},{"Start":"01:45.185 ","End":"01:48.620","Text":"So continuing, I want to substitute back"},{"Start":"01:48.620 ","End":"01:52.220","Text":"now that I have asterisk and the double asterisk,"},{"Start":"01:52.220 ","End":"01:53.870","Text":"I want to substitute,"},{"Start":"01:53.870 ","End":"01:57.880","Text":"and what I\u0027m going to do is put the A(t) here,"},{"Start":"01:57.880 ","End":"01:59.930","Text":"and the rest of it here,"},{"Start":"01:59.930 ","End":"02:01.640","Text":"the double asterisk here,"},{"Start":"02:01.640 ","End":"02:04.550","Text":"so what we end up with is,"},{"Start":"02:04.550 ","End":"02:10.010","Text":"A(t) is just t so that\u0027s that bit and this expression"},{"Start":"02:10.010 ","End":"02:15.920","Text":"we already found here is 2t^t and we have a+C from here."},{"Start":"02:15.920 ","End":"02:20.360","Text":"This is now y in terms of t. Let me just simplify this a"},{"Start":"02:20.360 ","End":"02:24.380","Text":"bit and get that multiplying out y equals to 2t,"},{"Start":"02:24.380 ","End":"02:25.640","Text":"because the e^-t,"},{"Start":"02:25.640 ","End":"02:29.005","Text":"e^t cancel plus C e^-t."},{"Start":"02:29.005 ","End":"02:33.205","Text":"This stage is where we introduce the initial condition."},{"Start":"02:33.205 ","End":"02:36.510","Text":"We told that y of 0 is 1,"},{"Start":"02:36.510 ","End":"02:39.555","Text":"so y=1, t=0."},{"Start":"02:39.555 ","End":"02:43.800","Text":"So 1=2 times 0+C e^-0."},{"Start":"02:43.800 ","End":"02:45.720","Text":"This is 0, this is C,"},{"Start":"02:45.720 ","End":"02:50.155","Text":"so 1=C, and if C=1,"},{"Start":"02:50.155 ","End":"02:53.420","Text":"then we just put it here, C=1,"},{"Start":"02:53.420 ","End":"02:58.460","Text":"and we get 2t+1e^-t and this is it."},{"Start":"02:58.460 ","End":"03:02.855","Text":"I just have to return the debt of showing you how I did that integral."},{"Start":"03:02.855 ","End":"03:07.440","Text":"If you remember the integral was this,"},{"Start":"03:07.440 ","End":"03:11.040","Text":"and the way to do it is by parts,"},{"Start":"03:11.040 ","End":"03:13.700","Text":"just to let this equal u and this v,"},{"Start":"03:13.700 ","End":"03:17.720","Text":"and then use the formula for integration by parts and just in"},{"Start":"03:17.720 ","End":"03:22.190","Text":"case you\u0027ve forgotten that it\u0027s the integral of,"},{"Start":"03:22.190 ","End":"03:24.380","Text":"well, there\u0027s 2 versions of it."},{"Start":"03:24.380 ","End":"03:30.870","Text":"We can say udv=uv minus the integral of vdu,"},{"Start":"03:30.870 ","End":"03:36.180","Text":"but there\u0027s the other variation where the integral of uv\u0027"},{"Start":"03:36.180 ","End":"03:44.700","Text":"dx=uv minus the integral of vu\u0027 dx."},{"Start":"03:44.700 ","End":"03:48.880","Text":"Very similar, and we are done."}],"Thumbnail":null,"ID":7681},{"Watched":false,"Name":"Exercise 6","Duration":"3m 8s","ChapterTopicVideoID":7608,"CourseChapterTopicPlaylistID":4223,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.145","Text":"Here we have another differential equation to solve and it\u0027s clearly linear."},{"Start":"00:05.145 ","End":"00:09.405","Text":"We have an extra condition that sin x is bigger than 0."},{"Start":"00:09.405 ","End":"00:16.290","Text":"For example, this would happen if we took x between 0 and Pi or 180 degrees,"},{"Start":"00:16.290 ","End":"00:18.465","Text":"then the sine would be positive."},{"Start":"00:18.465 ","End":"00:20.159","Text":"Let\u0027s get onto the solution."},{"Start":"00:20.159 ","End":"00:24.390","Text":"First of all, the general formula for linear differential equations."},{"Start":"00:24.390 ","End":"00:29.160","Text":"We have it in this form and we just have to say what a and b are here,"},{"Start":"00:29.160 ","End":"00:31.695","Text":"and then use the formula for the solution."},{"Start":"00:31.695 ","End":"00:35.865","Text":"In our case, a(x) is the cotangent of x."},{"Start":"00:35.865 ","End":"00:43.170","Text":"It\u0027s placed after the y and b(x) is the right-hand side is 5e^cos x."},{"Start":"00:43.170 ","End":"00:46.405","Text":"Let\u0027s first of all compute a(x)."},{"Start":"00:46.405 ","End":"00:50.540","Text":"I\u0027ll call that asterisk and the other side computation,"},{"Start":"00:50.540 ","End":"00:52.595","Text":"call this double asterisk."},{"Start":"00:52.595 ","End":"00:56.210","Text":"Asterisk is computing a(x),"},{"Start":"00:56.210 ","End":"01:01.715","Text":"which is just a convention for the integral of little a(x)."},{"Start":"01:01.715 ","End":"01:03.185","Text":"That\u0027s our convention."},{"Start":"01:03.185 ","End":"01:05.570","Text":"This is equal to the integral of"},{"Start":"01:05.570 ","End":"01:12.145","Text":"cotangent xdx and that\u0027s 1 of those integral table lookups."},{"Start":"01:12.145 ","End":"01:14.840","Text":"It\u0027s possible to do it with substitution,"},{"Start":"01:14.840 ","End":"01:19.265","Text":"but it\u0027s a lookup and the answer is ln(sin x)."},{"Start":"01:19.265 ","End":"01:23.015","Text":"For capital A, we don\u0027t add a C, it\u0027s not necessary."},{"Start":"01:23.015 ","End":"01:25.175","Text":"We\u0027ve discussed this before."},{"Start":"01:25.175 ","End":"01:28.580","Text":"Now about the double asterisk,"},{"Start":"01:28.580 ","End":"01:36.175","Text":"b(x) is what we wrote here 5e^cos x and then we need the e^A(x),"},{"Start":"01:36.175 ","End":"01:38.430","Text":"so it\u0027s e to the power of."},{"Start":"01:38.430 ","End":"01:41.540","Text":"I just copied natural log (sin x) here."},{"Start":"01:41.540 ","End":"01:45.260","Text":"We can simplify because the e to"},{"Start":"01:45.260 ","End":"01:47.270","Text":"the power of natural log of something is"},{"Start":"01:47.270 ","End":"01:49.685","Text":"just that something because they are inverse of each other,"},{"Start":"01:49.685 ","End":"01:54.735","Text":"so e^ln(sin x) is just sine x."},{"Start":"01:54.735 ","End":"01:58.020","Text":"Now I have 5e^cos x sine xdx."},{"Start":"01:58.020 ","End":"02:01.550","Text":"I\u0027m going to use the fact formula that if I"},{"Start":"02:01.550 ","End":"02:05.495","Text":"have e to the power of something and its derivative alongside,"},{"Start":"02:05.495 ","End":"02:08.195","Text":"then the integral of that is just e to the power of."},{"Start":"02:08.195 ","End":"02:12.950","Text":"We almost have that because if we want to take f as cosine x,"},{"Start":"02:12.950 ","End":"02:15.800","Text":"its derivative f prime is minus sine x."},{"Start":"02:15.800 ","End":"02:21.725","Text":"No problem, put minus sine x and put an extra minus in front and it will all work out."},{"Start":"02:21.725 ","End":"02:25.520","Text":"Now we can use this formula and we get that this is e to the"},{"Start":"02:25.520 ","End":"02:29.755","Text":"f with the minus 5 at the front, e^cos x."},{"Start":"02:29.755 ","End":"02:33.900","Text":"Now we can continue with our substitution,"},{"Start":"02:33.900 ","End":"02:36.035","Text":"but the formula for the final answer,"},{"Start":"02:36.035 ","End":"02:39.575","Text":"we\u0027ve got minus A^x."},{"Start":"02:39.575 ","End":"02:43.710","Text":"That\u0027s minus the ln(sin x) from"},{"Start":"02:43.710 ","End":"02:48.020","Text":"here and we\u0027ve got the whole expression here that was our double asterisk."},{"Start":"02:48.020 ","End":"02:54.230","Text":"That\u0027s minus 5e^cos x plus C. Again,"},{"Start":"02:54.230 ","End":"02:57.680","Text":"we have e to the power of natural log,"},{"Start":"02:57.680 ","End":"02:59.540","Text":"but because it\u0027s a minus,"},{"Start":"02:59.540 ","End":"03:02.690","Text":"we\u0027re going to get 1/sin x."},{"Start":"03:02.690 ","End":"03:04.610","Text":"We\u0027ve done this manipulation before,"},{"Start":"03:04.610 ","End":"03:09.120","Text":"I won\u0027t get into it and this will be our final answer."}],"Thumbnail":null,"ID":7682},{"Watched":false,"Name":"Exercise 7","Duration":"3m 50s","ChapterTopicVideoID":7609,"CourseChapterTopicPlaylistID":4223,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.920","Text":"In this exercise we have to solve the linear differential equation that\u0027s here;"},{"Start":"00:05.920 ","End":"00:07.950","Text":"the linear of the first-order,"},{"Start":"00:07.950 ","End":"00:13.980","Text":"and we\u0027re given the condition that sine x is bigger than 0 this could happen for example,"},{"Start":"00:13.980 ","End":"00:19.635","Text":"if we said that 0 is less than x is less than Pi."},{"Start":"00:19.635 ","End":"00:21.420","Text":"Between 0 and 190 degrees,"},{"Start":"00:21.420 ","End":"00:22.560","Text":"the sine is positive."},{"Start":"00:22.560 ","End":"00:24.510","Text":"I prefer the limitation on x,"},{"Start":"00:24.510 ","End":"00:26.219","Text":"but in general."},{"Start":"00:26.219 ","End":"00:27.980","Text":"This is linear."},{"Start":"00:27.980 ","End":"00:30.830","Text":"I\u0027m going to remind you of the formular."},{"Start":"00:30.830 ","End":"00:37.070","Text":"The template, the pattern of a linear differential equation is y\u0027 plus some function of"},{"Start":"00:37.070 ","End":"00:43.310","Text":"x times y equals another function of x and this formula provides the general solution."},{"Start":"00:43.310 ","End":"00:47.465","Text":"First of all, let\u0027s just rewrite it in a more convenient way."},{"Start":"00:47.465 ","End":"00:50.900","Text":"I\u0027ll put the cotangent in front together with"},{"Start":"00:50.900 ","End":"00:55.460","Text":"the minus and so we can see exactly what a(x) is in here,"},{"Start":"00:55.460 ","End":"01:00.050","Text":"and b(x) is just the constant function 1 even a constant is a function of x."},{"Start":"01:00.050 ","End":"01:04.850","Text":"What I want to do is to compute two pieces separately,"},{"Start":"01:04.850 ","End":"01:07.040","Text":"it\u0027s just more organized that way."},{"Start":"01:07.040 ","End":"01:13.970","Text":"I\u0027m going to compute A(x) and I\u0027m going to compute this integral up to here."},{"Start":"01:13.970 ","End":"01:17.525","Text":"One of them I like to call the asterisk,"},{"Start":"01:17.525 ","End":"01:19.396","Text":"that\u0027s the first one."},{"Start":"01:19.396 ","End":"01:22.820","Text":"The second one I will double asterisk."},{"Start":"01:22.820 ","End":"01:26.235","Text":"A(x) is the integral of A(x) dx,"},{"Start":"01:26.235 ","End":"01:31.600","Text":"and A(x) is according to this minus twice cotangent of x"},{"Start":"01:31.600 ","End":"01:38.360","Text":"so we get to this point and the integral of cotangent of x is natural log of sine x."},{"Start":"01:38.360 ","End":"01:40.520","Text":"At least if sine x is positive,"},{"Start":"01:40.520 ","End":"01:42.080","Text":"you can take my word for it,"},{"Start":"01:42.080 ","End":"01:44.375","Text":"or you can look it up in the integral table,"},{"Start":"01:44.375 ","End":"01:47.375","Text":"or maybe I\u0027ll say a few words on this."},{"Start":"01:47.375 ","End":"01:52.925","Text":"Basically, it\u0027s because the integral of cotangent x,"},{"Start":"01:52.925 ","End":"01:59.745","Text":"cotangent is cosine x over sine x and this folds into the pattern of"},{"Start":"01:59.745 ","End":"02:07.260","Text":"f\u0027 over f which is natural log of f. In our case,"},{"Start":"02:07.260 ","End":"02:09.740","Text":"f is sine, f\u0027 is cosine,"},{"Start":"02:09.740 ","End":"02:15.560","Text":"so we get the natural log of the sine and that\u0027s the general idea."},{"Start":"02:15.560 ","End":"02:17.205","Text":"The second one;"},{"Start":"02:17.205 ","End":"02:20.230","Text":"b(x) is just 1,"},{"Start":"02:20.230 ","End":"02:23.460","Text":"so we have e^A(x)."},{"Start":"02:23.460 ","End":"02:25.070","Text":"That\u0027s what we just found here,"},{"Start":"02:25.070 ","End":"02:26.915","Text":"that\u0027s copied into here."},{"Start":"02:26.915 ","End":"02:29.090","Text":"How do we do this integral?"},{"Start":"02:29.090 ","End":"02:32.540","Text":"We\u0027re going to use the general template that"},{"Start":"02:32.540 ","End":"02:36.230","Text":"e^minus k natural log of something is just"},{"Start":"02:36.230 ","End":"02:40.176","Text":"1 over that something to the k and k here being 2."},{"Start":"02:40.176 ","End":"02:46.430","Text":"Applying this, we get here the integral of one 1 sine squared x dx."},{"Start":"02:46.430 ","End":"02:49.705","Text":"Once again, this is an immediate or a table lookup"},{"Start":"02:49.705 ","End":"02:54.095","Text":"integral and the answer is minus cotangent x."},{"Start":"02:54.095 ","End":"02:56.000","Text":"Now let\u0027s put it all together."},{"Start":"02:56.000 ","End":"03:05.810","Text":"This formula here, e^minus A(x) from here is minus 2 natural log sine x."},{"Start":"03:05.810 ","End":"03:09.160","Text":"Well, we have all this double asterisk here,"},{"Start":"03:09.160 ","End":"03:13.025","Text":"so it just turns out to be minus cotangent x. I can copy from there,"},{"Start":"03:13.025 ","End":"03:16.370","Text":"and that\u0027s the plus c here. We\u0027re almost done."},{"Start":"03:16.370 ","End":"03:19.940","Text":"There\u0027s a little bit of simplification to do and that is"},{"Start":"03:19.940 ","End":"03:24.800","Text":"that once again we can use this formula,"},{"Start":"03:24.800 ","End":"03:31.785","Text":"so the minus cancels with the minus and then I\u0027ve got k being minus 2."},{"Start":"03:31.785 ","End":"03:37.230","Text":"What we get is instead of 1 over sin(x)^2;"},{"Start":"03:37.230 ","End":"03:39.525","Text":"sine x is the box,"},{"Start":"03:39.525 ","End":"03:44.988","Text":"we get it on the numerator because when k is negative it pushes it to the numerator."},{"Start":"03:44.988 ","End":"03:48.230","Text":"The rest of it is as it was before,"},{"Start":"03:48.230 ","End":"03:51.090","Text":"and this is our answer."}],"Thumbnail":null,"ID":7683},{"Watched":false,"Name":"Exercise 8","Duration":"3m 15s","ChapterTopicVideoID":7610,"CourseChapterTopicPlaylistID":4223,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"Here we have a differential equation with an initial condition."},{"Start":"00:04.170 ","End":"00:07.230","Text":"It\u0027s actually linear of the first-order, but not quite."},{"Start":"00:07.230 ","End":"00:09.315","Text":"We just need to manipulate a little bit."},{"Start":"00:09.315 ","End":"00:15.300","Text":"Also, note that the dependent variable is not y it\u0027s z."},{"Start":"00:15.300 ","End":"00:18.330","Text":"Don\u0027t have to worry that z is there instead of y."},{"Start":"00:18.330 ","End":"00:20.070","Text":"Let\u0027s do some manipulation."},{"Start":"00:20.070 ","End":"00:22.515","Text":"First let\u0027s remember what the pattern is,"},{"Start":"00:22.515 ","End":"00:25.395","Text":"z\u0027 plus some function (x),"},{"Start":"00:25.395 ","End":"00:29.835","Text":"z = b(x) and the solution z = this."},{"Start":"00:29.835 ","End":"00:33.465","Text":"This is not quite in that form by dividing everything by x^2,"},{"Start":"00:33.465 ","End":"00:35.940","Text":"I get z\u0027 plus 2 over x,"},{"Start":"00:35.940 ","End":"00:39.465","Text":"z = cosine x over x^2."},{"Start":"00:39.465 ","End":"00:42.225","Text":"This is my a(x) as here,"},{"Start":"00:42.225 ","End":"00:43.710","Text":"this is the b(x)."},{"Start":"00:43.710 ","End":"00:45.935","Text":"Now we want to compute some things."},{"Start":"00:45.935 ","End":"00:49.070","Text":"I like compute separately A(x) and"},{"Start":"00:49.070 ","End":"00:53.450","Text":"separately this integral and then throw them together in this formula."},{"Start":"00:53.450 ","End":"00:57.065","Text":"A(x) is the indefinite integral of a(x) dx."},{"Start":"00:57.065 ","End":"01:00.635","Text":"Here a(x) is 2 over x."},{"Start":"01:00.635 ","End":"01:03.725","Text":"It gives us twice natural log(x)."},{"Start":"01:03.725 ","End":"01:06.970","Text":"We have to add that x is not 0."},{"Start":"01:06.970 ","End":"01:08.630","Text":"Now the other one,"},{"Start":"01:08.630 ","End":"01:10.880","Text":"this expression double asterisk,"},{"Start":"01:10.880 ","End":"01:13.730","Text":"I\u0027ll call it, b(x) e to the A(x)."},{"Start":"01:13.730 ","End":"01:15.740","Text":"So b(x) is from here,"},{"Start":"01:15.740 ","End":"01:19.625","Text":"cosine x over x^2 e to the power of."},{"Start":"01:19.625 ","End":"01:21.610","Text":"Now we need to capital A(x),"},{"Start":"01:21.610 ","End":"01:22.660","Text":"which is just right here."},{"Start":"01:22.660 ","End":"01:25.340","Text":"These 2 natural log(x) that goes in here."},{"Start":"01:25.340 ","End":"01:29.275","Text":"What we get is cosine x over x^2."},{"Start":"01:29.275 ","End":"01:33.080","Text":"Here, using this trick with the logarithms,"},{"Start":"01:33.080 ","End":"01:36.860","Text":"It\u0027s e to the natural log of this squared,"},{"Start":"01:36.860 ","End":"01:39.440","Text":"so it\u0027s absolute value of x^2."},{"Start":"01:39.440 ","End":"01:45.840","Text":"However, the absolute value of x^2 is the same as x^2 in both cases,"},{"Start":"01:45.840 ","End":"01:47.310","Text":"it\u0027s not going to be negative."},{"Start":"01:47.310 ","End":"01:49.860","Text":"This actually cancels with this."},{"Start":"01:49.860 ","End":"01:56.045","Text":"What we get is just the integral of cosine x dx,"},{"Start":"01:56.045 ","End":"01:58.565","Text":"and this is sine(x)."},{"Start":"01:58.565 ","End":"02:01.595","Text":"Now we throw all these bits together with"},{"Start":"02:01.595 ","End":"02:07.115","Text":"the asterisk and the double-asterisk into this formula."},{"Start":"02:07.115 ","End":"02:13.660","Text":"A(x) here is twice natural log of absolute value of x."},{"Start":"02:13.660 ","End":"02:16.440","Text":"That\u0027s here, just about see it."},{"Start":"02:16.440 ","End":"02:19.095","Text":"This whole thing was sine x."},{"Start":"02:19.095 ","End":"02:24.890","Text":"Again, we\u0027re going to use the trick with each of the power of."},{"Start":"02:24.890 ","End":"02:29.060","Text":"We get 1 over absolute value of x^2."},{"Start":"02:29.060 ","End":"02:32.735","Text":"But like we did before, we can throw out the absolute value of x when it\u0027s squared."},{"Start":"02:32.735 ","End":"02:34.705","Text":"This is what we end up getting."},{"Start":"02:34.705 ","End":"02:37.190","Text":"If we didn\u0027t have the initial condition, we\u0027d stop here."},{"Start":"02:37.190 ","End":"02:38.750","Text":"This would be the general solution."},{"Start":"02:38.750 ","End":"02:41.480","Text":"But in our case, we have an initial condition."},{"Start":"02:41.480 ","End":"02:44.150","Text":"That is, that z(Pi) is 0,"},{"Start":"02:44.150 ","End":"02:47.165","Text":"means I plug in 0 for z,"},{"Start":"02:47.165 ","End":"02:55.190","Text":"and I plugin Pi for x. I get 0 = 1 over Pi^2 sine Pi plus C. Now,"},{"Start":"02:55.190 ","End":"02:58.310","Text":"sine Pi is 0,"},{"Start":"02:58.310 ","End":"03:06.470","Text":"so I get that C over Pi^2 is 0 and C is 0."},{"Start":"03:06.470 ","End":"03:08.680","Text":"If I put this C in here,"},{"Start":"03:08.680 ","End":"03:13.905","Text":"then all I get is that z is sine x over x^2."},{"Start":"03:13.905 ","End":"03:16.570","Text":"Now we\u0027re done."}],"Thumbnail":null,"ID":7684}],"ID":4223},{"Name":"Ricatti Equations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Ricatti Equations","Duration":"4m 31s","ChapterTopicVideoID":7639,"CourseChapterTopicPlaylistID":4225,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.470","Text":"In this exercise we\u0027re asked to say what a Riccati equation"},{"Start":"00:04.470 ","End":"00:09.195","Text":"is in the field of ordinary differential equations and explain how to solve it."},{"Start":"00:09.195 ","End":"00:12.690","Text":"Well, this is really a tutorial disguised as an exercise."},{"Start":"00:12.690 ","End":"00:17.790","Text":"It\u0027s just a format for me to explain to you what is a Riccati equation."},{"Start":"00:17.790 ","End":"00:21.060","Text":"Here\u0027s what the Riccati equation looks like."},{"Start":"00:21.060 ","End":"00:26.160","Text":"It\u0027s y\u0027 equals some function of x plus another function of x,"},{"Start":"00:26.160 ","End":"00:27.780","Text":"y plus another function of x,"},{"Start":"00:27.780 ","End":"00:29.895","Text":"y times y squared."},{"Start":"00:29.895 ","End":"00:36.000","Text":"The functions q_0 and q_2 here must not be the 0 functions."},{"Start":"00:36.000 ","End":"00:41.115","Text":"The simple reason for each of them if q_2 is 0,"},{"Start":"00:41.115 ","End":"00:46.760","Text":"then we just get a linear equation this equals this plus this."},{"Start":"00:46.760 ","End":"00:48.260","Text":"Just bringing this to the other side,"},{"Start":"00:48.260 ","End":"00:49.550","Text":"you\u0027ll see it\u0027s linear."},{"Start":"00:49.550 ","End":"00:52.130","Text":"If q_0 is 0,"},{"Start":"00:52.130 ","End":"00:55.370","Text":"then if you bring this to the other side,"},{"Start":"00:55.370 ","End":"00:58.820","Text":"you will see that we\u0027ve got a Bernoulli equation."},{"Start":"00:58.820 ","End":"01:05.450","Text":"These 2 cases we know how to solve so we\u0027ll assume that here these 2 functions are not 0."},{"Start":"01:05.450 ","End":"01:07.810","Text":"Middle one could be."},{"Start":"01:07.870 ","End":"01:10.400","Text":"How do we solve such a thing?"},{"Start":"01:10.400 ","End":"01:13.910","Text":"Well, there\u0027s no real way of actually solving it."},{"Start":"01:13.910 ","End":"01:17.240","Text":"The next best thing we can do is say that if you happen to"},{"Start":"01:17.240 ","End":"01:21.035","Text":"make an educated guess and find one solution,"},{"Start":"01:21.035 ","End":"01:24.550","Text":"we can use that to find the general solution."},{"Start":"01:24.550 ","End":"01:26.765","Text":"When I\u0027m writing the steps,"},{"Start":"01:26.765 ","End":"01:29.870","Text":"the first step is to guess a solution."},{"Start":"01:29.870 ","End":"01:32.180","Text":"But it\u0027s not just a guess."},{"Start":"01:32.180 ","End":"01:34.340","Text":"It might be an educated guess and I\u0027ll give you"},{"Start":"01:34.340 ","End":"01:37.429","Text":"some tips of how to maybe look for solutions,"},{"Start":"01:37.429 ","End":"01:39.730","Text":"but nothing is guaranteed here."},{"Start":"01:39.730 ","End":"01:44.180","Text":"The other thing is that often it\u0027s given as part of the question."},{"Start":"01:44.180 ","End":"01:50.170","Text":"Most often the question includes find the general solution given a particular solution."},{"Start":"01:50.170 ","End":"01:54.530","Text":"The second step is to solve this equation,"},{"Start":"01:54.530 ","End":"01:57.890","Text":"which is the linear equation and then the letter z,"},{"Start":"01:57.890 ","End":"01:59.779","Text":"I didn\u0027t want to use y."},{"Start":"01:59.779 ","End":"02:03.440","Text":"We just notice that it uses that q_1 and q_2,"},{"Start":"02:03.440 ","End":"02:05.930","Text":"q_0 doesn\u0027t come into this."},{"Start":"02:05.930 ","End":"02:08.165","Text":"Just compute these expressions,"},{"Start":"02:08.165 ","End":"02:10.965","Text":"q_1 and q_2 are from here,"},{"Start":"02:10.965 ","End":"02:13.489","Text":"y_1 is our particular solution."},{"Start":"02:13.489 ","End":"02:14.915","Text":"These are all functions of x,"},{"Start":"02:14.915 ","End":"02:16.670","Text":"and so we get a linear function,"},{"Start":"02:16.670 ","End":"02:18.740","Text":"z\u0027 plus some function of x,"},{"Start":"02:18.740 ","End":"02:20.420","Text":"z equals some function of x."},{"Start":"02:20.420 ","End":"02:22.594","Text":"Here we get the general solution,"},{"Start":"02:22.594 ","End":"02:24.620","Text":"meaning that when we find z,"},{"Start":"02:24.620 ","End":"02:26.915","Text":"it will contain a constant."},{"Start":"02:26.915 ","End":"02:30.890","Text":"Once we have this general z for this equation,"},{"Start":"02:30.890 ","End":"02:35.690","Text":"the general solution to the original equation is that y"},{"Start":"02:35.690 ","End":"02:40.655","Text":"equal our particular solution plus all these solutions."},{"Start":"02:40.655 ","End":"02:42.230","Text":"This contains a constant,"},{"Start":"02:42.230 ","End":"02:45.300","Text":"so we get many solutions for y."},{"Start":"02:45.300 ","End":"02:50.570","Text":"There is 1 exceptional solution just somehow not written here."},{"Start":"02:50.570 ","End":"02:54.710","Text":"The solution is this or y equals y_1."},{"Start":"02:54.710 ","End":"02:57.650","Text":"Because our original solution is not contained here,"},{"Start":"02:57.650 ","End":"03:01.700","Text":"there\u0027s no z I can put here to make 1/z equal to 0,"},{"Start":"03:01.700 ","End":"03:06.050","Text":"so we have our original solution plus all these."},{"Start":"03:06.050 ","End":"03:11.150","Text":"This is the general and this is general infinity plus 1 solutions if you like."},{"Start":"03:11.150 ","End":"03:12.590","Text":"Those are the steps."},{"Start":"03:12.590 ","End":"03:16.550","Text":"I\u0027ll just mention what I said before that"},{"Start":"03:16.550 ","End":"03:20.884","Text":"in practice on exam questions or homework assignments,"},{"Start":"03:20.884 ","End":"03:23.810","Text":"very often I can\u0027t say how many percent of the times,"},{"Start":"03:23.810 ","End":"03:27.980","Text":"but a particular solution will be given in the question so you won\u0027t have to guess."},{"Start":"03:27.980 ","End":"03:31.640","Text":"If not, suggestion is to try a solution,"},{"Start":"03:31.640 ","End":"03:33.260","Text":"something of this form,"},{"Start":"03:33.260 ","End":"03:40.340","Text":"something exponential ax^b or with base ea^bx or something like that."},{"Start":"03:40.340 ","End":"03:47.360","Text":"Sometimes a particular solution might be obvious that y equals 0 will be a solution."},{"Start":"03:47.360 ","End":"03:51.020","Text":"That\u0027s basically it except that I would just like to show you"},{"Start":"03:51.020 ","End":"03:55.505","Text":"the little summary table that I\u0027ll be including with the exercises."},{"Start":"03:55.505 ","End":"03:58.385","Text":"This is the condensed version of all the above."},{"Start":"03:58.385 ","End":"04:00.410","Text":"This is the Riccati equation,"},{"Start":"04:00.410 ","End":"04:01.955","Text":"what it looks like."},{"Start":"04:01.955 ","End":"04:07.160","Text":"Y_1 is a particular solution and the general solution is y equals y_1 plus 1/z."},{"Start":"04:07.160 ","End":"04:12.290","Text":"z is the general solution to this linear equation,"},{"Start":"04:12.290 ","End":"04:16.610","Text":"except that we also have y equals y_1,"},{"Start":"04:16.610 ","End":"04:21.055","Text":"of course, because that was the first solution that we found."},{"Start":"04:21.055 ","End":"04:23.960","Text":"That\u0027s the theory."},{"Start":"04:23.960 ","End":"04:31.710","Text":"Let\u0027s do some exercises and you can always return to this for the steps. That\u0027s it."}],"Thumbnail":null,"ID":7708},{"Watched":false,"Name":"Exercise 1","Duration":"10m 19s","ChapterTopicVideoID":7636,"CourseChapterTopicPlaylistID":4225,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.560","Text":"In this exercise, we have the following differential equation to solve."},{"Start":"00:04.560 ","End":"00:08.040","Text":"I\u0027m going to show you that this differential equation"},{"Start":"00:08.040 ","End":"00:12.045","Text":"is of type Riccati equation and I\u0027ll solve it as such."},{"Start":"00:12.045 ","End":"00:18.300","Text":"First, I\u0027ll give you a summary of Riccati equations."},{"Start":"00:18.300 ","End":"00:20.460","Text":"First of all, the definition,"},{"Start":"00:20.460 ","End":"00:28.395","Text":"it\u0027s of the form y\u0027 equals some f(x) another f(x) times y and other f(x) times y^2."},{"Start":"00:28.395 ","End":"00:30.090","Text":"If you look at it, that\u0027s what we have here,"},{"Start":"00:30.090 ","End":"00:33.150","Text":"but I\u0027ll spell out in a moment what the q_0,"},{"Start":"00:33.150 ","End":"00:35.220","Text":"q_1, and q_2 are."},{"Start":"00:35.220 ","End":"00:36.465","Text":"You have to first of all,"},{"Start":"00:36.465 ","End":"00:40.170","Text":"guess a solution of y_1(x),"},{"Start":"00:40.170 ","End":"00:43.190","Text":"then we make the following substitution."},{"Start":"00:43.190 ","End":"00:45.890","Text":"The substitution leads to this equation,"},{"Start":"00:45.890 ","End":"00:49.540","Text":"which is a linear differential equation in z."},{"Start":"00:49.540 ","End":"00:52.820","Text":"When we\u0027ve got the general solution for z,"},{"Start":"00:52.820 ","End":"00:58.699","Text":"we substitute in here and we get the general solution for y."},{"Start":"00:58.699 ","End":"01:02.135","Text":"Let\u0027s first really verify that this is a Riccati equation."},{"Start":"01:02.135 ","End":"01:06.885","Text":"If I let this q_0(x) be e^2x,"},{"Start":"01:06.885 ","End":"01:09.517","Text":"and I\u0027ll let q_1 equals,"},{"Start":"01:09.517 ","End":"01:12.350","Text":"and I just change the fraction to a decimal here,"},{"Start":"01:12.350 ","End":"01:14.960","Text":"no real change 2.5,"},{"Start":"01:14.960 ","End":"01:19.220","Text":"that\u0027s q_1 is f(x) times y and here I can write it"},{"Start":"01:19.220 ","End":"01:25.020","Text":"as 1y^2 and 1 is a constant function, but it\u0027s f(x)."},{"Start":"01:25.020 ","End":"01:26.920","Text":"It could be considered as q_2."},{"Start":"01:26.920 ","End":"01:34.580","Text":"We do have a Riccati equation here and I just expanded this bracket to be easier for me,"},{"Start":"01:34.580 ","End":"01:37.490","Text":"we somehow have to come up with a particular solution."},{"Start":"01:37.490 ","End":"01:39.110","Text":"In the majority of cases,"},{"Start":"01:39.110 ","End":"01:41.120","Text":"it\u0027s actually given in the question."},{"Start":"01:41.120 ","End":"01:43.475","Text":"I\u0027m going to make things a bit tougher."},{"Start":"01:43.475 ","End":"01:44.930","Text":"In the theory clip,"},{"Start":"01:44.930 ","End":"01:49.100","Text":"I gave a hint that if you\u0027re not given a particular solution,"},{"Start":"01:49.100 ","End":"01:57.620","Text":"you might try a function of the form ae^bx or ax^b."},{"Start":"01:57.620 ","End":"01:59.495","Text":"That\u0027s what we\u0027re going to do here."},{"Start":"01:59.495 ","End":"02:02.060","Text":"After looking at it a bit,"},{"Start":"02:02.060 ","End":"02:10.990","Text":"we see that the best to try would be y_1 equals ae^x."},{"Start":"02:10.990 ","End":"02:12.260","Text":"I\u0027m just narrowing it down."},{"Start":"02:12.260 ","End":"02:20.870","Text":"I\u0027m using this form with b is 1 because the derivative is also ae^x down here."},{"Start":"02:20.870 ","End":"02:23.615","Text":"Then the y\u0027 will cancel with y,"},{"Start":"02:23.615 ","End":"02:26.270","Text":"and also e^x^2 is equal e^2x,"},{"Start":"02:26.270 ","End":"02:27.875","Text":"which matches with this."},{"Start":"02:27.875 ","End":"02:33.600","Text":"We\u0027re going to go with y equals ae^x is a particular solution."},{"Start":"02:33.600 ","End":"02:36.649","Text":"I wrote it down here, we\u0027ll just call it y."},{"Start":"02:36.649 ","End":"02:39.770","Text":"See if we can solve with y equals ae^x."},{"Start":"02:39.770 ","End":"02:42.529","Text":"The derivative of this is this thing itself."},{"Start":"02:42.529 ","End":"02:48.950","Text":"We\u0027re still missing a. I\u0027ll substitute in this equation and this is good"},{"Start":"02:48.950 ","End":"02:56.180","Text":"because you see this cancels with this and all the exponents are 2x."},{"Start":"02:56.180 ","End":"02:59.990","Text":"Now I\u0027m going to take e^2x outside the brackets."},{"Start":"02:59.990 ","End":"03:01.640","Text":"This is what we get,"},{"Start":"03:01.640 ","End":"03:03.500","Text":"just taking a bracket."},{"Start":"03:03.500 ","End":"03:07.580","Text":"Now notice that e to the power of something is never 0."},{"Start":"03:07.580 ","End":"03:09.095","Text":"It\u0027s always positive."},{"Start":"03:09.095 ","End":"03:12.275","Text":"This quadratic must equal 0."},{"Start":"03:12.275 ","End":"03:17.910","Text":"If we use the formula a is the not x and they\u0027ll be two solutions,"},{"Start":"03:17.910 ","End":"03:20.115","Text":"a_1 and a_2,"},{"Start":"03:20.115 ","End":"03:22.415","Text":"and there\u0027s going to be the formula with the square root."},{"Start":"03:22.415 ","End":"03:24.560","Text":"Anyway, you know how to solve quadratics."},{"Start":"03:24.560 ","End":"03:26.650","Text":"I\u0027ll just go to the last stage,"},{"Start":"03:26.650 ","End":"03:30.200","Text":"call it, a_1,2 meaning both solutions."},{"Start":"03:30.200 ","End":"03:31.775","Text":"If we take the minus,"},{"Start":"03:31.775 ","End":"03:34.805","Text":"we get minus 8/4 is minus 2."},{"Start":"03:34.805 ","End":"03:35.900","Text":"If we take the plus,"},{"Start":"03:35.900 ","End":"03:37.445","Text":"you get minus 2/4,"},{"Start":"03:37.445 ","End":"03:38.825","Text":"which is minus 1/2,"},{"Start":"03:38.825 ","End":"03:41.045","Text":"That\u0027ll be our a_1 and a_2,"},{"Start":"03:41.045 ","End":"03:44.510","Text":"but we only need one particular solution."},{"Start":"03:44.510 ","End":"03:45.830","Text":"I\u0027ll choose this one."},{"Start":"03:45.830 ","End":"03:47.420","Text":"No particular reason."},{"Start":"03:47.420 ","End":"03:49.415","Text":"I\u0027ll just fancy going with this one."},{"Start":"03:49.415 ","End":"03:53.390","Text":"Answer y_1 is just ae^x,"},{"Start":"03:53.390 ","End":"03:57.035","Text":"but I took a as minus 0.5,"},{"Start":"03:57.035 ","End":"03:59.795","Text":"which is the same as minus 1/2."},{"Start":"03:59.795 ","End":"04:02.810","Text":"This is the particular solution."},{"Start":"04:02.810 ","End":"04:05.735","Text":"According to the theory,"},{"Start":"04:05.735 ","End":"04:12.875","Text":"if we make a substitution that the general y equals the particular y plus 1/z,"},{"Start":"04:12.875 ","End":"04:20.880","Text":"then z satisfies this equation."},{"Start":"04:20.880 ","End":"04:24.350","Text":"We want to now find the general solution."},{"Start":"04:24.350 ","End":"04:27.360","Text":"We have q_1 and q_2."},{"Start":"04:27.360 ","End":"04:31.670","Text":"If you go back, you\u0027ll see that this was our q_1 and"},{"Start":"04:31.670 ","End":"04:35.945","Text":"q_2 was the constant function 1 and y_1."},{"Start":"04:35.945 ","End":"04:38.960","Text":"I don\u0027t know why I\u0027m going back from fraction to decimal."},{"Start":"04:38.960 ","End":"04:42.560","Text":"Let\u0027s meanwhile write it also as a decimal."},{"Start":"04:42.560 ","End":"04:45.748","Text":"This bit is the y_1,"},{"Start":"04:45.748 ","End":"04:49.435","Text":"and q_2 once again is 1."},{"Start":"04:49.435 ","End":"04:51.815","Text":"I can simplify this a bit."},{"Start":"04:51.815 ","End":"04:53.380","Text":"Here\u0027s the one from here,"},{"Start":"04:53.380 ","End":"04:56.105","Text":"and I\u0027ve got 2.5e^x,"},{"Start":"04:56.105 ","End":"04:59.390","Text":"and this is minus 1e^x."},{"Start":"04:59.390 ","End":"05:03.750","Text":"Altogether, 1.5e^x times z is minus 1,"},{"Start":"05:03.750 ","End":"05:06.530","Text":"and this fits the pattern of a linear equation."},{"Start":"05:06.530 ","End":"05:08.240","Text":"If I let this be a(x),"},{"Start":"05:08.240 ","End":"05:10.710","Text":"this b(x), I\u0027m done."},{"Start":"05:10.710 ","End":"05:15.370","Text":"I\u0027m going to remind you of the formula for solving a linear equation."},{"Start":"05:15.370 ","End":"05:17.180","Text":"Thereby is other than using the formula,"},{"Start":"05:17.180 ","End":"05:20.540","Text":"we\u0027re going to use the formula and it doesn\u0027t say here,"},{"Start":"05:20.540 ","End":"05:26.285","Text":"but it\u0027s understood that A Is the integral of a."},{"Start":"05:26.285 ","End":"05:29.015","Text":"There\u0027s actually two integrals to solve."},{"Start":"05:29.015 ","End":"05:32.090","Text":"This, I\u0027ll call it asterisk,"},{"Start":"05:32.090 ","End":"05:33.725","Text":"and then I have another integral,"},{"Start":"05:33.725 ","End":"05:37.160","Text":"this part to here is double asterisk,"},{"Start":"05:37.160 ","End":"05:41.840","Text":"so I\u0027ll do these two integrals on the next page."},{"Start":"05:41.840 ","End":"05:43.250","Text":"We are on a new page,"},{"Start":"05:43.250 ","End":"05:46.910","Text":"and this is the equation we have to solve linear in z."},{"Start":"05:46.910 ","End":"05:50.450","Text":"As I said, the linear equation gives us two integrals."},{"Start":"05:50.450 ","End":"05:53.570","Text":"I called them asterisk and double asterisk."},{"Start":"05:53.570 ","End":"05:55.310","Text":"The first one is A,"},{"Start":"05:55.310 ","End":"05:57.215","Text":"which is the integral of a,"},{"Start":"05:57.215 ","End":"06:00.470","Text":"which is this and this is an easy integral."},{"Start":"06:00.470 ","End":"06:04.640","Text":"1 becomes x,1.5e^x is just itself."},{"Start":"06:04.640 ","End":"06:09.800","Text":"We don\u0027t put the constants and these integrals just as a constant at the end."},{"Start":"06:09.800 ","End":"06:14.534","Text":"The second integral and they called the second one double asterisk,"},{"Start":"06:14.534 ","End":"06:20.885","Text":"is this, the integral of be^A and the A,"},{"Start":"06:20.885 ","End":"06:22.430","Text":"I take it from here."},{"Start":"06:22.430 ","End":"06:28.370","Text":"We\u0027ve got the integral of minus 1 is b,"},{"Start":"06:28.370 ","End":"06:30.050","Text":"this was a here,"},{"Start":"06:30.050 ","End":"06:31.640","Text":"I put that in here."},{"Start":"06:31.640 ","End":"06:36.170","Text":"I\u0027m going to use this template formula, not immediately,"},{"Start":"06:36.170 ","End":"06:39.260","Text":"but if I have the integral of e to the power"},{"Start":"06:39.260 ","End":"06:42.680","Text":"of something and I have the derivative of that something,"},{"Start":"06:42.680 ","End":"06:47.390","Text":"then the integral is just e to the power of this function."},{"Start":"06:47.390 ","End":"06:49.220","Text":"This is what I\u0027m aiming for,"},{"Start":"06:49.220 ","End":"06:53.150","Text":"and I need to do a little bit of algebra to get this into this form."},{"Start":"06:53.150 ","End":"07:00.515","Text":"The first step is to use the rules for exponents that this plus becomes a product."},{"Start":"07:00.515 ","End":"07:06.050","Text":"Now I\u0027m aiming to get it to look like this."},{"Start":"07:06.050 ","End":"07:08.630","Text":"If this is my f here,"},{"Start":"07:08.630 ","End":"07:11.570","Text":"this exponent here, 1.5e^x."},{"Start":"07:11.570 ","End":"07:17.540","Text":"This is like my f. What I want is to have f\u0027 here,"},{"Start":"07:17.540 ","End":"07:23.580","Text":"but f\u0027 is 1.5e^x"},{"Start":"07:23.580 ","End":"07:28.400","Text":"if this is f. That\u0027s why I\u0027ve put the 1.5 in here."},{"Start":"07:28.400 ","End":"07:32.900","Text":"Now, this is the derivative of this highlighted this"},{"Start":"07:32.900 ","End":"07:39.360","Text":"f\u0027 is the derivative of 1.5e^x is 1.5e^x."},{"Start":"07:40.450 ","End":"07:43.355","Text":"But because I put a 1.5 here,"},{"Start":"07:43.355 ","End":"07:46.040","Text":"I compensate by dividing by 1.5."},{"Start":"07:46.040 ","End":"07:47.285","Text":"And then all is well,"},{"Start":"07:47.285 ","End":"07:50.945","Text":"I also took the constant in front of the integral sign."},{"Start":"07:50.945 ","End":"07:53.195","Text":"Now I apply the formula."},{"Start":"07:53.195 ","End":"07:57.095","Text":"The integral is e^f,"},{"Start":"07:57.095 ","End":"08:01.115","Text":"besides the integral there was still a minus 1/1.5 here."},{"Start":"08:01.115 ","End":"08:04.160","Text":"Now the equation for the answer z is off screen."},{"Start":"08:04.160 ","End":"08:05.780","Text":"Let me just go and look at it."},{"Start":"08:05.780 ","End":"08:08.900","Text":"Basically it\u0027s e to the minus asterisk"},{"Start":"08:08.900 ","End":"08:12.785","Text":"times double asterisk plus C. I\u0027ll just make a note of that."},{"Start":"08:12.785 ","End":"08:16.970","Text":"Z equals e to the minus asterisk it\u0027s"},{"Start":"08:16.970 ","End":"08:21.320","Text":"just symbolic times double asterisk plus the constant."},{"Start":"08:21.320 ","End":"08:23.360","Text":"That\u0027s how it was."},{"Start":"08:23.360 ","End":"08:28.535","Text":"In our case here we have e to the minus the asterisk was a(x),"},{"Start":"08:28.535 ","End":"08:31.325","Text":"it\u0027s x plus 1.5e^x, that\u0027s here."},{"Start":"08:31.325 ","End":"08:34.580","Text":"The double asterisk integral is here,"},{"Start":"08:34.580 ","End":"08:37.370","Text":"and then plus the constant."},{"Start":"08:37.370 ","End":"08:40.175","Text":"I just need a bit more space."},{"Start":"08:40.175 ","End":"08:42.590","Text":"Then the general solution for y,"},{"Start":"08:42.590 ","End":"08:45.740","Text":"with the help of the general solution for z, is this,"},{"Start":"08:45.740 ","End":"08:50.405","Text":"that y is y_1 plus 1/z, not precisely,"},{"Start":"08:50.405 ","End":"08:54.190","Text":"or y equals y_1,"},{"Start":"08:54.190 ","End":"08:56.884","Text":"because our particular solution is also a solution."},{"Start":"08:56.884 ","End":"08:59.215","Text":"I can\u0027t get 1/z to be 0,"},{"Start":"08:59.215 ","End":"09:01.700","Text":"so I write this in separately."},{"Start":"09:01.700 ","End":"09:06.720","Text":"Y_1(x) was minus 0.5e^x,"},{"Start":"09:06.720 ","End":"09:10.500","Text":"1 over just copied z(x),"},{"Start":"09:10.500 ","End":"09:17.180","Text":"or y(x) is just the original minus 0.5e^x."},{"Start":"09:17.180 ","End":"09:19.310","Text":"A bit of tidying up,"},{"Start":"09:19.310 ","End":"09:21.650","Text":"we simplify, we get this."},{"Start":"09:21.650 ","End":"09:23.760","Text":"I\u0027ll just say a few words."},{"Start":"09:23.760 ","End":"09:26.360","Text":"I break this up into a product."},{"Start":"09:26.360 ","End":"09:32.060","Text":"Get e^-x, e^1.5, e^x."},{"Start":"09:32.060 ","End":"09:36.910","Text":"e^ minus x goes to the top of e^x and e^ minus1.5x"},{"Start":"09:36.910 ","End":"09:42.540","Text":"cancels with this and also 1/1.5 is 2/3."},{"Start":"09:42.540 ","End":"09:44.940","Text":"1 divided by 1 1/2 is 2/3,"},{"Start":"09:44.940 ","End":"09:53.115","Text":"and this e^ minus 1.5 goes now after the C. This is the answer."},{"Start":"09:53.115 ","End":"09:54.720","Text":"Y equals this,"},{"Start":"09:54.720 ","End":"10:02.810","Text":"or y(x) equals minus 0.5e^x."},{"Start":"10:02.810 ","End":"10:05.075","Text":"Like I said, I\u0027ve seen often,"},{"Start":"10:05.075 ","End":"10:08.990","Text":"this particular solution is forgotten somehow."},{"Start":"10:08.990 ","End":"10:11.390","Text":"This is the first one we found."},{"Start":"10:11.390 ","End":"10:15.020","Text":"This is the really general solution where C can be"},{"Start":"10:15.020 ","End":"10:20.670","Text":"anything and we also have this particular solution. We\u0027re done."}],"Thumbnail":null,"ID":7709},{"Watched":false,"Name":"Exercise 2","Duration":"6m 43s","ChapterTopicVideoID":7637,"CourseChapterTopicPlaylistID":4225,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.590","Text":"In this exercise, we\u0027re given this differential equation."},{"Start":"00:04.590 ","End":"00:08.550","Text":"I claim that this is a Riccati equation,"},{"Start":"00:08.550 ","End":"00:11.715","Text":"double c and 1t."},{"Start":"00:11.715 ","End":"00:15.345","Text":"The essence of Riccati equations is this."},{"Start":"00:15.345 ","End":"00:18.060","Text":"This is the definition of a Riccati equation,"},{"Start":"00:18.060 ","End":"00:19.965","Text":"one that looks like this."},{"Start":"00:19.965 ","End":"00:24.090","Text":"What we do is we have to somehow, by guess,"},{"Start":"00:24.090 ","End":"00:26.880","Text":"or otherwise get a private solution,"},{"Start":"00:26.880 ","End":"00:29.070","Text":"a particular solution y1,"},{"Start":"00:29.070 ","End":"00:34.410","Text":"and then we make the substitution y equals y1 plus 1 over z,"},{"Start":"00:34.410 ","End":"00:36.720","Text":"where z is a new variable,"},{"Start":"00:36.720 ","End":"00:41.420","Text":"then we solve the following equation for z."},{"Start":"00:41.420 ","End":"00:43.925","Text":"It\u0027s a linear differential equation."},{"Start":"00:43.925 ","End":"00:46.760","Text":"When we found the general solution for this,"},{"Start":"00:46.760 ","End":"00:48.605","Text":"we substitute it back here."},{"Start":"00:48.605 ","End":"00:51.260","Text":"I get the general solution for y,"},{"Start":"00:51.260 ","End":"00:53.960","Text":"except that there is an additional solution,"},{"Start":"00:53.960 ","End":"01:00.125","Text":"which is the original solution y equals y1 also, it\u0027s often omitted."},{"Start":"01:00.125 ","End":"01:03.090","Text":"But of course, 1 over z can never be 0,"},{"Start":"01:03.090 ","End":"01:08.735","Text":"so this is the extra one that\u0027s not included in this."},{"Start":"01:08.735 ","End":"01:10.700","Text":"Let\u0027s get to it."},{"Start":"01:10.700 ","End":"01:13.250","Text":"Our original equation, I rewrote it here,"},{"Start":"01:13.250 ","End":"01:18.005","Text":"and we\u0027ve labeled this as q naught, q1 and q2."},{"Start":"01:18.005 ","End":"01:21.660","Text":"That way we see that it really is of this form,"},{"Start":"01:21.660 ","End":"01:24.105","Text":"so it is a Riccati equation."},{"Start":"01:24.105 ","End":"01:29.330","Text":"Now, the hard part is getting a particular solution y1."},{"Start":"01:29.330 ","End":"01:33.560","Text":"In most cases, it\u0027s given as part of the question,"},{"Start":"01:33.560 ","End":"01:39.575","Text":"but I\u0027ve made things a bit harder and haven\u0027t given you a private solution, a particular."},{"Start":"01:39.575 ","End":"01:40.940","Text":"In the tutorial clip,"},{"Start":"01:40.940 ","End":"01:44.765","Text":"I mentioned that you might want to try one of the following,"},{"Start":"01:44.765 ","End":"01:49.740","Text":"either ax to the power of b with some a and b,"},{"Start":"01:49.740 ","End":"01:55.490","Text":"or ae to the power of bx."},{"Start":"01:55.490 ","End":"01:58.505","Text":"Now, after thinking about this a bit,"},{"Start":"01:58.505 ","End":"02:03.125","Text":"I suggest that we try this form with b is 1."},{"Start":"02:03.125 ","End":"02:04.730","Text":"If y is a x,"},{"Start":"02:04.730 ","End":"02:05.960","Text":"it\u0027s a polynomial,"},{"Start":"02:05.960 ","End":"02:09.140","Text":"and y squared is also going to be a polynomial,"},{"Start":"02:09.140 ","End":"02:11.600","Text":"and it\u0027s going to be of degree 2."},{"Start":"02:11.600 ","End":"02:13.330","Text":"These are all going to be up to degree 2."},{"Start":"02:13.330 ","End":"02:15.195","Text":"That\u0027s why I\u0027m taking b as 1,"},{"Start":"02:15.195 ","End":"02:17.510","Text":"but a we still have to discover."},{"Start":"02:17.510 ","End":"02:21.898","Text":"We\u0027re going to try y=ax,"},{"Start":"02:21.898 ","End":"02:25.025","Text":"and the derivative y\u0027 equals a."},{"Start":"02:25.025 ","End":"02:27.170","Text":"Yes, there\u0027s some guesswork here."},{"Start":"02:27.170 ","End":"02:31.775","Text":"There is no exact way of getting a particular solution."},{"Start":"02:31.775 ","End":"02:32.900","Text":"We try these,"},{"Start":"02:32.900 ","End":"02:34.580","Text":"and it\u0027s a bit of trial and error."},{"Start":"02:34.580 ","End":"02:37.640","Text":"As I said, it\u0027s usually given in the question itself,"},{"Start":"02:37.640 ","End":"02:39.020","Text":"which is making it a bit tougher."},{"Start":"02:39.020 ","End":"02:44.695","Text":"Let\u0027s try this, substitute y and y\u0027 in this equation."},{"Start":"02:44.695 ","End":"02:46.845","Text":"Y\u0027 is a,"},{"Start":"02:46.845 ","End":"02:50.145","Text":"and wherever we see y we put ax,"},{"Start":"02:50.145 ","End":"02:52.510","Text":"so our z is this times x."},{"Start":"02:52.510 ","End":"02:54.215","Text":"After multiplying it out,"},{"Start":"02:54.215 ","End":"02:57.365","Text":"we get the following equation."},{"Start":"02:57.365 ","End":"02:59.645","Text":"Let me just get some more space here."},{"Start":"02:59.645 ","End":"03:02.660","Text":"Bringing everything to the right-hand side and collecting terms,"},{"Start":"03:02.660 ","End":"03:05.795","Text":"we see it\u0027s a quadratic in x,"},{"Start":"03:05.795 ","End":"03:08.960","Text":"the something plus something x plus something x squared."},{"Start":"03:08.960 ","End":"03:11.345","Text":"Now, in order for this to be 0,"},{"Start":"03:11.345 ","End":"03:13.190","Text":"we\u0027re talking about the 0 function,"},{"Start":"03:13.190 ","End":"03:14.660","Text":"not for a particular x,"},{"Start":"03:14.660 ","End":"03:16.250","Text":"then all of these have to be 0."},{"Start":"03:16.250 ","End":"03:17.630","Text":"We have to have 0 here,"},{"Start":"03:17.630 ","End":"03:19.145","Text":"we have to have 0 here,"},{"Start":"03:19.145 ","End":"03:20.975","Text":"and we have to have 0 here."},{"Start":"03:20.975 ","End":"03:25.700","Text":"Usually we won\u0027t find the solution 3 equations in 1 unknown."},{"Start":"03:25.700 ","End":"03:30.305","Text":"But here, clearly we have that a=minus 1."},{"Start":"03:30.305 ","End":"03:32.855","Text":"If you check, it will make everything 0."},{"Start":"03:32.855 ","End":"03:40.585","Text":"We found a, so now our particular solution is y= minus x."},{"Start":"03:40.585 ","End":"03:42.590","Text":"That\u0027s up, particular solution,"},{"Start":"03:42.590 ","End":"03:44.210","Text":"y1 minus x,"},{"Start":"03:44.210 ","End":"03:46.760","Text":"and now we make the substitution above,"},{"Start":"03:46.760 ","End":"03:49.070","Text":"and this is the substitution."},{"Start":"03:49.070 ","End":"03:52.865","Text":"We don\u0027t do anything with it just yet until we find z,"},{"Start":"03:52.865 ","End":"03:58.175","Text":"but the theory guarantees that z satisfies the equation."},{"Start":"03:58.175 ","End":"04:01.545","Text":"This one here, it involves q1 and q2."},{"Start":"04:01.545 ","End":"04:04.890","Text":"Q naught is not used here and y1."},{"Start":"04:04.890 ","End":"04:07.275","Text":"Let\u0027s substitute everything."},{"Start":"04:07.275 ","End":"04:10.950","Text":"You\u0027ll just look back and see what q1 and q2 are."},{"Start":"04:10.950 ","End":"04:13.545","Text":"I\u0027ll just copied q1 was this,"},{"Start":"04:13.545 ","End":"04:17.115","Text":"q2 was minus 1,"},{"Start":"04:17.115 ","End":"04:20.470","Text":"y1 we said is minus x,"},{"Start":"04:20.470 ","End":"04:22.910","Text":"and also the other side,"},{"Start":"04:22.910 ","End":"04:27.035","Text":"minus 1, that\u0027s minus q2."},{"Start":"04:27.035 ","End":"04:30.560","Text":"Q2 was substituted here and here."},{"Start":"04:30.560 ","End":"04:36.560","Text":"This is a linear differential equation in z. Simplify this,"},{"Start":"04:36.560 ","End":"04:41.550","Text":"notice that the 2x minus 2x, everything cancels."},{"Start":"04:41.550 ","End":"04:44.075","Text":"If you check it, we\u0027re just left with this."},{"Start":"04:44.075 ","End":"04:46.865","Text":"Let\u0027s call this a of x,"},{"Start":"04:46.865 ","End":"04:49.835","Text":"and this we\u0027ll call b of x."},{"Start":"04:49.835 ","End":"04:55.465","Text":"Then we have the formula for linear equations,"},{"Start":"04:55.465 ","End":"05:02.630","Text":"and here is the formula in general where A is the integral of a,"},{"Start":"05:02.630 ","End":"05:04.340","Text":"it doesn\u0027t say, but that\u0027s standard,"},{"Start":"05:04.340 ","End":"05:06.860","Text":"and we have 2 integrations to perform."},{"Start":"05:06.860 ","End":"05:10.070","Text":"We first of all have to find A by integrating a,"},{"Start":"05:10.070 ","End":"05:11.885","Text":"and then we have another integral here."},{"Start":"05:11.885 ","End":"05:14.470","Text":"Let\u0027s continue on the next page."},{"Start":"05:14.470 ","End":"05:17.075","Text":"Here we are. This was our equation."},{"Start":"05:17.075 ","End":"05:20.315","Text":"The first integral is defined A."},{"Start":"05:20.315 ","End":"05:22.475","Text":"The integral of a,"},{"Start":"05:22.475 ","End":"05:23.990","Text":"little a is minus 1,"},{"Start":"05:23.990 ","End":"05:26.570","Text":"so we just get minus x."},{"Start":"05:26.570 ","End":"05:28.250","Text":"The next integral."},{"Start":"05:28.250 ","End":"05:31.580","Text":"If you look back at the formula is this."},{"Start":"05:31.580 ","End":"05:34.870","Text":"Now we do have A of x,"},{"Start":"05:34.870 ","End":"05:37.010","Text":"b is still here, it\u0027s minus 1,"},{"Start":"05:37.010 ","End":"05:42.700","Text":"and here we have e to the power of this was minus x from here."},{"Start":"05:42.700 ","End":"05:44.923","Text":"This is minus e to the minus x,"},{"Start":"05:44.923 ","End":"05:47.330","Text":"and the integral is just e to the minus x."},{"Start":"05:47.330 ","End":"05:49.040","Text":"We don\u0027t add the constants."},{"Start":"05:49.040 ","End":"05:51.020","Text":"At this stage it\u0027s 1 at the end."},{"Start":"05:51.020 ","End":"05:53.540","Text":"Then I refer you back to the formula."},{"Start":"05:53.540 ","End":"05:57.365","Text":"It said e to the minus A here."},{"Start":"05:57.365 ","End":"06:00.065","Text":"It\u0027s minus, minus x, so it\u0027s x,"},{"Start":"06:00.065 ","End":"06:06.830","Text":"and inside we just had this integral plus c. That gives us the general form of z."},{"Start":"06:06.830 ","End":"06:08.270","Text":"But we\u0027re not looking for z,"},{"Start":"06:08.270 ","End":"06:10.445","Text":"we\u0027re looking for y. Yeah,"},{"Start":"06:10.445 ","End":"06:12.574","Text":"one more step was to simplify."},{"Start":"06:12.574 ","End":"06:14.750","Text":"Multiply this out and we get this."},{"Start":"06:14.750 ","End":"06:18.430","Text":"Now, y is equal to, as written here,"},{"Start":"06:18.430 ","End":"06:21.135","Text":"y1 plus 1 over z,"},{"Start":"06:21.135 ","End":"06:23.895","Text":"and y1, if you recall,"},{"Start":"06:23.895 ","End":"06:26.020","Text":"was minus x, z,"},{"Start":"06:26.020 ","End":"06:27.635","Text":"I take from here,"},{"Start":"06:27.635 ","End":"06:29.885","Text":"this is the general solution,"},{"Start":"06:29.885 ","End":"06:34.250","Text":"except that it doesn\u0027t cover the original particular solution which"},{"Start":"06:34.250 ","End":"06:38.720","Text":"is minus x. Y could be one of these,"},{"Start":"06:38.720 ","End":"06:40.775","Text":"so some value of c, or this."},{"Start":"06:40.775 ","End":"06:43.470","Text":"That\u0027s the general solution."}],"Thumbnail":null,"ID":7710},{"Watched":false,"Name":"Exercise 3","Duration":"5m 17s","ChapterTopicVideoID":7638,"CourseChapterTopicPlaylistID":4225,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"In this exercise, we\u0027re given this differential equation to solve."},{"Start":"00:03.750 ","End":"00:06.225","Text":"It doesn\u0027t look familiar at first,"},{"Start":"00:06.225 ","End":"00:08.850","Text":"but I claim that this is actually"},{"Start":"00:08.850 ","End":"00:14.280","Text":"a Riccati equation with just some little bit of algebra on this, and we\u0027ll see."},{"Start":"00:14.280 ","End":"00:18.315","Text":"I\u0027m going to summarize the information about Riccati equations."},{"Start":"00:18.315 ","End":"00:22.080","Text":"This is the general shape of a Riccati equation."},{"Start":"00:22.080 ","End":"00:27.540","Text":"The thing about Riccati equations is that we have to guess a particular solution,"},{"Start":"00:27.540 ","End":"00:29.010","Text":"y_1, or often,"},{"Start":"00:29.010 ","End":"00:33.675","Text":"in fact mostly it\u0027s given in the question but not here."},{"Start":"00:33.675 ","End":"00:38.555","Text":"We have to guess y_1 or otherwise get to a particular solution."},{"Start":"00:38.555 ","End":"00:44.975","Text":"Then we make a substitution that the general y is y_1 plus 1/z."},{"Start":"00:44.975 ","End":"00:51.950","Text":"Then theory shows that z satisfies the following linear differential equation."},{"Start":"00:51.950 ","End":"00:53.760","Text":"We solve for z,"},{"Start":"00:53.760 ","End":"00:56.415","Text":"and it will contain some constant C,"},{"Start":"00:56.415 ","End":"01:00.595","Text":"and then we put it back here and we get a general solution for y."},{"Start":"01:00.595 ","End":"01:04.700","Text":"Except that there\u0027s another solution which is not covered by this,"},{"Start":"01:04.700 ","End":"01:09.335","Text":"and that\u0027s the particular solution because 1/z can\u0027t be 0,"},{"Start":"01:09.335 ","End":"01:13.565","Text":"so we have to add the original guess to this."},{"Start":"01:13.565 ","End":"01:16.280","Text":"If we just open up the brackets here,"},{"Start":"01:16.280 ","End":"01:21.680","Text":"this is what we get and we see that this is indeed a Riccati equation,"},{"Start":"01:21.680 ","End":"01:22.700","Text":"I\u0027ve even labeled it."},{"Start":"01:22.700 ","End":"01:24.350","Text":"The q_0 is this bit,"},{"Start":"01:24.350 ","End":"01:28.190","Text":"q_1 is this bit, q_2 is this."},{"Start":"01:28.190 ","End":"01:32.990","Text":"The next thing is we need a private or particular solution."},{"Start":"01:32.990 ","End":"01:40.010","Text":"There was a hint in the tutorial theoretical clip that you could try y_1 of the form"},{"Start":"01:40.010 ","End":"01:47.600","Text":"ae^bx, or ax^b."},{"Start":"01:47.600 ","End":"01:52.085","Text":"Generally, when it\u0027s a polynomial low order with the coefficients,"},{"Start":"01:52.085 ","End":"01:55.120","Text":"you\u0027ll try this with b=1,"},{"Start":"01:55.120 ","End":"01:59.410","Text":"I\u0027m going to help you and tell you try y=ax."},{"Start":"02:00.170 ","End":"02:03.215","Text":"If there is a solution of this form,"},{"Start":"02:03.215 ","End":"02:04.730","Text":"y\u0027 is a,"},{"Start":"02:04.730 ","End":"02:09.040","Text":"and now I\u0027ll substitute y and y\u0027 in this equation."},{"Start":"02:09.040 ","End":"02:11.025","Text":"Let\u0027s see what we get,"},{"Start":"02:11.025 ","End":"02:13.530","Text":"just make the substitution that y\u0027 is a."},{"Start":"02:13.530 ","End":"02:18.350","Text":"Whenever you see y we put a and we get this."},{"Start":"02:18.350 ","End":"02:21.100","Text":"Next, collect terms with the powers of x,"},{"Start":"02:21.100 ","End":"02:25.460","Text":"there\u0027s a constant term but there\u0027s no x term and there\u0027s an x^2 term."},{"Start":"02:25.460 ","End":"02:28.220","Text":"In any event, this is the 0 polynomial,"},{"Start":"02:28.220 ","End":"02:33.620","Text":"this has to be 0 and this has to be 0."},{"Start":"02:33.620 ","End":"02:39.470","Text":"In general not be a solution because it\u0027s two equations in one unknown a,"},{"Start":"02:39.470 ","End":"02:41.510","Text":"but in this case there is,"},{"Start":"02:41.510 ","End":"02:44.500","Text":"and the solution is clearly a=1."},{"Start":"02:44.500 ","End":"02:46.080","Text":"For this it has to be 1,"},{"Start":"02:46.080 ","End":"02:49.650","Text":"and if you check that 1 minus 2 plus 1 is also 0."},{"Start":"02:49.650 ","End":"02:56.645","Text":"If a is 1, then our particular solution is y_1 equals just x."},{"Start":"02:56.645 ","End":"02:58.760","Text":"Now we have a particular solution,"},{"Start":"02:58.760 ","End":"03:05.345","Text":"remember that the next step is to look for solutions of the form y_1 plus 1/z,"},{"Start":"03:05.345 ","End":"03:11.990","Text":"where y_1 is this and z is an unknown function of x."},{"Start":"03:11.990 ","End":"03:17.405","Text":"But we know from the theory that it satisfies a linear differential equation,"},{"Start":"03:17.405 ","End":"03:20.285","Text":"and this is it which I copied from above."},{"Start":"03:20.285 ","End":"03:24.055","Text":"Notice that we have all the quantities we need q_1,"},{"Start":"03:24.055 ","End":"03:26.699","Text":"and we have q_1 here."},{"Start":"03:26.699 ","End":"03:31.500","Text":"We need q_2 twice here and here,"},{"Start":"03:31.500 ","End":"03:33.120","Text":"and we have q_2 here,"},{"Start":"03:33.120 ","End":"03:34.755","Text":"which is just 1."},{"Start":"03:34.755 ","End":"03:38.205","Text":"We also have y_1,"},{"Start":"03:38.205 ","End":"03:41.465","Text":"here it is, y_1 is just x."},{"Start":"03:41.465 ","End":"03:43.565","Text":"If I plug all that in,"},{"Start":"03:43.565 ","End":"03:45.590","Text":"get some more space,"},{"Start":"03:45.590 ","End":"03:48.365","Text":"then you can check q_1,"},{"Start":"03:48.365 ","End":"03:50.630","Text":"q_2 is 1,"},{"Start":"03:50.630 ","End":"03:54.825","Text":"q_2 here, and we know that y_1 is x."},{"Start":"03:54.825 ","End":"03:56.190","Text":"This is what we get,"},{"Start":"03:56.190 ","End":"04:00.300","Text":"and after simplifying all we get is this,"},{"Start":"04:00.300 ","End":"04:02.920","Text":"see the 2x cancels with the 2x."},{"Start":"04:03.230 ","End":"04:06.890","Text":"Normally, we treat this as a linear equation,"},{"Start":"04:06.890 ","End":"04:10.370","Text":"and technically it is a linear differential equation,"},{"Start":"04:10.370 ","End":"04:12.425","Text":"but because of the 0 here,"},{"Start":"04:12.425 ","End":"04:15.855","Text":"it\u0027s like all we have is z\u0027 is minus 1,"},{"Start":"04:15.855 ","End":"04:17.990","Text":"it doesn\u0027t make any sense to solve it using"},{"Start":"04:17.990 ","End":"04:23.675","Text":"that heavy formula or the mechanism for linear when we have a special case here."},{"Start":"04:23.675 ","End":"04:26.210","Text":"If z\u0027 is minus 1,"},{"Start":"04:26.210 ","End":"04:31.955","Text":"then the straightforward integration gives us that z is minus x plus the C,"},{"Start":"04:31.955 ","End":"04:34.480","Text":"and now that we have z,"},{"Start":"04:34.480 ","End":"04:36.390","Text":"we know that y,"},{"Start":"04:36.390 ","End":"04:37.920","Text":"if you look above,"},{"Start":"04:37.920 ","End":"04:41.250","Text":"y was y_1 plus 1/z,"},{"Start":"04:41.250 ","End":"04:44.610","Text":"and in our case, y_1 was x,"},{"Start":"04:44.610 ","End":"04:50.520","Text":"that gives us that y is x plus 1 over what\u0027s written here."},{"Start":"04:50.520 ","End":"04:54.110","Text":"That\u0027s usually given as a general solution,"},{"Start":"04:54.110 ","End":"04:56.300","Text":"but in most books,"},{"Start":"04:56.300 ","End":"05:00.305","Text":"they forget to also include the original guess,"},{"Start":"05:00.305 ","End":"05:07.255","Text":"which was that y(x)=x to be consistent,"},{"Start":"05:07.255 ","End":"05:09.440","Text":"is also a solution."},{"Start":"05:09.440 ","End":"05:11.750","Text":"We\u0027ve got infinity plus 1 solutions,"},{"Start":"05:11.750 ","End":"05:17.329","Text":"any value of z here plus this solution. We\u0027re done."}],"Thumbnail":null,"ID":7711},{"Watched":false,"Name":"Exercise 4","Duration":"13m 33s","ChapterTopicVideoID":7635,"CourseChapterTopicPlaylistID":4225,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.290","Text":"In this exercise, we have the following differential equation"},{"Start":"00:04.290 ","End":"00:09.945","Text":"and I claim that this is a Riccati equation."},{"Start":"00:09.945 ","End":"00:12.135","Text":"I\u0027m going to summarize what that is."},{"Start":"00:12.135 ","End":"00:16.440","Text":"If you remember that Riccati equation is where the derivative"},{"Start":"00:16.440 ","End":"00:21.600","Text":"is some function of x and then something times y,"},{"Start":"00:21.600 ","End":"00:22.920","Text":"and then something times y^2."},{"Start":"00:22.920 ","End":"00:26.579","Text":"All these are functions of x and when we have such an equation,"},{"Start":"00:26.579 ","End":"00:29.295","Text":"I\u0027m just looking quickly, we can see that this is of that form."},{"Start":"00:29.295 ","End":"00:35.250","Text":"Here\u0027s something y plus something y^2 something. So functions of x."},{"Start":"00:35.250 ","End":"00:42.136","Text":"The way we do this is we have to somehow be given or guess a particular solution y1,"},{"Start":"00:42.136 ","End":"00:47.960","Text":"and then we look for a general solution of the form y equals y1 plus 1 over z,"},{"Start":"00:47.960 ","End":"00:51.710","Text":"where z is a new variable and according to this theory,"},{"Start":"00:51.710 ","End":"00:56.945","Text":"z has to satisfy this equation where the q\u0027s are taken from here."},{"Start":"00:56.945 ","End":"01:04.625","Text":"At the end, when we found what z is in terms of some constant general solution,"},{"Start":"01:04.625 ","End":"01:06.373","Text":"then we plug it back in here,"},{"Start":"01:06.373 ","End":"01:11.000","Text":"and we get a general solution for y and is also the extra solution,"},{"Start":"01:11.000 ","End":"01:14.810","Text":"which is the one that we guessed, y equals y1."},{"Start":"01:14.810 ","End":"01:17.660","Text":"Actually usually it\u0027s given as part of the question."},{"Start":"01:17.660 ","End":"01:21.560","Text":"I\u0027ll meet you halfway and say that there\u0027s"},{"Start":"01:21.560 ","End":"01:25.520","Text":"going to be a particular solution of the form y equals ax."},{"Start":"01:25.520 ","End":"01:28.985","Text":"In the tutorial, I mentioned that if you\u0027re not given,"},{"Start":"01:28.985 ","End":"01:33.950","Text":"you want to try (y=ax)^b,"},{"Start":"01:33.950 ","End":"01:39.730","Text":"often works, or (ae)^ bx."},{"Start":"01:39.730 ","End":"01:42.440","Text":"In this case, we\u0027re using this form with b as 1."},{"Start":"01:42.440 ","End":"01:46.452","Text":"It\u0027s been lucky for us in the past and the previous exercises,"},{"Start":"01:46.452 ","End":"01:48.185","Text":"so we\u0027ll try this again."},{"Start":"01:48.185 ","End":"01:50.210","Text":"Let\u0027s get back to the equation."},{"Start":"01:50.210 ","End":"01:52.370","Text":"First of all, it is a Riccati equation."},{"Start":"01:52.370 ","End":"01:55.100","Text":"I mumbled something about it before, but let\u0027s see."},{"Start":"01:55.100 ","End":"01:59.735","Text":"Precisely if we let this be q naught of x,"},{"Start":"01:59.735 ","End":"02:04.565","Text":"and this bit here including the minus is q1(x),"},{"Start":"02:04.565 ","End":"02:08.105","Text":"and the 2 cos(x) I bring in front is the q2,"},{"Start":"02:08.105 ","End":"02:11.720","Text":"then we see that it fits the definition of"},{"Start":"02:11.720 ","End":"02:16.025","Text":"a Riccati equation with this q naught, q1, q2."},{"Start":"02:16.025 ","End":"02:17.975","Text":"As I said, we\u0027re going to guess,"},{"Start":"02:17.975 ","End":"02:21.725","Text":"try to find a particular solution of this form."},{"Start":"02:21.725 ","End":"02:24.560","Text":"We just have to find the a. y is ax,"},{"Start":"02:24.560 ","End":"02:25.730","Text":"so y\u0027 is a,"},{"Start":"02:25.730 ","End":"02:26.780","Text":"so we can substitute."},{"Start":"02:26.780 ","End":"02:29.915","Text":"We\u0027ve got y\u0027 and we\u0027ve got y,"},{"Start":"02:29.915 ","End":"02:31.250","Text":"and we\u0027ve got y^2,"},{"Start":"02:31.250 ","End":"02:32.450","Text":"which is the square of this."},{"Start":"02:32.450 ","End":"02:34.445","Text":"If we substitute all that,"},{"Start":"02:34.445 ","End":"02:37.030","Text":"then we get all this,"},{"Start":"02:37.030 ","End":"02:39.050","Text":"and I spared you all the details,"},{"Start":"02:39.050 ","End":"02:40.775","Text":"you can check it if you want."},{"Start":"02:40.775 ","End":"02:45.215","Text":"I can write this in a more convenient form as follows."},{"Start":"02:45.215 ","End":"02:46.640","Text":"I bring the a over,"},{"Start":"02:46.640 ","End":"02:48.620","Text":"that gives me 1 minus a."},{"Start":"02:48.620 ","End":"02:52.550","Text":"Here I have the x with the minus ax is here."},{"Start":"02:52.550 ","End":"02:59.755","Text":"Whoops, there\u0027s an x^2 here."},{"Start":"02:59.755 ","End":"03:03.000","Text":"All the x^2 cos(x) I collect."},{"Start":"03:03.000 ","End":"03:04.530","Text":"So from here,"},{"Start":"03:04.530 ","End":"03:05.655","Text":"I have 2,"},{"Start":"03:05.655 ","End":"03:07.920","Text":"from here I have minus 4a,"},{"Start":"03:07.920 ","End":"03:09.945","Text":"and from here I have (2a)^2."},{"Start":"03:09.945 ","End":"03:12.550","Text":"Sorry, this is an x^2 there."},{"Start":"03:12.550 ","End":"03:15.234","Text":"Now, this function here has to be the 0 function."},{"Start":"03:15.234 ","End":"03:16.923","Text":"So this should be 0, and this should be zero,"},{"Start":"03:16.923 ","End":"03:18.257","Text":"and this should be zero,"},{"Start":"03:18.257 ","End":"03:26.622","Text":"and there is a value of a which makes all this 0 and that is a equals 1."},{"Start":"03:26.622 ","End":"03:29.258","Text":"You can check."},{"Start":"03:29.258 ","End":"03:30.350","Text":"Here and here is clear."},{"Start":"03:30.350 ","End":"03:33.505","Text":"a is 1, get 2 minus 4 plus 2 is 0."},{"Start":"03:33.505 ","End":"03:34.820","Text":"So a is 1."},{"Start":"03:34.820 ","End":"03:37.490","Text":"That means that y is just x."},{"Start":"03:37.490 ","End":"03:39.695","Text":"So we have our particular solution,"},{"Start":"03:39.695 ","End":"03:42.785","Text":"y1 is equal to x."},{"Start":"03:42.785 ","End":"03:45.530","Text":"Now we want to find the general solution."},{"Start":"03:45.530 ","End":"03:47.300","Text":"So if you remember,"},{"Start":"03:47.300 ","End":"03:49.190","Text":"I\u0027ll just get some space here,"},{"Start":"03:49.190 ","End":"03:52.413","Text":"we make the following substitution,"},{"Start":"03:52.413 ","End":"03:58.230","Text":"and the theory states that z(x) will satisfy the following,"},{"Start":"03:58.230 ","End":"04:01.425","Text":"and this is a linear equation."},{"Start":"04:01.425 ","End":"04:05.400","Text":"In z we have q1 and q2 above."},{"Start":"04:05.400 ","End":"04:07.430","Text":"We need q2 twice here and here."},{"Start":"04:07.430 ","End":"04:11.390","Text":"We substitute those and we have y1 over here."},{"Start":"04:11.390 ","End":"04:13.925","Text":"So altogether after the substitution,"},{"Start":"04:13.925 ","End":"04:15.410","Text":"this is what we get."},{"Start":"04:15.410 ","End":"04:17.330","Text":"The q1 is this bit,"},{"Start":"04:17.330 ","End":"04:18.620","Text":"the q2 is here,"},{"Start":"04:18.620 ","End":"04:22.480","Text":"the y1 is here."},{"Start":"04:22.480 ","End":"04:26.710","Text":"Again, the q2 is here, 2cos(x),"},{"Start":"04:26.710 ","End":"04:33.429","Text":"and this is now a linear differential equation where after simplification,"},{"Start":"04:33.429 ","End":"04:38.420","Text":"you see we have here minus 4x cos(x),"},{"Start":"04:38.420 ","End":"04:41.635","Text":"and here 2 times 2 is 4x cos(x)."},{"Start":"04:41.635 ","End":"04:43.800","Text":"This cancels with this."},{"Start":"04:43.800 ","End":"04:46.605","Text":"All we\u0027re left with is this."},{"Start":"04:46.605 ","End":"04:53.540","Text":"That\u0027s linear with a and x being minus 1 and b(x) being minus 2 cos(x)."},{"Start":"04:53.540 ","End":"04:58.880","Text":"There is the standard formula for solving linear differential equations,"},{"Start":"04:58.880 ","End":"05:01.040","Text":"which is this, where if you remember,"},{"Start":"05:01.040 ","End":"05:02.990","Text":"we don\u0027t usually state it each time,"},{"Start":"05:02.990 ","End":"05:06.770","Text":"but that big A is the integral of little a."},{"Start":"05:06.770 ","End":"05:09.095","Text":"So we have two integrals to solve."},{"Start":"05:09.095 ","End":"05:10.880","Text":"One is to find big A,"},{"Start":"05:10.880 ","End":"05:12.380","Text":"and then we have this integral here."},{"Start":"05:12.380 ","End":"05:14.540","Text":"Let\u0027s continue on the next page."},{"Start":"05:14.540 ","End":"05:19.220","Text":"Here again is our differential equation for z."},{"Start":"05:19.220 ","End":"05:22.280","Text":"Here\u0027s the first of 2 integrals we have to solve."},{"Start":"05:22.280 ","End":"05:24.140","Text":"First of all, capital A,"},{"Start":"05:24.140 ","End":"05:25.520","Text":"which is the integral of little a,"},{"Start":"05:25.520 ","End":"05:27.905","Text":"which is minus 1 is minus x."},{"Start":"05:27.905 ","End":"05:31.535","Text":"The other integral is this in general,"},{"Start":"05:31.535 ","End":"05:35.180","Text":"now b is minus 2 cos x,"},{"Start":"05:35.180 ","End":"05:38.870","Text":"and capital A we found here is minus x."},{"Start":"05:38.870 ","End":"05:43.915","Text":"This is what we get and I\u0027ll just take the constant out of the integral."},{"Start":"05:43.915 ","End":"05:48.140","Text":"Now, this is definitely not a straightforward integral,"},{"Start":"05:48.140 ","End":"05:55.025","Text":"but it has been solved in calculus 1 and I\u0027ll do it again at the end."},{"Start":"05:55.025 ","End":"05:57.590","Text":"We\u0027ll continue with the exercise,"},{"Start":"05:57.590 ","End":"06:01.910","Text":"not break the flow and at the end they\u0027ll show you how we do this."},{"Start":"06:01.910 ","End":"06:04.085","Text":"I\u0027ll just quote the answer now,"},{"Start":"06:04.085 ","End":"06:06.650","Text":"this is the solution after the minus 2."},{"Start":"06:06.650 ","End":"06:08.600","Text":"This part is the solution."},{"Start":"06:08.600 ","End":"06:11.060","Text":"We\u0027ll get another set I\u0027ll show you at the end."},{"Start":"06:11.060 ","End":"06:14.420","Text":"The 2.5 cancel and instead of the minus,"},{"Start":"06:14.420 ","End":"06:16.745","Text":"I will reverse the order of the subtraction."},{"Start":"06:16.745 ","End":"06:19.760","Text":"So that\u0027s the second integral and we don\u0027t put"},{"Start":"06:19.760 ","End":"06:22.970","Text":"the constants in as a single constant at the end."},{"Start":"06:22.970 ","End":"06:30.980","Text":"I\u0027m going to remind you that z is given by this formula and we have both integrals."},{"Start":"06:30.980 ","End":"06:32.870","Text":"We have capital A here,"},{"Start":"06:32.870 ","End":"06:35.770","Text":"and we have this bit over here."},{"Start":"06:35.770 ","End":"06:41.015","Text":"We get z equals z to the minus minus x, which is just x."},{"Start":"06:41.015 ","End":"06:48.140","Text":"Then this bit from here plus the c and that\u0027s the general solution for z."},{"Start":"06:48.140 ","End":"06:54.680","Text":"But first I\u0027m going to just simplify this."},{"Start":"06:54.680 ","End":"06:59.130","Text":"This if I multiply out e to the x with e to the minus x, it just cancels."},{"Start":"06:59.130 ","End":"07:04.100","Text":"I\u0027ve got cos(x) minus sin(x) and Ce^x."},{"Start":"07:04.100 ","End":"07:05.590","Text":"As I was saying,"},{"Start":"07:05.590 ","End":"07:13.440","Text":"y is y1 plus 1 over z. y1 was x,"},{"Start":"07:13.440 ","End":"07:16.370","Text":"if you recall, and 1 over this,"},{"Start":"07:16.370 ","End":"07:20.060","Text":"and this is the general solution for y, not quite."},{"Start":"07:20.060 ","End":"07:24.140","Text":"There is also the possibility that y(x) is"},{"Start":"07:24.140 ","End":"07:29.010","Text":"the original particular solution that we found x for some reason."},{"Start":"07:29.010 ","End":"07:33.221","Text":"It\u0027s not covered by this because you can\u0027t have 1 over something to give you 0."},{"Start":"07:33.221 ","End":"07:37.340","Text":"That\u0027s like original one is somehow not included in the general,"},{"Start":"07:37.340 ","End":"07:39.623","Text":"so we just add it manually,"},{"Start":"07:39.623 ","End":"07:45.690","Text":"and this is the general solution for any value of C, this, or this."},{"Start":"07:45.690 ","End":"07:52.625","Text":"We\u0027re not done because I still owe you the difficult integral that we had before."},{"Start":"07:52.625 ","End":"07:55.240","Text":"So I\u0027ll do that on the following page."},{"Start":"07:55.240 ","End":"07:59.240","Text":"Here we are with what I owe you is to show you"},{"Start":"07:59.240 ","End":"08:04.745","Text":"the integral of this expression and I said it\u0027s not that easy."},{"Start":"08:04.745 ","End":"08:08.960","Text":"We actually do it using integration by parts twice."},{"Start":"08:08.960 ","End":"08:12.679","Text":"I\u0027d like to remind you of the formula for integration by parts."},{"Start":"08:12.679 ","End":"08:18.080","Text":"One way is to say that the integral of udv is equals"},{"Start":"08:18.080 ","End":"08:24.506","Text":"to uv minus the integral of vdu."},{"Start":"08:24.506 ","End":"08:29.540","Text":"There\u0027s a variant on this where we don\u0027t write dv,"},{"Start":"08:29.540 ","End":"08:32.630","Text":"but we write v prime dx."},{"Start":"08:32.630 ","End":"08:40.228","Text":"Otherwise, what we have is that the integral of uv prime dx for both u,"},{"Start":"08:40.228 ","End":"08:45.990","Text":"and v are functions of x is equals to uv minus the"},{"Start":"08:45.990 ","End":"08:52.060","Text":"integral of v times u\u0027dx."},{"Start":"08:52.060 ","End":"08:54.230","Text":"It\u0027s just a more expanded form of this,"},{"Start":"08:54.230 ","End":"08:57.010","Text":"and I\u0027ll use this variant."},{"Start":"08:57.010 ","End":"09:00.705","Text":"So what\u0027s u and what\u0027s v?"},{"Start":"09:00.705 ","End":"09:04.010","Text":"Hang on."},{"Start":"09:04.010 ","End":"09:06.380","Text":"There\u0027s yet another slight variation."},{"Start":"09:06.380 ","End":"09:10.880","Text":"Sometimes the reverse u and v. Well,"},{"Start":"09:10.880 ","End":"09:13.400","Text":"I could bring this to the other side and this to the other side,"},{"Start":"09:13.400 ","End":"09:20.330","Text":"and I could write it as the integral of u prime vdx equals"},{"Start":"09:20.330 ","End":"09:29.280","Text":"uv minus the integral of uv prime dx."},{"Start":"09:29.280 ","End":"09:35.780","Text":"It is just swapping u and v. I noticed that I wrote it according to this variant."},{"Start":"09:35.780 ","End":"09:43.790","Text":"So I\u0027ll let this be u prime and this be v. According to the formula,"},{"Start":"09:43.790 ","End":"09:47.315","Text":"it\u0027s equal to uv minus integral of uv prime."},{"Start":"09:47.315 ","End":"09:52.805","Text":"Now, if v is e^minus x, I\u0027ll expand a bit."},{"Start":"09:52.805 ","End":"09:56.480","Text":"If u prime is cos(x),"},{"Start":"09:56.480 ","End":"10:02.195","Text":"then that gives me that u is sin(x)."},{"Start":"10:02.195 ","End":"10:07.205","Text":"If v is e^minus x,"},{"Start":"10:07.205 ","End":"10:13.925","Text":"then that gives me the v prime is minus e^minus x."},{"Start":"10:13.925 ","End":"10:16.783","Text":"In one case I\u0027m integrating,"},{"Start":"10:16.783 ","End":"10:18.740","Text":"and in one case I\u0027m differentiating."},{"Start":"10:18.740 ","End":"10:20.840","Text":"In the event we get the missing 2,"},{"Start":"10:20.840 ","End":"10:22.760","Text":"there\u0027s four possibilities, u, u\u0027,"},{"Start":"10:22.760 ","End":"10:27.050","Text":"v, v\u0027, we start with 2 and we take the other 2 and then we just plug it in."},{"Start":"10:27.050 ","End":"10:28.747","Text":"Here\u0027s u, here\u0027s v,"},{"Start":"10:28.747 ","End":"10:30.305","Text":"here\u0027s again u,"},{"Start":"10:30.305 ","End":"10:32.060","Text":"and here\u0027s v prime."},{"Start":"10:32.060 ","End":"10:36.210","Text":"Notice that this second integral here,"},{"Start":"10:36.210 ","End":"10:42.635","Text":"the minus with the minus cancels and I have the integral of sin(x)e to the minus x,"},{"Start":"10:42.635 ","End":"10:45.046","Text":"which is very similar to this,"},{"Start":"10:45.046 ","End":"10:48.170","Text":"and I\u0027m going to do this on the next line."},{"Start":"10:48.170 ","End":"10:51.755","Text":"If I started out with sine(x)e to the minus x,"},{"Start":"10:51.755 ","End":"10:53.587","Text":"I won\u0027t go into all the details."},{"Start":"10:53.587 ","End":"10:59.120","Text":"Again, here we see what is u prime and v and here\u0027s u and here\u0027s v prime,"},{"Start":"10:59.120 ","End":"11:00.950","Text":"and we get something similar."},{"Start":"11:00.950 ","End":"11:03.786","Text":"Now it looks like we\u0027re going around in circles"},{"Start":"11:03.786 ","End":"11:08.090","Text":"because if we take the integral of cos(x)e to the x,"},{"Start":"11:08.090 ","End":"11:11.555","Text":"we need the integral of sin(x)e to the minus x."},{"Start":"11:11.555 ","End":"11:14.210","Text":"If we want sine(x)e to the minus x,"},{"Start":"11:14.210 ","End":"11:21.165","Text":"eventually we get back to cos(x)e to the minus x plus 1 minus."},{"Start":"11:21.165 ","End":"11:24.068","Text":"So how is it that we\u0027re not going around in circles?"},{"Start":"11:24.068 ","End":"11:27.305","Text":"Well, there is a nuance here."},{"Start":"11:27.305 ","End":"11:30.595","Text":"Notice that here it\u0027s a minus, but here,"},{"Start":"11:30.595 ","End":"11:36.358","Text":"this is a plus if we multiply out and here we have 3 minuses to minus,"},{"Start":"11:36.358 ","End":"11:37.925","Text":"and this is a crucial difference."},{"Start":"11:37.925 ","End":"11:40.355","Text":"It enables us to do some algebra."},{"Start":"11:40.355 ","End":"11:45.530","Text":"Let me first of all summarize what we have so far without these multiple minuses."},{"Start":"11:45.530 ","End":"11:47.810","Text":"The first one we have,"},{"Start":"11:47.810 ","End":"11:52.865","Text":"I\u0027m just copying it but here the minus with the minus gives me a plus."},{"Start":"11:52.865 ","End":"11:57.080","Text":"Like, we have a minus in front for one thing,"},{"Start":"11:57.080 ","End":"12:01.070","Text":"but this integral, we have minus minus minus,"},{"Start":"12:01.070 ","End":"12:03.170","Text":"so it\u0027s just a single minus."},{"Start":"12:03.170 ","End":"12:07.010","Text":"Now, what we can do is we haven\u0027t gotten anywhere as"},{"Start":"12:07.010 ","End":"12:11.330","Text":"far as the integral of cos(x)e to the minus x or sin(x)e to the minus x,"},{"Start":"12:11.330 ","End":"12:13.325","Text":"but I do have them twice."},{"Start":"12:13.325 ","End":"12:18.185","Text":"It\u0027s as if I had two equations and two unknowns for each of these two integrals."},{"Start":"12:18.185 ","End":"12:19.790","Text":"For example, I could add or subtract"},{"Start":"12:19.790 ","End":"12:23.600","Text":"these two equations or I could just use substitution."},{"Start":"12:23.600 ","End":"12:27.020","Text":"So for example, if I take the top equation and"},{"Start":"12:27.020 ","End":"12:31.489","Text":"replace the sine that\u0027s here with what\u0027s here,"},{"Start":"12:31.489 ","End":"12:33.050","Text":"that\u0027s what\u0027s in brackets,"},{"Start":"12:33.050 ","End":"12:36.785","Text":"it\u0027s this, and then if I open up the brackets,"},{"Start":"12:36.785 ","End":"12:42.680","Text":"what we get is because we had twice cos(x)e to the minus x but here it\u0027s with a minus."},{"Start":"12:42.680 ","End":"12:45.140","Text":"On the other side if I bring it to this side,"},{"Start":"12:45.140 ","End":"12:49.220","Text":"I get twice this integral and on the right-hand side,"},{"Start":"12:49.220 ","End":"12:50.975","Text":"I\u0027ve got this and this,"},{"Start":"12:50.975 ","End":"12:56.570","Text":"which is just e to the minus x outside the brackets, sin(x) minus cos(x)."},{"Start":"12:56.570 ","End":"13:02.060","Text":"Now all I have to do is divide by 2 and that gives me the integral of cos(x)e"},{"Start":"13:02.060 ","End":"13:07.564","Text":"to the minus x is 1.5 of what\u0027s written here and that\u0027s that integral."},{"Start":"13:07.564 ","End":"13:12.185","Text":"By the way, we could just as easily have done the substitution the other way,"},{"Start":"13:12.185 ","End":"13:17.525","Text":"and substituted this here if we wanted the integral of sin(x)e to the minus x,"},{"Start":"13:17.525 ","End":"13:19.955","Text":"but that\u0027s not what we want in this case."},{"Start":"13:19.955 ","End":"13:21.920","Text":"Then we\u0027d get something similar."},{"Start":"13:21.920 ","End":"13:24.155","Text":"I think we\u0027d get it with a plus here or something."},{"Start":"13:24.155 ","End":"13:25.520","Text":"Anyway, it\u0027s not what\u0027s needed."},{"Start":"13:25.520 ","End":"13:27.103","Text":"This is the result,"},{"Start":"13:27.103 ","End":"13:29.690","Text":"and this is what I used earlier."},{"Start":"13:29.690 ","End":"13:33.420","Text":"Now I\u0027ve settled that debt."}],"Thumbnail":null,"ID":7712}],"ID":4225},{"Name":"Existence and Uniqueness","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Existence and Uniqueness","Duration":"17m 37s","ChapterTopicVideoID":7644,"CourseChapterTopicPlaylistID":4226,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.980","Text":"In this clip, we\u0027ll be talking about an existence and uniqueness theorem."},{"Start":"00:04.980 ","End":"00:06.680","Text":"These are very common in mathematics,"},{"Start":"00:06.680 ","End":"00:10.995","Text":"existence and uniqueness for first-order ordinary differential equations."},{"Start":"00:10.995 ","End":"00:17.309","Text":"Actually, initial value problems differential equations with an initial condition."},{"Start":"00:17.309 ","End":"00:21.915","Text":"I\u0027ll start by considering 3 different initial value problems."},{"Start":"00:21.915 ","End":"00:25.590","Text":"Here\u0027s the first, y\u0027 equals y over x,"},{"Start":"00:25.590 ","End":"00:30.375","Text":"that\u0027s the equation and the initial condition, y(1) equals 2."},{"Start":"00:30.375 ","End":"00:35.460","Text":"Now the second where you notice it\u0027s the same equation part,"},{"Start":"00:35.460 ","End":"00:38.385","Text":"but a different initial condition,"},{"Start":"00:38.385 ","End":"00:43.865","Text":"this time y(0) equals 0 and the third example,"},{"Start":"00:43.865 ","End":"00:50.000","Text":"yet again the same equation and with yet another initial condition."},{"Start":"00:50.000 ","End":"00:51.440","Text":"Now in each of these cases,"},{"Start":"00:51.440 ","End":"00:54.200","Text":"I\u0027m looking for y as a function of x,"},{"Start":"00:54.200 ","End":"00:59.660","Text":"say y equals y(x) but the difference is in the first problem,"},{"Start":"00:59.660 ","End":"01:02.450","Text":"I wanted to pass through the point 1,"},{"Start":"01:02.450 ","End":"01:05.885","Text":"2 when x is 1, y is 2."},{"Start":"01:05.885 ","End":"01:12.265","Text":"The second problem, I want this function to pass through the point 0, 0."},{"Start":"01:12.265 ","End":"01:14.180","Text":"In the third example,"},{"Start":"01:14.180 ","End":"01:19.190","Text":"I want this function to pass through the point 0,1."},{"Start":"01:19.190 ","End":"01:20.810","Text":"It will look similar,"},{"Start":"01:20.810 ","End":"01:24.095","Text":"but there is an essential difference between all 3 of them."},{"Start":"01:24.095 ","End":"01:28.730","Text":"Now it\u0027s easy to compute the general solution for this part, which is the same here,"},{"Start":"01:28.730 ","End":"01:37.400","Text":"here and here and the solution is y=cx for some any constant c, y=cx,"},{"Start":"01:37.400 ","End":"01:42.140","Text":"then y\u0027 is just equal to c. On the other hand,"},{"Start":"01:42.140 ","End":"01:46.565","Text":"y over x is cx over x,"},{"Start":"01:46.565 ","End":"01:52.100","Text":"which is also equal to c so we see that these both sides of the equation are equal,"},{"Start":"01:52.100 ","End":"01:54.340","Text":"so this satisfies it."},{"Start":"01:54.340 ","End":"01:59.975","Text":"Now let\u0027s see what we get if we apply the initial condition."},{"Start":"01:59.975 ","End":"02:01.925","Text":"In the first case,"},{"Start":"02:01.925 ","End":"02:05.500","Text":"we get the result y=2x and let me explain,"},{"Start":"02:05.500 ","End":"02:06.740","Text":"I want to highlight this."},{"Start":"02:06.740 ","End":"02:08.135","Text":"This is the important thing."},{"Start":"02:08.135 ","End":"02:10.100","Text":"Now, each of the 3 cases,"},{"Start":"02:10.100 ","End":"02:12.350","Text":"we\u0027re going to have something y=cx."},{"Start":"02:12.350 ","End":"02:14.120","Text":"If it satisfies this,"},{"Start":"02:14.120 ","End":"02:15.310","Text":"I can put x=1,"},{"Start":"02:15.310 ","End":"02:18.740","Text":"y=2 and so I would get from here,"},{"Start":"02:18.740 ","End":"02:26.340","Text":"2=c times 1 and so c=2."},{"Start":"02:26.340 ","End":"02:28.620","Text":"If I put c=2 here,"},{"Start":"02:28.620 ","End":"02:29.955","Text":"then this is what I get."},{"Start":"02:29.955 ","End":"02:31.400","Text":"This is the solution,"},{"Start":"02:31.400 ","End":"02:37.295","Text":"and this is the only solution to this initial value problem, y =2x."},{"Start":"02:37.295 ","End":"02:42.380","Text":"Let\u0027s consider the second example because still see the initial condition here."},{"Start":"02:42.380 ","End":"02:47.285","Text":"In this case. The second case we have that y(0) is 0."},{"Start":"02:47.285 ","End":"02:50.255","Text":"If I put y=0, x=0 here,"},{"Start":"02:50.255 ","End":"02:57.170","Text":"I\u0027ll get 0=c times 0 and this is true for any c. For example,"},{"Start":"02:57.170 ","End":"03:00.470","Text":"if I let c=1, I get y=x."},{"Start":"03:00.470 ","End":"03:02.075","Text":"If I put c=2,"},{"Start":"03:02.075 ","End":"03:05.435","Text":"y=2x, just happens to be the same as here."},{"Start":"03:05.435 ","End":"03:10.565","Text":"Then any cx will do so we get an infinite number of solutions."},{"Start":"03:10.565 ","End":"03:18.225","Text":"Each of these satisfies this equation and when x is 0, y is 0."},{"Start":"03:18.225 ","End":"03:20.570","Text":"So far we\u0027ve had a unique solution,"},{"Start":"03:20.570 ","End":"03:22.805","Text":"an infinite number of solutions."},{"Start":"03:22.805 ","End":"03:24.845","Text":"Let\u0027s see the third case."},{"Start":"03:24.845 ","End":"03:27.160","Text":"The third case y(0) is 1."},{"Start":"03:27.160 ","End":"03:29.660","Text":"If I plug that into here,"},{"Start":"03:29.660 ","End":"03:32.060","Text":"means y=1 and c is 0,"},{"Start":"03:32.060 ","End":"03:35.570","Text":"I get 1=c times 0 and there is"},{"Start":"03:35.570 ","End":"03:40.790","Text":"no solution for c. There is no such value that we multiply by 0 gives 1."},{"Start":"03:40.790 ","End":"03:45.305","Text":"There are no solutions syndicated by the empty set notation."},{"Start":"03:45.305 ","End":"03:47.300","Text":"The solution set is empty."},{"Start":"03:47.300 ","End":"03:49.235","Text":"Here we have 1 solution,"},{"Start":"03:49.235 ","End":"03:53.005","Text":"infinite number of solutions, no solution."},{"Start":"03:53.005 ","End":"03:54.605","Text":"But that\u0027s not all."},{"Start":"03:54.605 ","End":"03:57.515","Text":"There are actually other possibilities."},{"Start":"03:57.515 ","End":"04:00.440","Text":"Let me first write down what we had so far, the situations."},{"Start":"04:00.440 ","End":"04:02.095","Text":"We\u0027ve had a unique solution,"},{"Start":"04:02.095 ","End":"04:05.735","Text":"about infinite number of solutions but no solution."},{"Start":"04:05.735 ","End":"04:10.340","Text":"But it\u0027s also possible that we have a finite number of solutions,"},{"Start":"04:10.340 ","End":"04:15.345","Text":"say 2 solutions and I\u0027ll give an example of that and let\u0027s see."},{"Start":"04:15.345 ","End":"04:17.825","Text":"If I look at the equation,"},{"Start":"04:17.825 ","End":"04:23.955","Text":"y\u0027= x over y and y(0) is 0,"},{"Start":"04:23.955 ","End":"04:26.675","Text":"this problem has just 2 solutions."},{"Start":"04:26.675 ","End":"04:29.660","Text":"Y= minus x or y= x."},{"Start":"04:29.660 ","End":"04:31.865","Text":"You can check that it satisfies."},{"Start":"04:31.865 ","End":"04:33.110","Text":"Take the second case,"},{"Start":"04:33.110 ","End":"04:38.775","Text":"y\u0027 =1 and x over y is also 1 because x=y and"},{"Start":"04:38.775 ","End":"04:45.490","Text":"similarly here y\u0027 is minus 1 and x over y is also minus 1."},{"Start":"04:45.490 ","End":"04:47.525","Text":"There are 2 solutions."},{"Start":"04:47.525 ","End":"04:49.610","Text":"I hadn\u0027t really intended to, but I can show you how we would"},{"Start":"04:49.610 ","End":"04:52.295","Text":"solve this with the separation of variables."},{"Start":"04:52.295 ","End":"04:58.670","Text":"We could say that dy over dx=x over y and"},{"Start":"04:58.670 ","End":"05:05.765","Text":"then we could cross-multiply and get y dy=x dx."},{"Start":"05:05.765 ","End":"05:10.970","Text":"Then we could put an integral in front of each and get 1.5 y^2"},{"Start":"05:10.970 ","End":"05:17.060","Text":"=1.5 x^2 plus c. Since y(0) is 0,"},{"Start":"05:17.060 ","End":"05:18.799","Text":"I could plug in x=0,"},{"Start":"05:18.799 ","End":"05:23.690","Text":"y=0 and then I would get that c=0."},{"Start":"05:23.690 ","End":"05:30.710","Text":"From c=0, we would get that y^2=x^2"},{"Start":"05:30.710 ","End":"05:38.345","Text":"if I just multiply it by 2 and so y=plus or minus x."},{"Start":"05:38.345 ","End":"05:41.575","Text":"This gives us these 2 solutions."},{"Start":"05:41.575 ","End":"05:45.770","Text":"We can also get 2 solutions,"},{"Start":"05:45.770 ","End":"05:50.060","Text":"but it could also be any number other than 2 finite number."},{"Start":"05:50.060 ","End":"05:52.820","Text":"All these examples that give rise to"},{"Start":"05:52.820 ","End":"05:57.680","Text":"2 main questions that we\u0027re concerned about and they\u0027ll continue on the next page."},{"Start":"05:57.680 ","End":"05:59.585","Text":"Before the questions, the setup,"},{"Start":"05:59.585 ","End":"06:03.460","Text":"we\u0027re given an initial value problem, y\u0027 equals,"},{"Start":"06:03.460 ","End":"06:09.395","Text":"some expression in x and y and we\u0027re given an initial condition that when x is x0,"},{"Start":"06:09.395 ","End":"06:13.085","Text":"y is y0, x naught, y naught."},{"Start":"06:13.085 ","End":"06:17.135","Text":"Then we ask about existence and uniqueness."},{"Start":"06:17.135 ","End":"06:22.490","Text":"The existence just asks under what conditions is the problem has a solution,"},{"Start":"06:22.490 ","End":"06:23.870","Text":"1 or more solutions."},{"Start":"06:23.870 ","End":"06:28.595","Text":"Sometimes it\u0027s important to know whether there\u0027s a solution or not and not just how many."},{"Start":"06:28.595 ","End":"06:31.400","Text":"But the second question is,"},{"Start":"06:31.400 ","End":"06:33.695","Text":"assuming that there is a solution,"},{"Start":"06:33.695 ","End":"06:37.370","Text":"under what conditions is it unique or the only one,"},{"Start":"06:37.370 ","End":"06:40.880","Text":"sometimes you want to know that that\u0027s the only solution."},{"Start":"06:40.880 ","End":"06:42.830","Text":"To answer these 2 questions,"},{"Start":"06:42.830 ","End":"06:45.289","Text":"we have the theorem."},{"Start":"06:45.289 ","End":"06:48.230","Text":"The theorem is called the existence and uniqueness theorem for"},{"Start":"06:48.230 ","End":"06:51.395","Text":"first-order ordinary differential equations."},{"Start":"06:51.395 ","End":"06:56.450","Text":"In the moment I\u0027ll show you what that theorem says, there are 2 parts."},{"Start":"06:56.450 ","End":"06:59.270","Text":"Part 1 relate to existence,"},{"Start":"06:59.270 ","End":"07:03.530","Text":"and it says, have a sketch after the theorem,"},{"Start":"07:03.530 ","End":"07:10.310","Text":"that if we have the function f(x,y) continuous in some open rectangle,"},{"Start":"07:10.310 ","End":"07:12.980","Text":"rectangle means, let\u0027s describe it as follows."},{"Start":"07:12.980 ","End":"07:17.390","Text":"That x is between 2 numbers and y is between 2 numbers."},{"Start":"07:17.390 ","End":"07:20.795","Text":"Then the combination, it\u0027s a rectangle without the boundaries."},{"Start":"07:20.795 ","End":"07:22.670","Text":"Supposing that our x naught,"},{"Start":"07:22.670 ","End":"07:27.500","Text":"y naught from the problem is inside this rectangular region,"},{"Start":"07:27.500 ","End":"07:30.290","Text":"then I put in italics,"},{"Start":"07:30.290 ","End":"07:32.390","Text":"if then this is the f part."},{"Start":"07:32.390 ","End":"07:33.800","Text":"If this holds true,"},{"Start":"07:33.800 ","End":"07:38.675","Text":"then the initial value problem has a solution."},{"Start":"07:38.675 ","End":"07:44.960","Text":"It does a solution in an open interval it doesn\u0027t mean for every x between a and b,"},{"Start":"07:44.960 ","End":"07:51.285","Text":"but some subinterval of ab because x naught has to be in this interval,"},{"Start":"07:51.285 ","End":"07:54.990","Text":"2-part and then give a sketch so Part 2"},{"Start":"07:54.990 ","End":"07:58.880","Text":"related to the uniqueness also has an if-clause and then-clause."},{"Start":"07:58.880 ","End":"08:01.475","Text":"If in addition, meaning all the above,"},{"Start":"08:01.475 ","End":"08:04.175","Text":"f is continuous and the rectangle,"},{"Start":"08:04.175 ","End":"08:07.460","Text":"and if in addition the partial derivative with respect"},{"Start":"08:07.460 ","End":"08:10.970","Text":"to y is continuous in the rectangle,"},{"Start":"08:10.970 ","End":"08:13.415","Text":"then we know that the solution is unique."},{"Start":"08:13.415 ","End":"08:17.360","Text":"It\u0027s also implies that f has a partial derivative with respect to y."},{"Start":"08:17.360 ","End":"08:20.525","Text":"It\u0027s at least differentiable with respect to y."},{"Start":"08:20.525 ","End":"08:23.390","Text":"Now, it is a bit abstract,"},{"Start":"08:23.390 ","End":"08:25.640","Text":"and here\u0027s a sketch,"},{"Start":"08:25.640 ","End":"08:28.370","Text":"see if I can fit everything in just about."},{"Start":"08:28.370 ","End":"08:30.185","Text":"Here\u0027s an illustration,"},{"Start":"08:30.185 ","End":"08:33.395","Text":"the first part talks about a rectangle."},{"Start":"08:33.395 ","End":"08:35.395","Text":"Here\u0027s the rectangle R."},{"Start":"08:35.395 ","End":"08:38.760","Text":"It doesn\u0027t actually include the edges,"},{"Start":"08:38.760 ","End":"08:44.255","Text":"but it\u0027s from a to b for x and from c to d for y,"},{"Start":"08:44.255 ","End":"08:47.520","Text":"and it has to contain the point x naught,"},{"Start":"08:47.520 ","End":"08:52.710","Text":"y naught that was in the statement of the theorem in the initial condition,"},{"Start":"08:52.710 ","End":"08:54.264","Text":"and this is the f from the theorem."},{"Start":"08:54.264 ","End":"08:58.520","Text":"It has to be continuous throughout this rectangle,"},{"Start":"08:58.520 ","End":"09:02.325","Text":"then the theorem says that there is a solution,"},{"Start":"09:02.325 ","End":"09:06.659","Text":"and this is what\u0027s drawn here in this pink magenta,"},{"Start":"09:06.659 ","End":"09:10.565","Text":"whatever, this solution y as a function of x,"},{"Start":"09:10.565 ","End":"09:14.670","Text":"but not necessarily from a to b,"},{"Start":"09:14.670 ","End":"09:21.670","Text":"it could just be x being only in a subinterval i part of a b but containing x naught,"},{"Start":"09:21.670 ","End":"09:26.175","Text":"of course, and we are guaranteed that there is at least a partial solution,"},{"Start":"09:26.175 ","End":"09:28.050","Text":"for part of the interval."},{"Start":"09:28.050 ","End":"09:31.245","Text":"Now, this still doesn\u0027t talk about uniqueness."},{"Start":"09:31.245 ","End":"09:34.125","Text":"For uniqueness, if we have the extra condition"},{"Start":"09:34.125 ","End":"09:38.045","Text":"that this partial derivative with respect to y is continuous,"},{"Start":"09:38.045 ","End":"09:40.710","Text":"then we\u0027re guaranteed that this is unique,"},{"Start":"09:40.710 ","End":"09:44.340","Text":"otherwise we might have several going through this same point."},{"Start":"09:44.340 ","End":"09:49.419","Text":"Now this is a major theorem and I\u0027d like to make some remarks on it."},{"Start":"09:49.419 ","End":"09:54.230","Text":"The first remark is that"},{"Start":"09:54.230 ","End":"10:00.330","Text":"these conditions are what is called sufficient but not necessary conditions."},{"Start":"10:00.330 ","End":"10:07.500","Text":"Meaning if the conditions hold about the continuity of f or f with respect to y,"},{"Start":"10:07.500 ","End":"10:11.130","Text":"then we\u0027re guaranteed the existence of a unique solution."},{"Start":"10:11.130 ","End":"10:14.370","Text":"Suppose that both f and f_y are continuous,"},{"Start":"10:14.370 ","End":"10:16.065","Text":"then we\u0027re guaranteed a solution,"},{"Start":"10:16.065 ","End":"10:18.605","Text":"but the conditions may not be met."},{"Start":"10:18.605 ","End":"10:23.940","Text":"For example, f might not be continuous and still the problem might have"},{"Start":"10:23.940 ","End":"10:31.200","Text":"a unique solution and similarly for f with respect to y. I\u0027ll give an example of that,"},{"Start":"10:31.200 ","End":"10:33.975","Text":"that it\u0027s not necessary condition."},{"Start":"10:33.975 ","End":"10:37.265","Text":"The example is the following initial value problem,"},{"Start":"10:37.265 ","End":"10:43.905","Text":"we are given y\u0027 is the cube root of y+4 and that it passes through 0,0."},{"Start":"10:43.905 ","End":"10:47.745","Text":"I\u0027m not going to present the solution, doesn\u0027t really matter."},{"Start":"10:47.745 ","End":"10:50.520","Text":"You can try it by separation of variables,"},{"Start":"10:50.520 ","End":"10:57.165","Text":"but it has a unique solution even though the partial derivative with respect to y,"},{"Start":"10:57.165 ","End":"10:58.860","Text":"which is this,"},{"Start":"10:58.860 ","End":"11:01.670","Text":"is actually discontinuous at the 0,0."},{"Start":"11:01.670 ","End":"11:08.595","Text":"In fact, it\u0027s not even defined 0,0 because this is 1 over y^2/3, 1/0."},{"Start":"11:08.595 ","End":"11:11.865","Text":"Anyway, I\u0027m just saying that there are examples."},{"Start":"11:11.865 ","End":"11:14.820","Text":"Just because you find that the conditions are not met,"},{"Start":"11:14.820 ","End":"11:17.630","Text":"doesn\u0027t mean it doesn\u0027t have a unique solution."},{"Start":"11:17.630 ","End":"11:21.255","Text":"What we\u0027re saying is that the conditions are met, then it does."},{"Start":"11:21.255 ","End":"11:23.205","Text":"Now the next remark,"},{"Start":"11:23.205 ","End":"11:27.090","Text":"the second remark is to note that although"},{"Start":"11:27.090 ","End":"11:32.025","Text":"the theorem guarantees the existence of a solution in a certain interval,"},{"Start":"11:32.025 ","End":"11:35.080","Text":"it doesn\u0027t actually mention what the size is,"},{"Start":"11:35.080 ","End":"11:37.440","Text":"it may be very small."},{"Start":"11:37.440 ","End":"11:40.215","Text":"I\u0027ll just return to the diagram for a moment."},{"Start":"11:40.215 ","End":"11:41.674","Text":"Here\u0027s the diagram."},{"Start":"11:41.674 ","End":"11:44.070","Text":"Remember, the theorem guarantees that there is a solution"},{"Start":"11:44.070 ","End":"11:46.770","Text":"in an interval i around the x naught,"},{"Start":"11:46.770 ","End":"11:48.855","Text":"but i might be very small."},{"Start":"11:48.855 ","End":"11:53.745","Text":"It might be x-naught plus or minus 1/1,000 or less."},{"Start":"11:53.745 ","End":"11:55.680","Text":"It could be very small."},{"Start":"11:55.680 ","End":"11:57.960","Text":"It could be far from the whole interval a b,"},{"Start":"11:57.960 ","End":"11:59.530","Text":"or it might be the whole interval a b,"},{"Start":"11:59.530 ","End":"12:01.530","Text":"but it might be extremely tiny."},{"Start":"12:01.530 ","End":"12:04.365","Text":"That\u0027s something to note. Now, let\u0027s get back."},{"Start":"12:04.365 ","End":"12:06.360","Text":"I\u0027m going to give an example."},{"Start":"12:06.360 ","End":"12:09.020","Text":"Let\u0027s take the initial value problem where"},{"Start":"12:09.020 ","End":"12:15.630","Text":"y\u0027=5+5y^2 with the initial condition that y of naught equals naught."},{"Start":"12:15.630 ","End":"12:19.680","Text":"We\u0027re guaranteed that there\u0027s a unique solution because this function"},{"Start":"12:19.680 ","End":"12:23.505","Text":"is continuous and its derivative with respect to y is continuous,"},{"Start":"12:23.505 ","End":"12:27.120","Text":"and certainly around 0,0,"},{"Start":"12:27.120 ","End":"12:29.790","Text":"and there is guaranteed a unique solution."},{"Start":"12:29.790 ","End":"12:31.635","Text":"Before we do the computation,"},{"Start":"12:31.635 ","End":"12:35.580","Text":"you find that the answer is y equals tangent of 5x."},{"Start":"12:35.580 ","End":"12:39.645","Text":"I\u0027m not going to solve it, but you can do it by separation of variables."},{"Start":"12:39.645 ","End":"12:43.320","Text":"This only exists for when 5x is"},{"Start":"12:43.320 ","End":"12:49.230","Text":"between minus 90 degrees and plus 90 degrees or plus or minus Pi over 2,"},{"Start":"12:49.230 ","End":"12:50.460","Text":"and then we divide by 5."},{"Start":"12:50.460 ","End":"12:54.300","Text":"In other words, we get this interval from minus this to this,"},{"Start":"12:54.300 ","End":"12:56.960","Text":"which is roughly minus 0.3-0.3,"},{"Start":"12:56.960 ","End":"13:00.845","Text":"not very small, but if we took a larger number here,"},{"Start":"13:00.845 ","End":"13:02.955","Text":"this could be pretty small."},{"Start":"13:02.955 ","End":"13:05.390","Text":"When guaranteed the unique solution,"},{"Start":"13:05.390 ","End":"13:09.230","Text":"but it may only be a small interval around the point."},{"Start":"13:09.230 ","End":"13:12.130","Text":"My third remark relates to the earlier example."},{"Start":"13:12.130 ","End":"13:16.085","Text":"I just want to see that the example we had with one solution,"},{"Start":"13:16.085 ","End":"13:19.275","Text":"infinite solutions, no solutions are consistent with the theorem."},{"Start":"13:19.275 ","End":"13:22.235","Text":"In this first example that we had,"},{"Start":"13:22.235 ","End":"13:27.750","Text":"this is the function f(x and y),"},{"Start":"13:27.750 ","End":"13:31.650","Text":"and it\u0027s continuous everywhere except where x is 0."},{"Start":"13:31.650 ","End":"13:34.065","Text":"In other words, not at the y-axis,"},{"Start":"13:34.065 ","End":"13:37.730","Text":"but I could take a rectangle around 1,2."},{"Start":"13:37.730 ","End":"13:42.280","Text":"I could take the whole right half plane to the right of the y-axis,"},{"Start":"13:42.280 ","End":"13:46.145","Text":"or some smaller rectangle around 1,2 in it."},{"Start":"13:46.145 ","End":"13:50.585","Text":"There is a rectangle where this is continuous and that\u0027s the existence part."},{"Start":"13:50.585 ","End":"13:52.065","Text":"As for the uniqueness,"},{"Start":"13:52.065 ","End":"13:58.310","Text":"if I take the partial derivative with respect to y(x,y),"},{"Start":"13:58.310 ","End":"14:02.570","Text":"then this just turns out to be 1/x,"},{"Start":"14:02.570 ","End":"14:06.675","Text":"which is also continuous in the right-half plane"},{"Start":"14:06.675 ","End":"14:11.420","Text":"to the right of the y-axis or whatever rectangle we took around 1,2."},{"Start":"14:11.420 ","End":"14:13.275","Text":"The conditions are met,"},{"Start":"14:13.275 ","End":"14:17.315","Text":"and as expected, we do have a unique solution."},{"Start":"14:17.315 ","End":"14:19.085","Text":"In this second example,"},{"Start":"14:19.085 ","End":"14:21.350","Text":"we had y of naught equals naught."},{"Start":"14:21.350 ","End":"14:24.930","Text":"This function, this same f(x,y) is not even"},{"Start":"14:24.930 ","End":"14:28.950","Text":"defined at 0,0 and it\u0027s certainly not continuous there,"},{"Start":"14:28.950 ","End":"14:36.600","Text":"so we don\u0027t satisfy even the most basic condition of the theorem of the existence part,"},{"Start":"14:36.600 ","End":"14:39.240","Text":"f is not continuous at the point,"},{"Start":"14:39.240 ","End":"14:42.360","Text":"so we can\u0027t expect existence."},{"Start":"14:42.360 ","End":"14:44.940","Text":"Nevertheless, even though we can\u0027t expect it,"},{"Start":"14:44.940 ","End":"14:46.305","Text":"there is a solution."},{"Start":"14:46.305 ","End":"14:48.560","Text":"In fact, there are many solutions."},{"Start":"14:48.560 ","End":"14:52.680","Text":"I\u0027m just reiterating the fact that it\u0027s sufficient but not necessary,"},{"Start":"14:52.680 ","End":"14:57.515","Text":"but where it breaks down is if we consider the partial derivative with respect to y,"},{"Start":"14:57.515 ","End":"15:03.365","Text":"this is not continuous at the 0,0,"},{"Start":"15:03.365 ","End":"15:06.405","Text":"so we can\u0027t guarantee uniqueness."},{"Start":"15:06.405 ","End":"15:12.425","Text":"In fact, there is no uniqueness because we saw that every c satisfies."},{"Start":"15:12.425 ","End":"15:16.115","Text":"It doesn\u0027t satisfy continuity of either nevertheless,"},{"Start":"15:16.115 ","End":"15:17.975","Text":"we do at least get existence,"},{"Start":"15:17.975 ","End":"15:19.995","Text":"but we don\u0027t get uniqueness."},{"Start":"15:19.995 ","End":"15:23.040","Text":"The third example, this one,"},{"Start":"15:23.040 ","End":"15:26.280","Text":"just like in number 2, when x is 0,"},{"Start":"15:26.280 ","End":"15:27.930","Text":"which is what we have here again,"},{"Start":"15:27.930 ","End":"15:29.810","Text":"here at x=0, here x=0."},{"Start":"15:29.810 ","End":"15:33.000","Text":"Once again, we said that f is not continuous,"},{"Start":"15:33.000 ","End":"15:34.830","Text":"not even defined there,"},{"Start":"15:34.830 ","End":"15:36.575","Text":"but although a number 2,"},{"Start":"15:36.575 ","End":"15:39.260","Text":"we had existence, we just didn\u0027t get uniqueness,"},{"Start":"15:39.260 ","End":"15:41.460","Text":"here we don\u0027t even have existence,"},{"Start":"15:41.460 ","End":"15:45.825","Text":"which goes to show that if f is not continuous or it doesn\u0027t exist,"},{"Start":"15:45.825 ","End":"15:48.180","Text":"we might have a solutions,"},{"Start":"15:48.180 ","End":"15:51.285","Text":"we might not, we might even have unique solution."},{"Start":"15:51.285 ","End":"15:56.745","Text":"We just can\u0027t say because the condition is sufficient but not necessary."},{"Start":"15:56.745 ","End":"15:59.800","Text":"Here\u0027s an example where the condition wasn\u0027t met and"},{"Start":"15:59.800 ","End":"16:03.540","Text":"indeed we did not even have existence of a solution."},{"Start":"16:03.540 ","End":"16:07.480","Text":"Anyway, everything\u0027s consistent with the theorem and we\u0027ve"},{"Start":"16:07.480 ","End":"16:09.095","Text":"reiterated the difference between"},{"Start":"16:09.095 ","End":"16:12.365","Text":"a necessary and sufficient condition. I talked quite a bit."},{"Start":"16:12.365 ","End":"16:15.420","Text":"Let me just summarize in writing some of the things I said."},{"Start":"16:15.420 ","End":"16:17.560","Text":"Well, first of all,"},{"Start":"16:17.560 ","End":"16:21.870","Text":"we noted that the partial derivative of f with respect to y,"},{"Start":"16:21.870 ","End":"16:24.995","Text":"f with respect to y(x and y) is 1/x."},{"Start":"16:24.995 ","End":"16:27.435","Text":"Like we said, in case 1,"},{"Start":"16:27.435 ","End":"16:32.520","Text":"older theorems conditions are satisfied and indeed we have a unique solution."},{"Start":"16:32.520 ","End":"16:34.605","Text":"In 2 and 3,"},{"Start":"16:34.605 ","End":"16:40.560","Text":"the conditions aren\u0027t satisfied since f with respect to y isn\u0027t continuous."},{"Start":"16:40.560 ","End":"16:43.515","Text":"In fact, f itself is continuous,"},{"Start":"16:43.515 ","End":"16:46.185","Text":"and so basically anything can happen."},{"Start":"16:46.185 ","End":"16:49.380","Text":"Let\u0027s also relate to the other example we had"},{"Start":"16:49.380 ","End":"16:52.895","Text":"when they were two solutions, that was this."},{"Start":"16:52.895 ","End":"16:57.405","Text":"We had this and we had two solutions, y equals minus x and y=x,"},{"Start":"16:57.405 ","End":"17:02.310","Text":"and in this example also has 0,0,"},{"Start":"17:02.310 ","End":"17:05.290","Text":"neither f nor f with respect to y."},{"Start":"17:05.290 ","End":"17:07.620","Text":"Meaning even if f were continuous,"},{"Start":"17:07.620 ","End":"17:10.505","Text":"f_y is not,"},{"Start":"17:10.505 ","End":"17:15.285","Text":"so we wouldn\u0027t have been surprised if there was no solution at all,"},{"Start":"17:15.285 ","End":"17:16.930","Text":"and as it turned out that there is,"},{"Start":"17:16.930 ","End":"17:19.045","Text":"it\u0027s just not unique."},{"Start":"17:19.045 ","End":"17:22.965","Text":"That\u0027s also consistent with the theorem."},{"Start":"17:22.965 ","End":"17:28.005","Text":"That\u0027s it for the theory on this existence and uniqueness theorem,"},{"Start":"17:28.005 ","End":"17:32.210","Text":"hopefully it\u0027ll become a bit clearer with the worked examples,"},{"Start":"17:32.210 ","End":"17:37.530","Text":"the solved exercises that follow this tutorial. Meanwhile, we\u0027re done."}],"Thumbnail":null,"ID":7713},{"Watched":false,"Name":"Exercise 1","Duration":"5m 12s","ChapterTopicVideoID":7643,"CourseChapterTopicPlaylistID":4226,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"In this exercise, we\u0027re given an initial value problem."},{"Start":"00:03.900 ","End":"00:08.310","Text":"Here\u0027s the differential equation and here\u0027s the initial condition."},{"Start":"00:08.310 ","End":"00:12.680","Text":"In part a, we have to prove that there are 2 solutions,"},{"Start":"00:12.680 ","End":"00:17.735","Text":"that this y1(x) and this y2(x) are both solutions."},{"Start":"00:17.735 ","End":"00:19.345","Text":"Then in part b,"},{"Start":"00:19.345 ","End":"00:23.790","Text":"we have to show why this doesn\u0027t contradict the Uniqueness Theorem."},{"Start":"00:23.790 ","End":"00:26.370","Text":"Let\u0027s start with y1,"},{"Start":"00:26.370 ","End":"00:30.760","Text":"and we have to show 2 things at the initial condition is met,"},{"Start":"00:30.760 ","End":"00:36.170","Text":"the differential equation is satisfied so y1, which is this."},{"Start":"00:36.170 ","End":"00:38.930","Text":"First of all, this is easy to check."},{"Start":"00:38.930 ","End":"00:41.300","Text":"Y1(2), we just plug in 2 here,"},{"Start":"00:41.300 ","End":"00:43.985","Text":"we get minus 2 plus 1 minus 1,"},{"Start":"00:43.985 ","End":"00:45.760","Text":"that\u0027s all very well."},{"Start":"00:45.760 ","End":"00:50.090","Text":"The next part is to show that the differential equation is satisfied."},{"Start":"00:50.090 ","End":"00:53.690","Text":"Instead of y, I put minus x plus 1,"},{"Start":"00:53.690 ","End":"00:56.675","Text":"so that\u0027s here and here wherever y appears,"},{"Start":"00:56.675 ","End":"01:00.530","Text":"and we have to show that we get equality here."},{"Start":"01:00.530 ","End":"01:04.685","Text":"The left-hand side is the derivative of minus x plus 1 is minus 1."},{"Start":"01:04.685 ","End":"01:07.520","Text":"Mainly now we\u0027ll just be working on the right-hand side and"},{"Start":"01:07.520 ","End":"01:10.280","Text":"see at the end that we also get minus 1."},{"Start":"01:10.280 ","End":"01:15.905","Text":"First of all, I just open the brackets minus x plus 1."},{"Start":"01:15.905 ","End":"01:20.150","Text":"Next I noticed what\u0027s under the square root sign is a perfect square."},{"Start":"01:20.150 ","End":"01:23.210","Text":"Just check by multiplying this out and see that you get this."},{"Start":"01:23.210 ","End":"01:27.050","Text":"There are many techniques for factorizing these special binomials."},{"Start":"01:27.050 ","End":"01:30.835","Text":"We have the square root of something squared."},{"Start":"01:30.835 ","End":"01:36.110","Text":"In general, you can\u0027t immediately just dropped the square root and the square,"},{"Start":"01:36.110 ","End":"01:41.060","Text":"because the square root of a squared in general is equal"},{"Start":"01:41.060 ","End":"01:46.895","Text":"to the absolute value of a and this will equal a,"},{"Start":"01:46.895 ","End":"01:50.945","Text":"provided that a is bigger or equal to 0."},{"Start":"01:50.945 ","End":"01:54.260","Text":"That\u0027s what I\u0027ll do, I\u0027ll throw out the square root and the square,"},{"Start":"01:54.260 ","End":"01:57.650","Text":"and here just get 1/2 x minus 1."},{"Start":"01:57.650 ","End":"02:02.555","Text":"But that\u0027s going to be true under the condition that this,"},{"Start":"02:02.555 ","End":"02:03.830","Text":"which is like my a here,"},{"Start":"02:03.830 ","End":"02:05.360","Text":"is bigger or equal to 0,"},{"Start":"02:05.360 ","End":"02:07.490","Text":"and if you multiply by 2 and bring this over,"},{"Start":"02:07.490 ","End":"02:09.440","Text":"it means x is bigger or equal to 2,"},{"Start":"02:09.440 ","End":"02:12.935","Text":"which means that we just get in under the wire so to speak,"},{"Start":"02:12.935 ","End":"02:14.870","Text":"because we have x equals 2,"},{"Start":"02:14.870 ","End":"02:19.145","Text":"so we are just inside this range."},{"Start":"02:19.145 ","End":"02:21.680","Text":"This cancels with this,"},{"Start":"02:21.680 ","End":"02:25.970","Text":"and indeed we do get minus 1 is equal to minus 1."},{"Start":"02:25.970 ","End":"02:29.225","Text":"That\u0027s fine, as long as this is true."},{"Start":"02:29.225 ","End":"02:31.790","Text":"That\u0027s the first part,"},{"Start":"02:31.790 ","End":"02:34.495","Text":"the first solution, now let\u0027s go for y2."},{"Start":"02:34.495 ","End":"02:37.560","Text":"Now I just copied it. Y2(x) is this."},{"Start":"02:37.560 ","End":"02:40.380","Text":"Let\u0027s also check that at 2 it\u0027s equal to 1."},{"Start":"02:40.380 ","End":"02:45.330","Text":"If I plug in x equals 2 get minus a 1/4 2^2 which is minus 1,"},{"Start":"02:45.330 ","End":"02:46.440","Text":"and that\u0027s what I wanted it,"},{"Start":"02:46.440 ","End":"02:48.375","Text":"2 it\u0027s got to equal minus 1,"},{"Start":"02:48.375 ","End":"02:54.090","Text":"that\u0027s the initial condition and the next part is the equation part."},{"Start":"02:54.090 ","End":"02:59.165","Text":"As before I just took the differential equation and wherever I saw y,"},{"Start":"02:59.165 ","End":"03:01.580","Text":"I put in minus 1/4 x^2,"},{"Start":"03:01.580 ","End":"03:04.025","Text":"and so this is what we have to show."},{"Start":"03:04.025 ","End":"03:09.350","Text":"Now the derivative of the minus 1/4 x^2 is minus 1/2 x."},{"Start":"03:09.350 ","End":"03:11.750","Text":"On the right-hand side under the square root sign,"},{"Start":"03:11.750 ","End":"03:16.600","Text":"I have 0 and the square root of 0 is 0 regardless of x."},{"Start":"03:16.600 ","End":"03:22.410","Text":"this just kind of drops out and we have minus 1/2 x equals minus 1/2 x,"},{"Start":"03:22.410 ","End":"03:24.335","Text":"and this is true for all x."},{"Start":"03:24.335 ","End":"03:29.450","Text":"All in all, what we had before was true for x is bigger or equal to 2."},{"Start":"03:29.450 ","End":"03:33.065","Text":"This is for all x, but certainly for x bigger or equal to 2,"},{"Start":"03:33.065 ","End":"03:36.530","Text":"there are 2 solutions, y1 and y2."},{"Start":"03:36.530 ","End":"03:38.915","Text":"Now let\u0027s move on to part b."},{"Start":"03:38.915 ","End":"03:41.540","Text":"We\u0027re now going to do part b just to remind you,"},{"Start":"03:41.540 ","End":"03:44.180","Text":"this was the initial value problem."},{"Start":"03:44.180 ","End":"03:48.950","Text":"In part a, we saw that there was more than one solution and we"},{"Start":"03:48.950 ","End":"03:54.010","Text":"were asked to say why this doesn\u0027t contradict the Uniqueness Theorem."},{"Start":"03:54.010 ","End":"03:57.570","Text":"As you recall, the Uniqueness Theorem has certain conditions"},{"Start":"03:57.570 ","End":"04:01.820","Text":"that have to be met in order to guarantee that there\u0027ll be a unique solution."},{"Start":"04:01.820 ","End":"04:05.080","Text":"We\u0027re going to show that these conditions are not met."},{"Start":"04:05.080 ","End":"04:08.060","Text":"One of the conditions is that the partial derivative of f with"},{"Start":"04:08.060 ","End":"04:11.210","Text":"respect to y should be continuous at the point."},{"Start":"04:11.210 ","End":"04:13.310","Text":"I guess I should say that in our case,"},{"Start":"04:13.310 ","End":"04:16.335","Text":"f(xy) is what\u0027s written here."},{"Start":"04:16.335 ","End":"04:20.820","Text":"Just mark that f(xy) and x naught y naught,"},{"Start":"04:20.820 ","End":"04:24.780","Text":"is the point where x is 2,"},{"Start":"04:24.780 ","End":"04:26.234","Text":"y is minus 1."},{"Start":"04:26.234 ","End":"04:29.855","Text":"Let\u0027s see what is f with respect to y?"},{"Start":"04:29.855 ","End":"04:31.520","Text":"If we differentiate this,"},{"Start":"04:31.520 ","End":"04:34.760","Text":"remembering that x is a constant and so we get 1 over"},{"Start":"04:34.760 ","End":"04:38.540","Text":"twice the square root and the internal derivative is 1,"},{"Start":"04:38.540 ","End":"04:39.860","Text":"this is what we get."},{"Start":"04:39.860 ","End":"04:42.494","Text":"Oops, I wrote this backwards,"},{"Start":"04:42.494 ","End":"04:45.720","Text":"this should read 2 minus 1, sorry."},{"Start":"04:45.720 ","End":"04:47.280","Text":"If we plug in x is 2,"},{"Start":"04:47.280 ","End":"04:48.675","Text":"y is minus 1,"},{"Start":"04:48.675 ","End":"04:52.785","Text":"we get 1/4 2^2 minus 1."},{"Start":"04:52.785 ","End":"04:57.485","Text":"This is 0 so this expression here is 0,"},{"Start":"04:57.485 ","End":"04:59.630","Text":"so 0 on the denominator."},{"Start":"04:59.630 ","End":"05:03.740","Text":"This f with respect to y is not continuous,"},{"Start":"05:03.740 ","End":"05:05.495","Text":"it\u0027s not even defined there."},{"Start":"05:05.495 ","End":"05:07.220","Text":"That means the theorem doesn\u0027t apply,"},{"Start":"05:07.220 ","End":"05:10.625","Text":"so anything could have happened and we don\u0027t have Uniqueness."},{"Start":"05:10.625 ","End":"05:13.350","Text":"That should do it."}],"Thumbnail":null,"ID":7714},{"Watched":false,"Name":"Exercise 2","Duration":"5m 15s","ChapterTopicVideoID":7640,"CourseChapterTopicPlaylistID":4226,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.870","Text":"In this exercise, we\u0027re given an initial value problem."},{"Start":"00:03.870 ","End":"00:06.000","Text":"This is a differential equation and this is"},{"Start":"00:06.000 ","End":"00:11.154","Text":"the initial condition and we have to show 3 things."},{"Start":"00:11.154 ","End":"00:14.265","Text":"That it satisfies the condition of the existence theorem,"},{"Start":"00:14.265 ","End":"00:18.405","Text":"but it does not satisfy the conditions of the uniqueness theorem."},{"Start":"00:18.405 ","End":"00:21.540","Text":"But nevertheless, we have to show that it does have"},{"Start":"00:21.540 ","End":"00:24.915","Text":"a unique solution and defined what it is."},{"Start":"00:24.915 ","End":"00:29.550","Text":"This problem was actually mentioned in the tutorial clip to show that"},{"Start":"00:29.550 ","End":"00:35.655","Text":"the conditions of the existence and uniqueness theorem are sufficient but not necessary."},{"Start":"00:35.655 ","End":"00:39.360","Text":"Anyway, this is the problem again,"},{"Start":"00:39.360 ","End":"00:43.230","Text":"in the theorem, labeled these x-naught,"},{"Start":"00:43.230 ","End":"00:46.095","Text":"y-naught, and this was f(x) and y."},{"Start":"00:46.095 ","End":"00:52.595","Text":"Part a is straightforward enough because this function is continuous everywhere."},{"Start":"00:52.595 ","End":"00:57.230","Text":"Cube root has no problem with that and in particular,"},{"Start":"00:57.230 ","End":"01:01.370","Text":"it\u0027s going to be continuous at point 0, 0,"},{"Start":"01:01.370 ","End":"01:05.255","Text":"so we\u0027ve satisfied the conditions of the existence theorem,"},{"Start":"01:05.255 ","End":"01:08.240","Text":"which also implies that there exists a solution."},{"Start":"01:08.240 ","End":"01:10.090","Text":"As for part b,"},{"Start":"01:10.090 ","End":"01:12.425","Text":"notice that the partial derivative,"},{"Start":"01:12.425 ","End":"01:14.974","Text":"if you take f with respect to y,"},{"Start":"01:14.974 ","End":"01:18.410","Text":"this is what we get and it\u0027s not continuous at 0,"},{"Start":"01:18.410 ","End":"01:20.300","Text":"0 because when y is 0,"},{"Start":"01:20.300 ","End":"01:22.660","Text":"we have a denominator of 0,"},{"Start":"01:22.660 ","End":"01:26.810","Text":"so it does not satisfy the condition of the uniqueness theorem."},{"Start":"01:26.810 ","End":"01:28.970","Text":"Well, we\u0027ll get to part C,"},{"Start":"01:28.970 ","End":"01:30.800","Text":"which says that despite this,"},{"Start":"01:30.800 ","End":"01:33.245","Text":"even though it doesn\u0027t satisfy the condition,"},{"Start":"01:33.245 ","End":"01:35.960","Text":"it doesn\u0027t mean that the problem doesn\u0027t."},{"Start":"01:35.960 ","End":"01:39.215","Text":"In fact, the problem does have a unique solution anyway,"},{"Start":"01:39.215 ","End":"01:43.130","Text":"because this condition is sufficient but not necessary."},{"Start":"01:43.130 ","End":"01:47.180","Text":"Now, let\u0027s show that this initial value problem does have"},{"Start":"01:47.180 ","End":"01:53.630","Text":"a unique solution and we\u0027ll start by just solving the differential equation."},{"Start":"01:53.630 ","End":"01:57.500","Text":"We\u0027ll use dy by dx because I\u0027m going to use separation"},{"Start":"01:57.500 ","End":"02:01.400","Text":"of variables and it\u0027s not going to be an easy one,"},{"Start":"02:01.400 ","End":"02:05.415","Text":"so I\u0027ll need some space, so let\u0027s see."},{"Start":"02:05.415 ","End":"02:07.910","Text":"Start by as I said, separating the variables,"},{"Start":"02:07.910 ","End":"02:10.430","Text":"putting this in the denominator and the dx here."},{"Start":"02:10.430 ","End":"02:15.320","Text":"Next, we want to put an integral sign in front of E and there we are."},{"Start":"02:15.320 ","End":"02:19.160","Text":"I just revealed the solution to the integral,"},{"Start":"02:19.160 ","End":"02:20.930","Text":"but don\u0027t feel cheated."},{"Start":"02:20.930 ","End":"02:24.350","Text":"At the end, I\u0027ll show you how I did this integral for those who are"},{"Start":"02:24.350 ","End":"02:28.010","Text":"interested in the details but didn\u0027t want to stop the flow."},{"Start":"02:28.010 ","End":"02:29.840","Text":"This is the integral of the left-hand side."},{"Start":"02:29.840 ","End":"02:36.090","Text":"The right-hand side, of course is x plus c and I just got us some more space."},{"Start":"02:36.090 ","End":"02:41.150","Text":"Now, we want to check the initial condition that when x equals 0,"},{"Start":"02:41.150 ","End":"02:43.520","Text":"y equals 0, this is satisfied."},{"Start":"02:43.520 ","End":"02:47.360","Text":"What I mean is that will be a condition on c and that will help us"},{"Start":"02:47.360 ","End":"02:52.130","Text":"define the c and we\u0027ll show that there is only 1 value of c. Let\u0027s substitute."},{"Start":"02:52.130 ","End":"02:55.795","Text":"Wherever we see the cube root of y, that\u0027s 0."},{"Start":"02:55.795 ","End":"02:57.855","Text":"On the right-hand side, the x is 0."},{"Start":"02:57.855 ","End":"02:59.250","Text":"On the right-hand side we have c,"},{"Start":"02:59.250 ","End":"03:03.420","Text":"here I have the 3 and then I have 4^2 over 2 is 8,"},{"Start":"03:03.420 ","End":"03:07.035","Text":"8 times 4 is 32,"},{"Start":"03:07.035 ","End":"03:09.696","Text":"and the natural log of 4,"},{"Start":"03:09.696 ","End":"03:11.235","Text":"just I leave it as that,"},{"Start":"03:11.235 ","End":"03:13.775","Text":"4 is positive, so I don\u0027t need the absolute value."},{"Start":"03:13.775 ","End":"03:19.190","Text":"This is the expression I\u0027ve got for c. If I do this subtraction and then I multiply out,"},{"Start":"03:19.190 ","End":"03:20.660","Text":"this is what I have."},{"Start":"03:20.660 ","End":"03:23.615","Text":"Then I put this value of c here"},{"Start":"03:23.615 ","End":"03:27.950","Text":"and this gives me the only possible solution because we didn\u0027t"},{"Start":"03:27.950 ","End":"03:31.190","Text":"have any choices along the way and we\u0027re"},{"Start":"03:31.190 ","End":"03:35.525","Text":"basically done except that I still owe you that integral."},{"Start":"03:35.525 ","End":"03:38.060","Text":"For those who would like to,"},{"Start":"03:38.060 ","End":"03:40.370","Text":"you\u0027re welcome to stay."},{"Start":"03:40.370 ","End":"03:43.805","Text":"If you\u0027re not interested in the integral, then we\u0027re done."},{"Start":"03:43.805 ","End":"03:48.455","Text":"This was the integral that we had to do and we\u0027re going to do it with a substitution."},{"Start":"03:48.455 ","End":"03:51.536","Text":"What I\u0027m going to do is let the denominator,"},{"Start":"03:51.536 ","End":"03:54.350","Text":"cube root of y plus 4 be t,"},{"Start":"03:54.350 ","End":"03:57.690","Text":"and then I can get y by"},{"Start":"03:57.690 ","End":"04:03.365","Text":"subtracting 4 from t and then raising to the power of 3, that will give me y."},{"Start":"04:03.365 ","End":"04:05.975","Text":"If I differentiate both sides,"},{"Start":"04:05.975 ","End":"04:10.160","Text":"I\u0027ve got that dy is this expression dt."},{"Start":"04:10.160 ","End":"04:13.590","Text":"If I put dy here,"},{"Start":"04:13.590 ","End":"04:16.680","Text":"and the 3 I can take outside."},{"Start":"04:16.680 ","End":"04:18.780","Text":"On the denominator, I have t,"},{"Start":"04:18.780 ","End":"04:24.140","Text":"so this is the integral I get and I have everything in terms of t. t"},{"Start":"04:24.140 ","End":"04:31.085","Text":"minus 4^2 using the binomial expansion is t^2 minus 8t plus 16."},{"Start":"04:31.085 ","End":"04:37.110","Text":"Then I\u0027m going to divide this polynomial by t and this is what we get,"},{"Start":"04:37.110 ","End":"04:38.340","Text":"t^2 over t,"},{"Start":"04:38.340 ","End":"04:40.245","Text":"8t over t is 8,"},{"Start":"04:40.245 ","End":"04:42.000","Text":"and so on and so on."},{"Start":"04:42.000 ","End":"04:45.115","Text":"We\u0027ll do each piece separately."},{"Start":"04:45.115 ","End":"04:47.910","Text":"The integral of t is t^2 over 2,"},{"Start":"04:47.910 ","End":"04:52.360","Text":"the integral of minus 8 minus 8t and since 1 over t,"},{"Start":"04:52.360 ","End":"04:55.240","Text":"its integral is natural log of absolute value of t,"},{"Start":"04:55.240 ","End":"04:57.185","Text":"this is what we get."},{"Start":"04:57.185 ","End":"04:59.090","Text":"Then we substitute back,"},{"Start":"04:59.090 ","End":"05:04.980","Text":"I guess I should have reminded you that t is the cube root of y plus 4,"},{"Start":"05:04.980 ","End":"05:06.525","Text":"so everywhere we see t,"},{"Start":"05:06.525 ","End":"05:09.120","Text":"put cube root of y plus 4 here,"},{"Start":"05:09.120 ","End":"05:11.130","Text":"here, and here."},{"Start":"05:11.130 ","End":"05:13.530","Text":"That\u0027s the integral that I owe you,"},{"Start":"05:13.530 ","End":"05:15.910","Text":"so now we\u0027re really done."}],"Thumbnail":null,"ID":7715},{"Watched":false,"Name":"Exercise 3","Duration":"1m 58s","ChapterTopicVideoID":7641,"CourseChapterTopicPlaylistID":4226,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this exercise, we have to solve the problem."},{"Start":"00:03.630 ","End":"00:05.280","Text":"It\u0027s an initial value problem."},{"Start":"00:05.280 ","End":"00:07.125","Text":"This is the differential equation."},{"Start":"00:07.125 ","End":"00:09.510","Text":"This is the initial condition and we\u0027re going to"},{"Start":"00:09.510 ","End":"00:12.780","Text":"use the existence and uniqueness theorem."},{"Start":"00:12.780 ","End":"00:17.145","Text":"Let\u0027s denote this right-hand side is f(x, y)."},{"Start":"00:17.145 ","End":"00:19.980","Text":"Then basically we have the initial value problem"},{"Start":"00:19.980 ","End":"00:29.280","Text":"y\u0027=f(x,y) which is this and y(4) is 0."},{"Start":"00:29.280 ","End":"00:34.385","Text":"That\u0027s like my x_0 and y_0 in the theorem."},{"Start":"00:34.385 ","End":"00:40.095","Text":"Now notice that both of these functions are well-behaved everywhere, they\u0027re continuous."},{"Start":"00:40.095 ","End":"00:43.220","Text":"The problem has a unique solution."},{"Start":"00:43.220 ","End":"00:45.350","Text":"By the way, when I say everywhere, of course,"},{"Start":"00:45.350 ","End":"00:50.425","Text":"that includes the point (4,0)."},{"Start":"00:50.425 ","End":"00:53.100","Text":"Take any rectangle you want around (4,"},{"Start":"00:53.100 ","End":"00:56.585","Text":"0), f and f_y are continuous in that rectangle."},{"Start":"00:56.585 ","End":"00:58.400","Text":"There is a unique solution."},{"Start":"00:58.400 ","End":"01:00.695","Text":"Now, just by inspection,"},{"Start":"01:00.695 ","End":"01:05.365","Text":"one thing too often try and see if the trivial solution y=0 works."},{"Start":"01:05.365 ","End":"01:09.095","Text":"In fact, that\u0027s what I\u0027m going to show that y(x)=0 is a solution."},{"Start":"01:09.095 ","End":"01:13.280","Text":"Well, certainly it satisfies the initial condition because y of every x is 0,"},{"Start":"01:13.280 ","End":"01:15.275","Text":"so y(4) is also 0."},{"Start":"01:15.275 ","End":"01:19.150","Text":"If I plug in 0 in place of y everywhere,"},{"Start":"01:19.150 ","End":"01:22.700","Text":"the derivative of 0, the 0 function is 0."},{"Start":"01:22.700 ","End":"01:26.465","Text":"Now, cosine of π/2 is always 0."},{"Start":"01:26.465 ","End":"01:28.295","Text":"It doesn\u0027t matter what this is."},{"Start":"01:28.295 ","End":"01:30.260","Text":"Sine of 0 is 0."},{"Start":"01:30.260 ","End":"01:31.790","Text":"Regardless of what x is,"},{"Start":"01:31.790 ","End":"01:34.610","Text":"we have 0 times something and 0 times something,"},{"Start":"01:34.610 ","End":"01:37.255","Text":"so the right-hand side is also 0."},{"Start":"01:37.255 ","End":"01:44.075","Text":"We\u0027ve shown that 0 is a solution and also we\u0027ve established that"},{"Start":"01:44.075 ","End":"01:47.105","Text":"we have a unique solution because"},{"Start":"01:47.105 ","End":"01:51.170","Text":"the conditions are satisfied for the existence and uniqueness theorem."},{"Start":"01:51.170 ","End":"01:53.285","Text":"If 0 is a solution,"},{"Start":"01:53.285 ","End":"01:54.890","Text":"it is this solution,"},{"Start":"01:54.890 ","End":"01:58.290","Text":"the only solution. We\u0027re done."}],"Thumbnail":null,"ID":7716},{"Watched":false,"Name":"Exercise 4","Duration":"9m 54s","ChapterTopicVideoID":7642,"CourseChapterTopicPlaylistID":4226,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.790","Text":"This is a more challenging exercise in the use of the existence and uniqueness theorem."},{"Start":"00:05.790 ","End":"00:08.670","Text":"I wouldn\u0027t expect you to be able to solve it without help,"},{"Start":"00:08.670 ","End":"00:10.125","Text":"and that\u0027s what I\u0027m here for."},{"Start":"00:10.125 ","End":"00:11.325","Text":"Let\u0027s read it first."},{"Start":"00:11.325 ","End":"00:14.235","Text":"We\u0027re given the initial value problem."},{"Start":"00:14.235 ","End":"00:17.970","Text":"This is the differential equation and this is the initial condition."},{"Start":"00:17.970 ","End":"00:19.410","Text":"There are two things we have to show."},{"Start":"00:19.410 ","End":"00:23.940","Text":"First of all, that any solution of the problem is bounded from below."},{"Start":"00:23.940 ","End":"00:29.205","Text":"What that means is that if we have a solution y(x),"},{"Start":"00:29.205 ","End":"00:35.730","Text":"then y(x) is bigger or equal to some lower bound to some constant"},{"Start":"00:35.730 ","End":"00:42.545","Text":"for all x in the definition and obviously c is going to be less than or equal to 4,"},{"Start":"00:42.545 ","End":"00:43.730","Text":"because at 0 we\u0027re 4."},{"Start":"00:43.730 ","End":"00:48.020","Text":"For example, we might want to show that y(x) is bigger or equal to,"},{"Start":"00:48.020 ","End":"00:50.150","Text":"I don\u0027t know minus 7 or 0."},{"Start":"00:50.150 ","End":"00:53.060","Text":"In fact, I\u0027ll show that it\u0027s bigger or equal to 1."},{"Start":"00:53.060 ","End":"00:54.755","Text":"Clear that\u0027s what I\u0027m going to show,"},{"Start":"00:54.755 ","End":"00:57.185","Text":"we\u0027ll show that it\u0027s bounded from below."},{"Start":"00:57.185 ","End":"01:00.770","Text":"In Part b, we\u0027re going to show that any solution to"},{"Start":"01:00.770 ","End":"01:05.840","Text":"this problem is increasing in all its domain and we\u0027ll do this"},{"Start":"01:05.840 ","End":"01:10.820","Text":"by showing that the derivative is positive based on part"},{"Start":"01:10.820 ","End":"01:16.230","Text":"a. I\u0027m going to do a sketch solution and then I\u0027ll do something more formal."},{"Start":"01:16.230 ","End":"01:18.380","Text":"I need some axis,"},{"Start":"01:18.380 ","End":"01:24.065","Text":"y-axis and an x-axis."},{"Start":"01:24.065 ","End":"01:28.875","Text":"We\u0027re looking for a solution for some function y(x),"},{"Start":"01:28.875 ","End":"01:33.510","Text":"which solves this differential equation and which passes through"},{"Start":"01:33.510 ","End":"01:38.705","Text":"the point 0,4 which is on the y-axis,"},{"Start":"01:38.705 ","End":"01:41.090","Text":"x is 0, y is 4."},{"Start":"01:41.090 ","End":"01:46.625","Text":"Now, I claim there exists a unique solution through this point on some interval."},{"Start":"01:46.625 ","End":"01:55.500","Text":"Let\u0027s go back here and let me label this function as f(x) and y."},{"Start":"01:55.500 ","End":"01:58.820","Text":"I\u0027m going to use the existence and"},{"Start":"01:58.820 ","End":"02:04.610","Text":"uniqueness theorem because f (x y) is continuous everywhere."},{"Start":"02:04.610 ","End":"02:06.800","Text":"It\u0027s a polynomial in x and y,"},{"Start":"02:06.800 ","End":"02:11.615","Text":"and f with respect to y is also continuous everywhere."},{"Start":"02:11.615 ","End":"02:13.160","Text":"In particular at 0,"},{"Start":"02:13.160 ","End":"02:16.400","Text":"4 but everywhere any initial condition that we had that"},{"Start":"02:16.400 ","End":"02:20.850","Text":"be a unique solution based on the theorem."},{"Start":"02:20.850 ","End":"02:22.800","Text":"We know that through here,"},{"Start":"02:22.800 ","End":"02:24.695","Text":"let\u0027s see if I just sketch something."},{"Start":"02:24.695 ","End":"02:26.120","Text":"I don\u0027t know how it looks."},{"Start":"02:26.120 ","End":"02:30.980","Text":"But there is a unique solution through this point for some interval,"},{"Start":"02:30.980 ","End":"02:32.360","Text":"let\u0027s say from here to here."},{"Start":"02:32.360 ","End":"02:33.490","Text":"Well, I won\u0027t even indicate it."},{"Start":"02:33.490 ","End":"02:36.330","Text":"It\u0027s like this from here to here is an interval i,"},{"Start":"02:36.330 ","End":"02:37.790","Text":"may be very small,"},{"Start":"02:37.790 ","End":"02:38.870","Text":"it may be large,"},{"Start":"02:38.870 ","End":"02:41.455","Text":"but where there is a unique solution."},{"Start":"02:41.455 ","End":"02:47.505","Text":"Now, I\u0027m going to put that aside for a moment and next I\u0027m going to alter the problem."},{"Start":"02:47.505 ","End":"02:51.870","Text":"Instead of the initial condition y(0) equals 4,"},{"Start":"02:51.870 ","End":"02:56.245","Text":"we\u0027ll look at the condition y(0) equals 1."},{"Start":"02:56.245 ","End":"03:01.295","Text":"In other words, I\u0027m looking for a solution that goes through 0, 1."},{"Start":"03:01.295 ","End":"03:02.600","Text":"The reason I chose 1,"},{"Start":"03:02.600 ","End":"03:04.700","Text":"is I see here, there\u0027s a y minus 1,"},{"Start":"03:04.700 ","End":"03:10.240","Text":"and I claim that if I take a horizontal line y equals 1,"},{"Start":"03:10.240 ","End":"03:12.690","Text":"then this is going to be a solution."},{"Start":"03:12.690 ","End":"03:14.880","Text":"If I say y(x) equals 1."},{"Start":"03:14.880 ","End":"03:17.170","Text":"Because if y equals 1,"},{"Start":"03:17.170 ","End":"03:19.190","Text":"then doesn\u0027t matter what this other expression is,"},{"Start":"03:19.190 ","End":"03:23.575","Text":"I\u0027ll get this satisfied and it will satisfy the initial condition."},{"Start":"03:23.575 ","End":"03:25.130","Text":"To distinguish the solutions,"},{"Start":"03:25.130 ","End":"03:31.100","Text":"I\u0027ll let say this is y_2(x) and I\u0027ll label this function y_1(x),"},{"Start":"03:31.100 ","End":"03:37.415","Text":"y_2(x) extends everywhere y_1(x) is on some interval we don\u0027t know."},{"Start":"03:37.415 ","End":"03:43.640","Text":"Now if I prove to you that y_1(x) and y_2(x) never cut each other,"},{"Start":"03:43.640 ","End":"03:46.700","Text":"then this will have to be always above,"},{"Start":"03:46.700 ","End":"03:49.610","Text":"because it\u0027s above here 4 is bigger than 1,"},{"Start":"03:49.610 ","End":"03:52.265","Text":"and if it doesn\u0027t cross then it\u0027s going to be always above."},{"Start":"03:52.265 ","End":"03:55.090","Text":"I\u0027ll show you why these two can\u0027t cross."},{"Start":"03:55.090 ","End":"04:00.350","Text":"I\u0027m going to modify the graph and let\u0027s say that they do cross at somewhere here,"},{"Start":"04:00.350 ","End":"04:02.165","Text":"there is a crossing point."},{"Start":"04:02.165 ","End":"04:05.890","Text":"Let me label the point where they cross."},{"Start":"04:05.890 ","End":"04:07.440","Text":"The x or the points will be,"},{"Start":"04:07.440 ","End":"04:09.300","Text":"let\u0027s say x naught."},{"Start":"04:09.300 ","End":"04:13.820","Text":"This time I\u0027m going to take a third initial value problem, the same equation,"},{"Start":"04:13.820 ","End":"04:15.845","Text":"but this time instead of this,"},{"Start":"04:15.845 ","End":"04:21.875","Text":"let\u0027s take y(x) naught equals 1."},{"Start":"04:21.875 ","End":"04:26.435","Text":"What I claim is that the blue line and the green line,"},{"Start":"04:26.435 ","End":"04:31.355","Text":"y_1 and y_2 both satisfy this third initial value problem."},{"Start":"04:31.355 ","End":"04:37.180","Text":"The reason is that they both satisfy the equation by definition."},{"Start":"04:37.180 ","End":"04:39.725","Text":"Just look at the picture they cross."},{"Start":"04:39.725 ","End":"04:45.515","Text":"Therefore, y_1 and y_2 both satisfy the condition that at x naught,"},{"Start":"04:45.515 ","End":"04:46.580","Text":"they are equal to 1."},{"Start":"04:46.580 ","End":"04:49.360","Text":"I mean, that\u0027s what it means for this to cross this."},{"Start":"04:49.360 ","End":"04:52.265","Text":"Around this point here,"},{"Start":"04:52.265 ","End":"04:54.785","Text":"I don\u0027t have uniqueness,"},{"Start":"04:54.785 ","End":"04:58.780","Text":"but the uniqueness that we talked about with f and f with respect to y,"},{"Start":"04:58.780 ","End":"05:02.960","Text":"I mean the theorem is satisfied everywhere in the plane."},{"Start":"05:02.960 ","End":"05:07.730","Text":"I mean, these two are continuous everywhere."},{"Start":"05:07.730 ","End":"05:10.596","Text":"I also expect to get a unique solution through her,"},{"Start":"05:10.596 ","End":"05:12.245","Text":"so that\u0027s a contradiction."},{"Start":"05:12.245 ","End":"05:15.740","Text":"The contradiction came from the fact that I made these two cross,"},{"Start":"05:15.740 ","End":"05:17.970","Text":"so they don\u0027t cross."},{"Start":"05:18.850 ","End":"05:20.900","Text":"I want to restore the picture,"},{"Start":"05:20.900 ","End":"05:21.940","Text":"but you remember how it was before."},{"Start":"05:21.940 ","End":"05:26.470","Text":"They didn\u0027t cross and then we said that y_1(x) was always"},{"Start":"05:26.470 ","End":"05:32.010","Text":"above y_2(x) because 4 is bigger than 1 and they don\u0027t cross."},{"Start":"05:32.010 ","End":"05:33.600","Text":"This was above this."},{"Start":"05:33.600 ","End":"05:39.755","Text":"This means that this solution to the original problem is bounded from below,"},{"Start":"05:39.755 ","End":"05:42.230","Text":"and as I said bounded below by 1."},{"Start":"05:42.230 ","End":"05:45.485","Text":"It\u0027s even bigger than not even bigger or equal to."},{"Start":"05:45.485 ","End":"05:48.425","Text":"That solves Part a."},{"Start":"05:48.425 ","End":"05:55.835","Text":"Part b follows from the fact that the derivative y\u0027 I claim is positive y minus 1."},{"Start":"05:55.835 ","End":"05:59.390","Text":"If I look here I\u0027ll change the inequality to a strict inequality."},{"Start":"05:59.390 ","End":"06:03.845","Text":"Why? This is y the solution to this is what we called y_1,"},{"Start":"06:03.845 ","End":"06:06.830","Text":"y_1(x) is bigger than 1,"},{"Start":"06:06.830 ","End":"06:10.775","Text":"so y minus 1 is bigger than 0,"},{"Start":"06:10.775 ","End":"06:15.890","Text":"x^2 is bigger than 0 and y is bigger than 1."},{"Start":"06:15.890 ","End":"06:21.395","Text":"It\u0027s certainly bigger than 0 and raised to the power of 5 is still bigger than 0."},{"Start":"06:21.395 ","End":"06:23.285","Text":"The derivative is positive,"},{"Start":"06:23.285 ","End":"06:26.150","Text":"so the function is increasing."},{"Start":"06:26.150 ","End":"06:28.100","Text":"It doesn\u0027t look like it\u0027s increasing here,"},{"Start":"06:28.100 ","End":"06:30.185","Text":"but that\u0027s just the sketch."},{"Start":"06:30.185 ","End":"06:32.690","Text":"Now I\u0027m going do this a bit more formally."},{"Start":"06:32.690 ","End":"06:36.920","Text":"This was the picture proof and now let us write it down."},{"Start":"06:36.920 ","End":"06:39.860","Text":"I cleared the board, we\u0027ll start again."},{"Start":"06:39.860 ","End":"06:43.459","Text":"In Part a, I\u0027m going to call this function here f(x"},{"Start":"06:43.459 ","End":"06:47.090","Text":"y) and the partial derivative with respect to y."},{"Start":"06:47.090 ","End":"06:49.250","Text":"You can check this is what we get."},{"Start":"06:49.250 ","End":"06:50.510","Text":"It doesn\u0027t really matter."},{"Start":"06:50.510 ","End":"06:52.880","Text":"We know this, as I said, this is a polynomial"},{"Start":"06:52.880 ","End":"06:55.954","Text":"and the derivative is going to be a polynomial in two variables."},{"Start":"06:55.954 ","End":"06:59.030","Text":"These are basic functions going to be continuous."},{"Start":"06:59.030 ","End":"07:04.355","Text":"Both f and f with respect to y are continuous everywhere for any x and y."},{"Start":"07:04.355 ","End":"07:07.760","Text":"This continuity of f and f_y are"},{"Start":"07:07.760 ","End":"07:11.810","Text":"sufficient condition for the problem to have a unique solution."},{"Start":"07:11.810 ","End":"07:14.510","Text":"The original problem with y(0) equals 4,"},{"Start":"07:14.510 ","End":"07:17.125","Text":"and we\u0027ll call that solution y_1(x)."},{"Start":"07:17.125 ","End":"07:18.560","Text":"Then as we said before,"},{"Start":"07:18.560 ","End":"07:24.290","Text":"if we change the problem slightly to make the initial condition y(0) equals 1,"},{"Start":"07:24.290 ","End":"07:26.330","Text":"then it has an obvious solution."},{"Start":"07:26.330 ","End":"07:29.705","Text":"If we let y_2(x) be the constant 1,"},{"Start":"07:29.705 ","End":"07:32.345","Text":"then that solves this problem."},{"Start":"07:32.345 ","End":"07:33.710","Text":"Because if you let y equals 1,"},{"Start":"07:33.710 ","End":"07:37.520","Text":"the differential equation is met when y equals 1,"},{"Start":"07:37.520 ","End":"07:41.105","Text":"derivative of y is 0 and y minus 1 is also 0."},{"Start":"07:41.105 ","End":"07:43.735","Text":"We get 0 equals 0."},{"Start":"07:43.735 ","End":"07:48.260","Text":"Then remember we talked about how these two can never intersect,"},{"Start":"07:48.260 ","End":"07:49.700","Text":"and we did it by contradiction."},{"Start":"07:49.700 ","End":"07:51.304","Text":"Suppose they did intersect,"},{"Start":"07:51.304 ","End":"07:52.910","Text":"then for some x naught,"},{"Start":"07:52.910 ","End":"07:55.715","Text":"what we would have would be that if they\u0027re both"},{"Start":"07:55.715 ","End":"07:59.090","Text":"equal and the second one is constantly equal to 1,"},{"Start":"07:59.090 ","End":"08:03.095","Text":"then the problem with the third initial condition,"},{"Start":"08:03.095 ","End":"08:08.045","Text":"same equation but this time y(x) naught equals 1 would have two different solutions,"},{"Start":"08:08.045 ","End":"08:14.380","Text":"y1 and y2, because y1 and y2 will both be 1 when x is x naught."},{"Start":"08:14.380 ","End":"08:19.440","Text":"That contradiction shows that they don\u0027t intersect."},{"Start":"08:19.440 ","End":"08:22.320","Text":"The last part they don\u0027t intersect,"},{"Start":"08:22.320 ","End":"08:29.010","Text":"and y1 is above y2 when x is naught because this one is 4 and this one is 1,"},{"Start":"08:29.010 ","End":"08:30.945","Text":"and 4 is bigger than 1 of course."},{"Start":"08:30.945 ","End":"08:32.900","Text":"If one is above the other and they don\u0027t intersect,"},{"Start":"08:32.900 ","End":"08:35.495","Text":"it\u0027s always above the other,"},{"Start":"08:35.495 ","End":"08:37.610","Text":"so y_1(x) is bigger than y_2(x),"},{"Start":"08:37.610 ","End":"08:40.025","Text":"which means that y_1(x) is bigger than 1,"},{"Start":"08:40.025 ","End":"08:43.850","Text":"so y1 is bounded from below which answers part a,"},{"Start":"08:43.850 ","End":"08:46.535","Text":"and specifically it\u0027s bounded below by 1."},{"Start":"08:46.535 ","End":"08:50.900","Text":"Now Part b is just a triviality I\u0027ll just expose this."},{"Start":"08:50.900 ","End":"08:54.485","Text":"Part b, if I take the derivative of y_1,"},{"Start":"08:54.485 ","End":"08:55.790","Text":"just what we had above,"},{"Start":"08:55.790 ","End":"08:59.645","Text":"but I put y_1(x) instead of just y,"},{"Start":"08:59.645 ","End":"09:03.710","Text":"y_1(x) derivative, just plugging in as I said."},{"Start":"09:03.710 ","End":"09:06.215","Text":"Wherever you saw y, and where was it."},{"Start":"09:06.215 ","End":"09:09.155","Text":"Here instead of y, put y_1(x)."},{"Start":"09:09.155 ","End":"09:11.255","Text":"This is what we have."},{"Start":"09:11.255 ","End":"09:13.655","Text":"Now, this is bigger than 1,"},{"Start":"09:13.655 ","End":"09:15.980","Text":"so this minus this is positive,"},{"Start":"09:15.980 ","End":"09:20.735","Text":"like we said, and this is bigger than 1."},{"Start":"09:20.735 ","End":"09:22.550","Text":"Also it\u0027s positive."},{"Start":"09:22.550 ","End":"09:25.940","Text":"Here I meant that even when I take away 1, it\u0027ll be positive."},{"Start":"09:25.940 ","End":"09:27.350","Text":"Here this is positive,"},{"Start":"09:27.350 ","End":"09:30.635","Text":"this is non-negative, positive,"},{"Start":"09:30.635 ","End":"09:35.390","Text":"non-negative, positive until the power of 5,"},{"Start":"09:35.390 ","End":"09:38.690","Text":"so everything\u0027s positive, so it will be positive."},{"Start":"09:38.690 ","End":"09:42.890","Text":"A positive derivative means that the thing is increasing,"},{"Start":"09:42.890 ","End":"09:47.070","Text":"actually not for all x and its domain."},{"Start":"09:47.070 ","End":"09:49.280","Text":"That was what we were asked here in all its domain,"},{"Start":"09:49.280 ","End":"09:53.960","Text":"I\u0027m sorry, for all x in the domain. That\u0027s it."}],"Thumbnail":null,"ID":7717},{"Watched":false,"Name":"EUT - Lipschitz Version","Duration":"6m 54s","ChapterTopicVideoID":25328,"CourseChapterTopicPlaylistID":4226,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.130","Text":"I\u0027d like to remind you of the existence and uniqueness theorem for"},{"Start":"00:05.130 ","End":"00:10.425","Text":"the initial value problem because we\u0027re going to discuss some variations of it,"},{"Start":"00:10.425 ","End":"00:14.055","Text":"but you should remember what the original form is."},{"Start":"00:14.055 ","End":"00:16.470","Text":"Given the initial value problem,"},{"Start":"00:16.470 ","End":"00:18.180","Text":"y\u0027 equals f(x, y),"},{"Start":"00:18.180 ","End":"00:20.505","Text":"and y(x_0)= y_0,"},{"Start":"00:20.505 ","End":"00:24.810","Text":"theorem has the existence and uniqueness part separately."},{"Start":"00:24.810 ","End":"00:25.935","Text":"First, the existence."},{"Start":"00:25.935 ","End":"00:32.085","Text":"If f is continuous in the open rectangle which contains this point,"},{"Start":"00:32.085 ","End":"00:35.325","Text":"then the above problem has a solution,"},{"Start":"00:35.325 ","End":"00:39.180","Text":"at least one in some open interval contained in a,"},{"Start":"00:39.180 ","End":"00:42.420","Text":"b and containing x_0."},{"Start":"00:42.420 ","End":"00:47.255","Text":"The second part is the uniqueness aspect."},{"Start":"00:47.255 ","End":"00:50.810","Text":"If we have an extra condition that the derivative"},{"Start":"00:50.810 ","End":"00:54.620","Text":"with respect to y is continuous in the rectangle,"},{"Start":"00:54.620 ","End":"00:57.155","Text":"then we\u0027re guaranteed this solution is unique."},{"Start":"00:57.155 ","End":"00:58.880","Text":"It\u0027s not if and only if there are"},{"Start":"00:58.880 ","End":"01:02.690","Text":"other weaker conditions under which we can guarantee uniqueness,"},{"Start":"01:02.690 ","End":"01:04.025","Text":"and that will come soon."},{"Start":"01:04.025 ","End":"01:09.450","Text":"But before that, I want to show you that we can combine these two,"},{"Start":"01:09.450 ","End":"01:14.300","Text":"and they\u0027re often combined into one existence and uniqueness theorem,"},{"Start":"01:14.300 ","End":"01:15.590","Text":"and it goes as follows."},{"Start":"01:15.590 ","End":"01:23.285","Text":"If f and f_y are continuous in the rectangle containing the point,"},{"Start":"01:23.285 ","End":"01:29.595","Text":"then the initial value problem has a unique solution in some interval containing x_0."},{"Start":"01:29.595 ","End":"01:32.585","Text":"It\u0027s just a condensed version."},{"Start":"01:32.585 ","End":"01:34.255","Text":"Now the variations."},{"Start":"01:34.255 ","End":"01:38.480","Text":"I\u0027ll start with the motivation why we need the variation I\u0027m going to introduce."},{"Start":"01:38.480 ","End":"01:41.780","Text":"If we have the initial value problem, this one,"},{"Start":"01:41.780 ","End":"01:45.680","Text":"y\u0027 = x times absolute value of y,"},{"Start":"01:45.680 ","End":"01:47.650","Text":"and that\u0027s our f(x, y),"},{"Start":"01:47.650 ","End":"01:51.725","Text":"then there is a solution y = 0."},{"Start":"01:51.725 ","End":"01:54.110","Text":"Remember I told you that you often check."},{"Start":"01:54.110 ","End":"01:57.560","Text":"Often y = 0 is a solution to the differential equation."},{"Start":"01:57.560 ","End":"02:02.375","Text":"This is worth checking because usually pretty easy to see and it often works."},{"Start":"02:02.375 ","End":"02:10.075","Text":"Is a solution, but is it unique around the point 1, 0?"},{"Start":"02:10.075 ","End":"02:13.010","Text":"Now we can\u0027t use the regular existence and"},{"Start":"02:13.010 ","End":"02:18.725","Text":"uniqueness theorem because the derivative with respect to y isn\u0027t continuous."},{"Start":"02:18.725 ","End":"02:22.160","Text":"It doesn\u0027t even exist at this point 1, 0."},{"Start":"02:22.160 ","End":"02:25.205","Text":"It doesn\u0027t exist at any x, 0."},{"Start":"02:25.205 ","End":"02:27.140","Text":"I can tell you now it\u0027s a spoiler,"},{"Start":"02:27.140 ","End":"02:31.685","Text":"0 is a unique solution."},{"Start":"02:31.685 ","End":"02:34.625","Text":"But now we\u0027re going to have to use a different tool"},{"Start":"02:34.625 ","End":"02:38.390","Text":"because our regular uniqueness theorem doesn\u0027t work here."},{"Start":"02:38.390 ","End":"02:42.170","Text":"And we replace the requirement of continuity of the"},{"Start":"02:42.170 ","End":"02:46.055","Text":"partial derivative with respect to y with a different requirement."},{"Start":"02:46.055 ","End":"02:51.410","Text":"Now we come to the Lipschitz version of the existence and uniqueness theorem."},{"Start":"02:51.410 ","End":"02:54.020","Text":"Again, we have f(x,"},{"Start":"02:54.020 ","End":"02:57.545","Text":"y) continuous on an open rectangle."},{"Start":"02:57.545 ","End":"03:03.605","Text":"By the way, a or b or c or d could be infinity containing the point x_0, y_0."},{"Start":"03:03.605 ","End":"03:07.070","Text":"Suppose that f is continuous in"},{"Start":"03:07.070 ","End":"03:11.810","Text":"the rectangle and that f and if we\u0027re replacing the condition,"},{"Start":"03:11.810 ","End":"03:16.770","Text":"is Lipschitz continuous in y in each closed rectangle,"},{"Start":"03:16.770 ","End":"03:21.200","Text":"D contained in our open rectangle R?"},{"Start":"03:21.200 ","End":"03:23.945","Text":"If these two conditions are met,"},{"Start":"03:23.945 ","End":"03:29.795","Text":"then the initial value problem has a unique solution in some interval I subset of a,"},{"Start":"03:29.795 ","End":"03:32.515","Text":"b and containing x_0."},{"Start":"03:32.515 ","End":"03:34.800","Text":"Now there\u0027s an asterisk here,"},{"Start":"03:34.800 ","End":"03:39.195","Text":"I have to tell you what it means Lipschitz continuous in y."},{"Start":"03:39.195 ","End":"03:43.250","Text":"This definition, let f be defined in"},{"Start":"03:43.250 ","End":"03:48.065","Text":"a domain D. Talking about domain in the x, y plane,"},{"Start":"03:48.065 ","End":"03:52.710","Text":"we say that f is Lipschitz continuous in y if there"},{"Start":"03:52.710 ","End":"03:57.210","Text":"exists a constant M such that two points,"},{"Start":"03:57.210 ","End":"03:58.580","Text":"x, y_1, and x, y_2,"},{"Start":"03:58.580 ","End":"04:00.920","Text":"meaning they are vertically one above the other."},{"Start":"04:00.920 ","End":"04:05.600","Text":"The absolute value of the difference or distance between them is"},{"Start":"04:05.600 ","End":"04:10.880","Text":"less than or equal to m times the distance between y_1 and y_2,"},{"Start":"04:10.880 ","End":"04:13.770","Text":"and this is for all x, y_1, x, y_2,"},{"Start":"04:13.770 ","End":"04:17.135","Text":"in D, the same x but different y\u0027s."},{"Start":"04:17.135 ","End":"04:21.500","Text":"Of course, you can also define Lipschitz continuous in x and in general in"},{"Start":"04:21.500 ","End":"04:26.360","Text":"any number of variables Lipschitz continuous in a particular variable."},{"Start":"04:26.360 ","End":"04:30.440","Text":"But we\u0027re just concerned about in two dimensions continuous in y."},{"Start":"04:30.440 ","End":"04:31.850","Text":"That\u0027s the definition,"},{"Start":"04:31.850 ","End":"04:34.145","Text":"and that explains this theorem."},{"Start":"04:34.145 ","End":"04:36.515","Text":"Now let\u0027s get back to our problem."},{"Start":"04:36.515 ","End":"04:41.330","Text":"What we saw is that y = 0 is a solution,"},{"Start":"04:41.330 ","End":"04:44.330","Text":"but we didn\u0027t know how to show uniqueness."},{"Start":"04:44.330 ","End":"04:48.439","Text":"Now we\u0027re going to show that this has a unique solution using"},{"Start":"04:48.439 ","End":"04:53.070","Text":"the Lipschitz version of the existence and uniqueness theorem."},{"Start":"04:53.070 ","End":"04:58.610","Text":"We have to show two conditions and then we can get the result of the uniqueness."},{"Start":"04:58.610 ","End":"05:03.980","Text":"The first is that f is continuous in an open rectangle."},{"Start":"05:03.980 ","End":"05:07.715","Text":"We can take the rectangle to be the whole plane,"},{"Start":"05:07.715 ","End":"05:10.910","Text":"because x absolute value of y is continuous everywhere."},{"Start":"05:10.910 ","End":"05:14.210","Text":"Now we have to show that it\u0027s Lipschitz continuous in"},{"Start":"05:14.210 ","End":"05:18.130","Text":"y in any closed rectangle that\u0027s contained,"},{"Start":"05:18.130 ","End":"05:22.235","Text":"well, it\u0027s always going to be contained in R because that\u0027s the whole plane."},{"Start":"05:22.235 ","End":"05:28.130","Text":"Let\u0027s write our closed rectangle as x between Alpha and Beta,"},{"Start":"05:28.130 ","End":"05:31.969","Text":"inclusive and y between Gamma and Delta."},{"Start":"05:31.969 ","End":"05:33.920","Text":"Let\u0027s check the condition."},{"Start":"05:33.920 ","End":"05:36.380","Text":"We take the same x in two different y\u0027s,"},{"Start":"05:36.380 ","End":"05:38.785","Text":"so this difference is equal to,"},{"Start":"05:38.785 ","End":"05:41.235","Text":"looking at the definition,"},{"Start":"05:41.235 ","End":"05:49.000","Text":"it\u0027s x absolute value of y_1 minus x absolute value of y_2 in absolute value."},{"Start":"05:49.000 ","End":"05:53.810","Text":"We can take the absolute value of x outside."},{"Start":"05:53.810 ","End":"05:58.960","Text":"We take this x and this x outside the absolute value what we\u0027re left is this,"},{"Start":"05:58.960 ","End":"06:02.235","Text":"and this is less than or equal to there is a version of"},{"Start":"06:02.235 ","End":"06:05.445","Text":"the triangle inequality or here\u0027s a reminder,"},{"Start":"06:05.445 ","End":"06:11.180","Text":"so we get this is less than or equal to this because we\u0027re using this backwards."},{"Start":"06:11.180 ","End":"06:18.020","Text":"Now the absolute value of x is at most the greatest of these two absolute values."},{"Start":"06:18.020 ","End":"06:19.580","Text":"If x is between Alpha and Beta,"},{"Start":"06:19.580 ","End":"06:21.830","Text":"the absolute value of x has to be less"},{"Start":"06:21.830 ","End":"06:24.455","Text":"than or equal to the maximum of these two absolute value,"},{"Start":"06:24.455 ","End":"06:28.230","Text":"and just keep the y_1 minus y_2."},{"Start":"06:28.230 ","End":"06:35.555","Text":"This is what we can call M that we needed for the Lipschitz continuity in y."},{"Start":"06:35.555 ","End":"06:38.900","Text":"Now that we\u0027ve met the conditions of this version of"},{"Start":"06:38.900 ","End":"06:42.480","Text":"the existence and uniqueness theorem, then like we said,"},{"Start":"06:42.480 ","End":"06:46.395","Text":"y = 0 is a solution of the IVP,"},{"Start":"06:46.395 ","End":"06:48.300","Text":"initial value problem,"},{"Start":"06:48.300 ","End":"06:50.565","Text":"so it\u0027s the solution."},{"Start":"06:50.565 ","End":"06:54.840","Text":"With this example that concludes this clip."}],"Thumbnail":null,"ID":26145},{"Watched":false,"Name":"Exercise 5","Duration":"5m ","ChapterTopicVideoID":25323,"CourseChapterTopicPlaylistID":4226,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.665","Text":"In this exercise, we have the following initial value problem"},{"Start":"00:04.665 ","End":"00:09.240","Text":"with the differential equation and the initial condition."},{"Start":"00:09.240 ","End":"00:12.750","Text":"It\u0027s defined in the right half plane where x is positive."},{"Start":"00:12.750 ","End":"00:19.540","Text":"We have to show that it has a unique solution around the point x equals 1."},{"Start":"00:20.270 ","End":"00:24.585","Text":"Solution being y is a function of x, of course."},{"Start":"00:24.585 ","End":"00:27.360","Text":"Now, the function f(x,"},{"Start":"00:27.360 ","End":"00:29.175","Text":"y) which is this,"},{"Start":"00:29.175 ","End":"00:33.015","Text":"it\u0027s continuous in the right-half plane,"},{"Start":"00:33.015 ","End":"00:37.330","Text":"and it contains our point 1, Pi."},{"Start":"00:37.460 ","End":"00:42.480","Text":"If this partial derivative were continuous,"},{"Start":"00:42.480 ","End":"00:45.230","Text":"or even just continuous around this point,"},{"Start":"00:45.230 ","End":"00:49.580","Text":"then we could use the regular existence and uniqueness theorem."},{"Start":"00:49.580 ","End":"00:56.960","Text":"But this isn\u0027t because the derivative of f with respect to y doesn\u0027t exist at that point."},{"Start":"00:56.960 ","End":"00:59.075","Text":"If you substitute x equals 1,"},{"Start":"00:59.075 ","End":"01:02.405","Text":"f of 1, y is absolute value of sine y."},{"Start":"01:02.405 ","End":"01:06.185","Text":"As a function of y, it\u0027s not differentiable."},{"Start":"01:06.185 ","End":"01:09.125","Text":"Whenever y is a multiple of Pi,"},{"Start":"01:09.125 ","End":"01:14.255","Text":"you think of the graph it has like V-shaped corners."},{"Start":"01:14.255 ","End":"01:16.160","Text":"Whenever y is a multiple of Pi,"},{"Start":"01:16.160 ","End":"01:17.440","Text":"in particular y equals Pi,"},{"Start":"01:17.440 ","End":"01:22.500","Text":"so the derivative doesn\u0027t even exist, never mind continuous."},{"Start":"01:22.500 ","End":"01:25.340","Text":"We can\u0027t use the regular theorem,"},{"Start":"01:25.340 ","End":"01:28.115","Text":"but we could try the Lipschitz version."},{"Start":"01:28.115 ","End":"01:31.970","Text":"Let R be the rectangle,"},{"Start":"01:31.970 ","End":"01:35.750","Text":"which is semi-infinite rectangle as follows."},{"Start":"01:35.750 ","End":"01:39.060","Text":"It\u0027s the whole right half plane."},{"Start":"01:40.250 ","End":"01:42.439","Text":"We need 2 conditions."},{"Start":"01:42.439 ","End":"01:43.940","Text":"The first is that f(x,"},{"Start":"01:43.940 ","End":"01:48.680","Text":"y) should be continuous around point."},{"Start":"01:48.680 ","End":"01:51.260","Text":"In fact, it\u0027s continuous in the whole right plane,"},{"Start":"01:51.260 ","End":"01:55.010","Text":"its composition of elementary functions."},{"Start":"01:55.010 ","End":"01:59.885","Text":"Secondly, we have to show that f is Lipschitz continuous in y."},{"Start":"01:59.885 ","End":"02:03.169","Text":"Then in closed rectangle D that\u0027s contained"},{"Start":"02:03.169 ","End":"02:06.905","Text":"in R. I think it has to be in the right-half plane."},{"Start":"02:06.905 ","End":"02:14.330","Text":"Now we can write D as x between Alpha and Beta inclusive,"},{"Start":"02:14.330 ","End":"02:17.000","Text":"y between Gamma and Delta inclusive."},{"Start":"02:17.000 ","End":"02:19.460","Text":"These are some constants."},{"Start":"02:19.460 ","End":"02:22.880","Text":"Now for any 2 points, x, y_1,"},{"Start":"02:22.880 ","End":"02:24.050","Text":"and x, y_2,"},{"Start":"02:24.050 ","End":"02:27.485","Text":"they have the same x, different ys."},{"Start":"02:27.485 ","End":"02:30.140","Text":"Of course we\u0027re assuming x is positive."},{"Start":"02:30.140 ","End":"02:34.850","Text":"Then we have that the absolute value of f(x,"},{"Start":"02:34.850 ","End":"02:36.200","Text":"y_1) minus f(x,"},{"Start":"02:36.200 ","End":"02:38.960","Text":"y_2) is the absolute value of 1/x,"},{"Start":"02:38.960 ","End":"02:43.655","Text":"absolute value sine y_1 minus sine y_2."},{"Start":"02:43.655 ","End":"02:47.450","Text":"We can take the 1/x outside the brackets."},{"Start":"02:47.450 ","End":"02:50.165","Text":"Next is positive, so we don\u0027t need an absolute value."},{"Start":"02:50.165 ","End":"02:54.950","Text":"The absolute value of absolute value of sine y_1 minus the absolute value sign y_2."},{"Start":"02:54.950 ","End":"02:57.380","Text":"Now we\u0027re going to use the triangle inequality,"},{"Start":"02:57.380 ","End":"02:59.045","Text":"one of the versions."},{"Start":"02:59.045 ","End":"03:01.145","Text":"We should really read it backwards,"},{"Start":"03:01.145 ","End":"03:07.335","Text":"and then we can get a less than or equal to sine y_1 minus sine y_2 here."},{"Start":"03:07.335 ","End":"03:12.295","Text":"Would use the trigonometric formula for the difference of sines."},{"Start":"03:12.295 ","End":"03:15.625","Text":"I\u0027ll leave it to you to check if this is the correct formula."},{"Start":"03:15.625 ","End":"03:20.800","Text":"Then the absolute value of a product is the product of the absolute values."},{"Start":"03:20.800 ","End":"03:22.420","Text":"This could really be an equal,"},{"Start":"03:22.420 ","End":"03:24.115","Text":"so it doesn\u0027t matter."},{"Start":"03:24.115 ","End":"03:27.520","Text":"Now we can use another inequalities."},{"Start":"03:27.520 ","End":"03:30.760","Text":"Absolute value of sine Alpha is less than or equal to absolute value of"},{"Start":"03:30.760 ","End":"03:35.065","Text":"Alpha and absolute value of cosine less than or equal to 1."},{"Start":"03:35.065 ","End":"03:38.890","Text":"This we can just throw out to the less than or equal to 1 here."},{"Start":"03:38.890 ","End":"03:44.595","Text":"This is less than or equal to absolute value of y_1 minus y_2 over the 2."},{"Start":"03:44.595 ","End":"03:46.290","Text":"I already canceled the 2 out."},{"Start":"03:46.290 ","End":"03:49.160","Text":"I did over 2 here with the 2 here."},{"Start":"03:49.160 ","End":"03:53.580","Text":"Then because x is bigger or equal to Alpha,"},{"Start":"03:53.580 ","End":"03:57.110","Text":"1/x is less than or equal to 1 over Alpha."},{"Start":"03:57.110 ","End":"04:00.455","Text":"1 over Alpha is just a constant, positive,"},{"Start":"04:00.455 ","End":"04:08.250","Text":"call it M. We have that this minus this is less than or equal to M times this minus this."},{"Start":"04:08.710 ","End":"04:13.910","Text":"This means that the Lipschitz condition is satisfied."},{"Start":"04:13.910 ","End":"04:17.569","Text":"We can use the Lipschitz version of the theorem."},{"Start":"04:17.569 ","End":"04:25.760","Text":"Finally, because y equals Pi is a solution of the initial value problem,"},{"Start":"04:25.760 ","End":"04:29.620","Text":"Let\u0027s go back and see where the equation is."},{"Start":"04:29.620 ","End":"04:34.355","Text":"Yeah, if you take the constant function Pi,"},{"Start":"04:34.355 ","End":"04:38.260","Text":"then our function is 0 because sine of Pi is 0,"},{"Start":"04:38.260 ","End":"04:40.279","Text":"0 times 1/x is 0."},{"Start":"04:40.279 ","End":"04:42.305","Text":"If y equals Pi,"},{"Start":"04:42.305 ","End":"04:46.280","Text":"then the left-hand side and the right-hand side are both 0."},{"Start":"04:46.280 ","End":"04:50.890","Text":"It also satisfies the initial condition because y of everything is Pi."},{"Start":"04:50.890 ","End":"04:53.420","Text":"If we have a solution,"},{"Start":"04:53.420 ","End":"04:55.430","Text":"and by the theorem,"},{"Start":"04:55.430 ","End":"04:58.430","Text":"it\u0027s unique, then it\u0027s the solution."},{"Start":"04:58.430 ","End":"05:00.810","Text":"Okay. Done."}],"Thumbnail":null,"ID":26140},{"Watched":false,"Name":"The Interval of Existence and Uniqueness","Duration":"6m 58s","ChapterTopicVideoID":25327,"CourseChapterTopicPlaylistID":4226,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"In this clip, I\u0027ll show you yet another variation of"},{"Start":"00:03.060 ","End":"00:07.095","Text":"the existence and uniqueness theorem which has its uses."},{"Start":"00:07.095 ","End":"00:10.245","Text":"Let\u0027s first of all remember the regular theorem,"},{"Start":"00:10.245 ","End":"00:12.030","Text":"but in simplified form."},{"Start":"00:12.030 ","End":"00:16.140","Text":"We\u0027re given that f and f with respect to y"},{"Start":"00:16.140 ","End":"00:20.340","Text":"are both continuous in a rectangle containing the point,"},{"Start":"00:20.340 ","End":"00:23.870","Text":"then the IVP, the familiar one has"},{"Start":"00:23.870 ","End":"00:28.370","Text":"the unique solution on some interval I containing x naught."},{"Start":"00:28.370 ","End":"00:33.095","Text":"The thing is, that the theorem gives no estimate of the size of the interval I."},{"Start":"00:33.095 ","End":"00:38.840","Text":"We don\u0027t have any minimum interval size that\u0027s guaranteed to have a solution on it."},{"Start":"00:38.840 ","End":"00:42.560","Text":"Turns out if we give a little bit extra restriction on"},{"Start":"00:42.560 ","End":"00:44.750","Text":"these functions then we can get an estimate of"},{"Start":"00:44.750 ","End":"00:47.300","Text":"the size of the interval; a lower estimate."},{"Start":"00:47.300 ","End":"00:52.475","Text":"If we add both of these are bounded on a closed sub-rectangle,"},{"Start":"00:52.475 ","End":"00:57.505","Text":"besides being continuous, then we will get the estimate."},{"Start":"00:57.505 ","End":"01:00.200","Text":"Here\u0027s the variation theorem."},{"Start":"01:00.200 ","End":"01:02.825","Text":"Suppose that in addition,"},{"Start":"01:02.825 ","End":"01:04.865","Text":"there exists a rectangle,"},{"Start":"01:04.865 ","End":"01:10.565","Text":"this time a closed one contained within our open rectangle,"},{"Start":"01:10.565 ","End":"01:12.920","Text":"such that for all x,"},{"Start":"01:12.920 ","End":"01:15.650","Text":"y in this rectangle D,"},{"Start":"01:15.650 ","End":"01:19.190","Text":"f is bounded in absolute value by M and f"},{"Start":"01:19.190 ","End":"01:22.940","Text":"with respect to y is bounded by K. I guess I should have"},{"Start":"01:22.940 ","End":"01:30.560","Text":"said that f_y exists for some constants M and K. Suppose we have this extra condition,"},{"Start":"01:30.560 ","End":"01:33.950","Text":"then the initial value problem has the unique solution"},{"Start":"01:33.950 ","End":"01:37.322","Text":"on the interval I or at least on the interval I,"},{"Start":"01:37.322 ","End":"01:41.120","Text":"from x naught minus Alpha to x_0 plus Alpha."},{"Start":"01:41.120 ","End":"01:42.905","Text":"I have to tell you what Alpha is."},{"Start":"01:42.905 ","End":"01:47.770","Text":"Well, Alpha is the minimum of a and b over M."},{"Start":"01:47.770 ","End":"01:52.760","Text":"It doesn\u0027t actually take into account the constant K. Really,"},{"Start":"01:52.760 ","End":"01:56.285","Text":"we don\u0027t even need this because on a closed rectangle,"},{"Start":"01:56.285 ","End":"01:58.424","Text":"a continuous function will be bounded,"},{"Start":"01:58.424 ","End":"02:00.655","Text":"so we really need this part."},{"Start":"02:00.655 ","End":"02:02.265","Text":"That\u0027s the theorem,"},{"Start":"02:02.265 ","End":"02:04.545","Text":"and let\u0027s give an example."},{"Start":"02:04.545 ","End":"02:05.850","Text":"For the example,"},{"Start":"02:05.850 ","End":"02:08.610","Text":"we\u0027re given the following IVP."},{"Start":"02:08.610 ","End":"02:13.100","Text":"The differential equation part is y\u0027=y^2 minus x,"},{"Start":"02:13.100 ","End":"02:17.285","Text":"and the initial condition that when x is 0, y=1."},{"Start":"02:17.285 ","End":"02:19.460","Text":"Now, the function f,"},{"Start":"02:19.460 ","End":"02:21.680","Text":"which is y^2 minus x,"},{"Start":"02:21.680 ","End":"02:24.680","Text":"has a partial derivative with respect to y,"},{"Start":"02:24.680 ","End":"02:29.180","Text":"that\u0027s 2y, and both of these are continuous on the whole plane."},{"Start":"02:29.180 ","End":"02:32.930","Text":"We\u0027ll take our rectangle R to be everything."},{"Start":"02:32.930 ","End":"02:35.030","Text":"Let\u0027s choose a closed rectangle,"},{"Start":"02:35.030 ","End":"02:38.715","Text":"D arbitrary centered at 0, 1."},{"Start":"02:38.715 ","End":"02:44.360","Text":"We\u0027ll take plus or minus a for the x and plus or minus b for the y."},{"Start":"02:44.360 ","End":"02:46.840","Text":"Here\u0027s a picture, 0,"},{"Start":"02:46.840 ","End":"02:51.010","Text":"1 plus or minus b, and then plus or minus a, that\u0027s the rectangle."},{"Start":"02:51.010 ","End":"02:56.280","Text":"We want to show that f and f_y are bounded on D. We know"},{"Start":"02:56.280 ","End":"03:01.715","Text":"they are bounded on D because they are continuous functions on a compact set."},{"Start":"03:01.715 ","End":"03:04.265","Text":"But we want to know what are the bounds,"},{"Start":"03:04.265 ","End":"03:06.340","Text":"particularly for f. f(x,"},{"Start":"03:06.340 ","End":"03:08.980","Text":"y) is y^2 minus x,"},{"Start":"03:08.980 ","End":"03:13.220","Text":"and that\u0027s less than or equal to using triangle inequality."},{"Start":"03:13.220 ","End":"03:21.285","Text":"Now, y^2, since y is less than or equal to 1 plus b in absolute value,"},{"Start":"03:21.285 ","End":"03:23.430","Text":"I mean, got to be one of the 2 endpoints,"},{"Start":"03:23.430 ","End":"03:28.870","Text":"and obviously the 1 plus b has a greater absolute value than the 1 minus b."},{"Start":"03:28.870 ","End":"03:33.810","Text":"For x, the greatest absolute value doesn\u0027t matter here or here,"},{"Start":"03:33.810 ","End":"03:35.855","Text":"in both cases it comes out to be a,"},{"Start":"03:35.855 ","End":"03:42.255","Text":"and this constant we\u0027ll call M. Less important is the bound for f_y."},{"Start":"03:42.255 ","End":"03:43.650","Text":"We don\u0027t even need it really."},{"Start":"03:43.650 ","End":"03:47.190","Text":"For some reason it\u0027s customary to compute the bound K,"},{"Start":"03:47.190 ","End":"03:49.655","Text":"but only M appears in the formula."},{"Start":"03:49.655 ","End":"03:51.335","Text":"Anyway, it\u0027s 2y,"},{"Start":"03:51.335 ","End":"03:53.930","Text":"2y from here, an absolute value, like we said,"},{"Start":"03:53.930 ","End":"03:57.390","Text":"the absolute value of y has its greatest value here,"},{"Start":"03:57.390 ","End":"03:59.475","Text":"so twice 1 plus b."},{"Start":"03:59.475 ","End":"04:05.420","Text":"This is the M and this is the K. Now we can apply the theorem that tells us that"},{"Start":"04:05.420 ","End":"04:12.485","Text":"our problem has a unique solution on the interval from a minus Alpha to a plus Alpha,"},{"Start":"04:12.485 ","End":"04:17.780","Text":"a is 0 where Alpha is given by the formula that we had which"},{"Start":"04:17.780 ","End":"04:23.600","Text":"is the minimum of a and b over M. This is just the M from here."},{"Start":"04:23.600 ","End":"04:28.220","Text":"For instance, if you choose a=b=1,"},{"Start":"04:28.220 ","End":"04:30.500","Text":"then we get, if you compute this,"},{"Start":"04:30.500 ","End":"04:34.110","Text":"Alpha is the minimum of 1 and 1 over 5,"},{"Start":"04:34.110 ","End":"04:35.975","Text":"it\u0027s 1 plus 1^2 plus 1 is,"},{"Start":"04:35.975 ","End":"04:38.080","Text":"2^2 is 4, plus 1 is 5."},{"Start":"04:38.080 ","End":"04:41.310","Text":"The minimum of these 2 is 1/5."},{"Start":"04:41.310 ","End":"04:48.545","Text":"We\u0027re guaranteed a solution at least on the interval minus 0.2 to 0.2."},{"Start":"04:48.545 ","End":"04:52.580","Text":"Now some remarks. First of all,"},{"Start":"04:52.580 ","End":"04:55.790","Text":"note that the interval that we get;"},{"Start":"04:55.790 ","End":"04:59.030","Text":"this one that we\u0027re guaranteed to have a solution on,"},{"Start":"04:59.030 ","End":"05:01.160","Text":"depends on the choice of a and b."},{"Start":"05:01.160 ","End":"05:02.660","Text":"If you change a and b,"},{"Start":"05:02.660 ","End":"05:04.610","Text":"you could get a larger interval."},{"Start":"05:04.610 ","End":"05:06.815","Text":"For example, in this case,"},{"Start":"05:06.815 ","End":"05:10.610","Text":"if we chose instead of 1 and 1,"},{"Start":"05:10.610 ","End":"05:13.355","Text":"we chose a is a quarter and b=1,"},{"Start":"05:13.355 ","End":"05:18.835","Text":"then you would get Alpha to be bigger than minus 0.2-0.2."},{"Start":"05:18.835 ","End":"05:22.505","Text":"You look at this, you look at this we get a larger interval."},{"Start":"05:22.505 ","End":"05:25.625","Text":"Larger is always larger on both sides."},{"Start":"05:25.625 ","End":"05:27.530","Text":"If it\u0027s only larger on one side,"},{"Start":"05:27.530 ","End":"05:28.955","Text":"you could just take the union."},{"Start":"05:28.955 ","End":"05:31.370","Text":"You could always take the larger on the left"},{"Start":"05:31.370 ","End":"05:34.570","Text":"and the largest on the right if you have 2 such intervals."},{"Start":"05:34.570 ","End":"05:37.930","Text":"That\u0027s the first point, it depends on a and b."},{"Start":"05:37.930 ","End":"05:41.635","Text":"Now, suppose we find the largest interval,"},{"Start":"05:41.635 ","End":"05:44.080","Text":"the best combination of a and b,"},{"Start":"05:44.080 ","End":"05:46.615","Text":"the largest interval we can get in such a way."},{"Start":"05:46.615 ","End":"05:49.405","Text":"It doesn\u0027t mean that that\u0027s the best possible,"},{"Start":"05:49.405 ","End":"05:55.252","Text":"it could be that there\u0027s still a larger interval in which the IVP has a unique solution,"},{"Start":"05:55.252 ","End":"05:57.715","Text":"it\u0027s just not given by this theorem."},{"Start":"05:57.715 ","End":"05:59.725","Text":"We\u0027ll see this in an example."},{"Start":"05:59.725 ","End":"06:01.600","Text":"Also I want to point out in some cases,"},{"Start":"06:01.600 ","End":"06:07.315","Text":"we can use the theorem to show that the IVP has a solution on the whole real line."},{"Start":"06:07.315 ","End":"06:10.235","Text":"We\u0027ll see this in one of the examples, I believe."},{"Start":"06:10.235 ","End":"06:16.945","Text":"The idea is to show that there is a unique solution on x naught plus or minus Alpha."},{"Start":"06:16.945 ","End":"06:20.405","Text":"The Alpha can be chosen as large as we want."},{"Start":"06:20.405 ","End":"06:23.270","Text":"If you take the union of these and Alpha goes larger,"},{"Start":"06:23.270 ","End":"06:25.465","Text":"it goes to minus infinity, infinity."},{"Start":"06:25.465 ","End":"06:32.442","Text":"The last remark is that this theorem is sometimes presented in a short form as follows,"},{"Start":"06:32.442 ","End":"06:36.725","Text":"and I\u0027ll leave you to pause this and look at it."},{"Start":"06:36.725 ","End":"06:40.330","Text":"There\u0027s nothing new here, it\u0027s slightly condensed."},{"Start":"06:40.330 ","End":"06:41.990","Text":"Basically, here\u0027s the given,"},{"Start":"06:41.990 ","End":"06:47.885","Text":"then there\u0027s the if condition and the supplementary condition with the boundedness,"},{"Start":"06:47.885 ","End":"06:55.535","Text":"and then the conclusion where we guarantee an interval with minimum length here."},{"Start":"06:55.535 ","End":"06:58.950","Text":"That concludes this clip."}],"Thumbnail":null,"ID":26144},{"Watched":false,"Name":"Exercise 6","Duration":"6m 45s","ChapterTopicVideoID":25324,"CourseChapterTopicPlaylistID":4226,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"In this exercise, we consider the following initial value problem."},{"Start":"00:04.170 ","End":"00:11.010","Text":"Y\u0027 equals 5 plus y^2 and y(0) = 0. 3 Parts."},{"Start":"00:11.010 ","End":"00:14.685","Text":"Find an interval in which the problem has a unique solution."},{"Start":"00:14.685 ","End":"00:18.390","Text":"Secondly, find the largest interval for which"},{"Start":"00:18.390 ","End":"00:22.215","Text":"the existence and uniqueness theorem guarantees a unique solution."},{"Start":"00:22.215 ","End":"00:25.275","Text":"Thirdly, to show by direct computation,"},{"Start":"00:25.275 ","End":"00:28.380","Text":"there\u0027s an interval larger than the one in part"},{"Start":"00:28.380 ","End":"00:32.175","Text":"b on which the problem has a unique solution."},{"Start":"00:32.175 ","End":"00:34.620","Text":"We\u0027ll start with part a."},{"Start":"00:34.620 ","End":"00:41.270","Text":"Both functions f and derivative with respect to Y are continuous in the whole plane,"},{"Start":"00:41.270 ","End":"00:44.660","Text":"which we can write as a rectangle as follows."},{"Start":"00:44.660 ","End":"00:50.570","Text":"We\u0027ll take our closed rectangle D to be the following."},{"Start":"00:50.570 ","End":"00:54.290","Text":"Here\u0027s a reminder of the theorem that we\u0027re using."},{"Start":"00:54.290 ","End":"00:57.754","Text":"You can just pause and take a look at it to refresh your memory."},{"Start":"00:57.754 ","End":"01:02.060","Text":"I forgot to say that this d is a closed sub rectangular var."},{"Start":"01:02.060 ","End":"01:04.730","Text":"It contains the point 0,0,"},{"Start":"01:04.730 ","End":"01:06.785","Text":"which is the one we want."},{"Start":"01:06.785 ","End":"01:09.320","Text":"Now in the rectangle D,"},{"Start":"01:09.320 ","End":"01:12.690","Text":"that is, we have the absolute value of f (x,"},{"Start":"01:12.690 ","End":"01:18.740","Text":"y) is the absolute value of 5+5(y^2)."},{"Start":"01:18.740 ","End":"01:21.934","Text":"It\u0027s positive, dropped the absolute value."},{"Start":"01:21.934 ","End":"01:26.120","Text":"That\u0027s less than or equal to 5+ 5b^2."},{"Start":"01:26.120 ","End":"01:29.300","Text":"Since absolute value of y less than or equal to b."},{"Start":"01:29.300 ","End":"01:32.880","Text":"This will be the M from the theorem."},{"Start":"01:33.580 ","End":"01:37.790","Text":"The derivative with respect to y also has an upper bound."},{"Start":"01:37.790 ","End":"01:40.370","Text":"The absolute value, absolute value of 10y,"},{"Start":"01:40.370 ","End":"01:42.230","Text":"which is less than or =10b and this is the"},{"Start":"01:42.230 ","End":"01:47.840","Text":"k. Theorem guarantees unique solution on 0 minus Alpha,"},{"Start":"01:47.840 ","End":"01:49.295","Text":"0 plus Alpha,"},{"Start":"01:49.295 ","End":"01:56.495","Text":"where Alpha is the minimum of a and b over m. The m here is the m here."},{"Start":"01:56.495 ","End":"01:59.030","Text":"Now we\u0027ll take an example of a and b."},{"Start":"01:59.030 ","End":"02:00.740","Text":"Just pick anything."},{"Start":"02:00.740 ","End":"02:02.735","Text":"Can I just chose a= 1b,"},{"Start":"02:02.735 ","End":"02:05.075","Text":"=2, you might choose something else."},{"Start":"02:05.075 ","End":"02:10.860","Text":"Using this, we get that Alpha is the minimum of one, which is the a."},{"Start":"02:10.860 ","End":"02:14.690","Text":"Here we have b/5+b^2,"},{"Start":"02:14.690 ","End":"02:18.470","Text":"which comes out to be 2/25."},{"Start":"02:18.470 ","End":"02:24.965","Text":"The problem has a unique solution plus or minus, this is 0.08."},{"Start":"02:24.965 ","End":"02:28.520","Text":"This is our solution. That\u0027s part a."},{"Start":"02:28.520 ","End":"02:31.804","Text":"In part b, we\u0027ll find the largest such interval,"},{"Start":"02:31.804 ","End":"02:36.170","Text":"which means playing around with a and b to get the best we can."},{"Start":"02:36.170 ","End":"02:40.710","Text":"In general, we said we have the integral minus Alpha Alpha,"},{"Start":"02:40.710 ","End":"02:42.315","Text":"where Alpha is this."},{"Start":"02:42.315 ","End":"02:51.810","Text":"Let\u0027s see that I will show this below that the maximum from old b will be 1/10."},{"Start":"02:51.810 ","End":"02:54.225","Text":"It occurs when b=1,"},{"Start":"02:54.225 ","End":"02:56.485","Text":"actually only when b=1."},{"Start":"02:56.485 ","End":"02:58.160","Text":"This is the maximum of"},{"Start":"02:58.160 ","End":"03:03.455","Text":"just the second part in this expression where we take the minimum of a and this."},{"Start":"03:03.455 ","End":"03:05.780","Text":"Alpha, which is the minimum of these two,"},{"Start":"03:05.780 ","End":"03:09.020","Text":"is less than or equal to the minimum of a and 1/10."},{"Start":"03:09.020 ","End":"03:13.245","Text":"Because if this is less than or equal to this and the minimum is less than or equal to."},{"Start":"03:13.245 ","End":"03:15.915","Text":"Most it can be 0.1."},{"Start":"03:15.915 ","End":"03:18.180","Text":"We take the smallest of these two."},{"Start":"03:18.180 ","End":"03:21.230","Text":"We can actually get the 0.1,"},{"Start":"03:21.230 ","End":"03:24.005","Text":"which is the maximum if you choose b=1,"},{"Start":"03:24.005 ","End":"03:26.180","Text":"which gives us a 10th here."},{"Start":"03:26.180 ","End":"03:29.630","Text":"A bigger or equal to 1/10."},{"Start":"03:29.630 ","End":"03:31.220","Text":"If it\u0027s bigger or equal to,"},{"Start":"03:31.220 ","End":"03:34.160","Text":"then the minimum will be the 1/10."},{"Start":"03:34.160 ","End":"03:40.550","Text":"The largest interval that we\u0027re guaranteed a solution on is minus 0.1,"},{"Start":"03:40.550 ","End":"03:44.750","Text":"0.1, which is larger than the interval we had before."},{"Start":"03:44.750 ","End":"03:49.580","Text":"We had minus 0.08, instead 0.1."},{"Start":"03:49.580 ","End":"03:56.027","Text":"Now I\u0027ll show you this computation why this has 1/10 as the maximum,"},{"Start":"03:56.027 ","End":"03:59.660","Text":"so we have a function of b and I shouldn\u0027t have used the same letter F,"},{"Start":"03:59.660 ","End":"04:01.760","Text":"should have used a different letter, never mind."},{"Start":"04:01.760 ","End":"04:04.195","Text":"This f is a different one."},{"Start":"04:04.195 ","End":"04:07.070","Text":"The maximum will do with calculus."},{"Start":"04:07.070 ","End":"04:10.880","Text":"The derivative f\u0027 is this."},{"Start":"04:10.880 ","End":"04:14.270","Text":"Take the 5th out first and then we have b/1+ b^2."},{"Start":"04:14.270 ","End":"04:17.720","Text":"Using the quotient rule, we get this."},{"Start":"04:17.720 ","End":"04:20.470","Text":"If we set f\u0027(b)=0,"},{"Start":"04:20.470 ","End":"04:26.400","Text":"we get the numerator =1 and so b=1,"},{"Start":"04:26.400 ","End":"04:29.355","Text":"can\u0027t be minus 1 because b is positive."},{"Start":"04:29.355 ","End":"04:31.935","Text":"The plugin b=1,"},{"Start":"04:31.935 ","End":"04:36.600","Text":"we get that f(b) is 0.1."},{"Start":"04:36.600 ","End":"04:39.030","Text":"Now, why is it a maximum?"},{"Start":"04:39.030 ","End":"04:42.595","Text":"Because if b is bigger than 1,"},{"Start":"04:42.595 ","End":"04:48.530","Text":"then the derivative here is going to be negative."},{"Start":"04:48.530 ","End":"04:50.810","Text":"All the rest of it is positive except for this."},{"Start":"04:50.810 ","End":"04:52.820","Text":"If b is bigger than 1, this is negative."},{"Start":"04:52.820 ","End":"04:54.905","Text":"If b is less than 1,"},{"Start":"04:54.905 ","End":"04:58.249","Text":"then this comes out to be positive."},{"Start":"04:58.249 ","End":"05:00.920","Text":"We\u0027re increasing up to the point and then decreasing,"},{"Start":"05:00.920 ","End":"05:02.330","Text":"so it\u0027s a maximum."},{"Start":"05:02.330 ","End":"05:04.010","Text":"That\u0027s the completes the missing part,"},{"Start":"05:04.010 ","End":"05:06.260","Text":"the asterisk for this question."},{"Start":"05:06.260 ","End":"05:12.335","Text":"Now let\u0027s go on and do part c. Here again is what part c is."},{"Start":"05:12.335 ","End":"05:16.790","Text":"Direct computation means solving the differential equation."},{"Start":"05:16.790 ","End":"05:20.195","Text":"We have this, we want to solve this part first,"},{"Start":"05:20.195 ","End":"05:22.309","Text":"then we\u0027ll see about the initial condition."},{"Start":"05:22.309 ","End":"05:24.655","Text":"Write it as dy by dx."},{"Start":"05:24.655 ","End":"05:27.980","Text":"Then we can separate variables like so."},{"Start":"05:27.980 ","End":"05:29.780","Text":"We take the integral of both sides,"},{"Start":"05:29.780 ","End":"05:32.300","Text":"or first take out the 1/5."},{"Start":"05:32.300 ","End":"05:33.950","Text":"This is a famous integral,"},{"Start":"05:33.950 ","End":"05:35.975","Text":"or look it up in the integral tables."},{"Start":"05:35.975 ","End":"05:38.960","Text":"It comes out to be arc tangent of y."},{"Start":"05:38.960 ","End":"05:41.165","Text":"Here, x plus a constant."},{"Start":"05:41.165 ","End":"05:45.395","Text":"Now, we know the initial condition that when x is 0, y is 0."},{"Start":"05:45.395 ","End":"05:49.000","Text":"If we plug that in here, we get c is 0."},{"Start":"05:49.000 ","End":"05:52.650","Text":"Arc tangent of 0 is 0, so 0 is 0,"},{"Start":"05:52.650 ","End":"05:54.440","Text":"plus C is 0,"},{"Start":"05:54.440 ","End":"05:59.030","Text":"which means that arc tangent of y is 5(x)."},{"Start":"05:59.030 ","End":"06:02.950","Text":"If you bring the 5 over to the other side when C is 0."},{"Start":"06:02.950 ","End":"06:09.700","Text":"That means that y is tangent of 5x inverse function."},{"Start":"06:10.040 ","End":"06:17.530","Text":"The tangent is defined on minus Pi/2 to Pi/2,"},{"Start":"06:17.530 ","End":"06:19.405","Text":"and then it goes to infinity."},{"Start":"06:19.405 ","End":"06:20.800","Text":"This is largest interval."},{"Start":"06:20.800 ","End":"06:22.510","Text":"It\u0027s defined on continuous,"},{"Start":"06:22.510 ","End":"06:29.500","Text":"divide everything by 5 and get that x is between minus Pi/10 and Pi over 10."},{"Start":"06:29.500 ","End":"06:32.470","Text":"Pi is roughly 3.14."},{"Start":"06:32.470 ","End":"06:38.665","Text":"We get this interval and this is the largest it can be because the endpoints,"},{"Start":"06:38.665 ","End":"06:41.500","Text":"the function is not defined, not continuous."},{"Start":"06:41.500 ","End":"06:45.410","Text":"This is our largest interval and we\u0027re done."}],"Thumbnail":null,"ID":26141},{"Watched":false,"Name":"Exercise 7","Duration":"6m 44s","ChapterTopicVideoID":25325,"CourseChapterTopicPlaylistID":4226,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.325","Text":"In this exercise, we have the following initial value problem y\u0027 is"},{"Start":"00:05.325 ","End":"00:11.190","Text":"minus x/y and y(0) is 1."},{"Start":"00:11.190 ","End":"00:14.655","Text":"This is defined for y bigger than 0,"},{"Start":"00:14.655 ","End":"00:16.785","Text":"the upper half plane."},{"Start":"00:16.785 ","End":"00:20.730","Text":"Part a, find an interval in which the problem has a unique solution."},{"Start":"00:20.730 ","End":"00:22.050","Text":"In part b,"},{"Start":"00:22.050 ","End":"00:24.870","Text":"we find the largest interval for which"},{"Start":"00:24.870 ","End":"00:29.100","Text":"the theorem guarantees a unique solution and in part a,"},{"Start":"00:29.100 ","End":"00:31.440","Text":"we show by direct computation,"},{"Start":"00:31.440 ","End":"00:33.809","Text":"meaning solving the equation,"},{"Start":"00:33.809 ","End":"00:36.900","Text":"that there\u0027s a larger interval than the one in"},{"Start":"00:36.900 ","End":"00:40.795","Text":"b on which the problem has a unique solution."},{"Start":"00:40.795 ","End":"00:44.540","Text":"Part a, f which is minus x/y,"},{"Start":"00:44.540 ","End":"00:46.850","Text":"and the derivative of f with respect to y,"},{"Start":"00:46.850 ","End":"00:51.440","Text":"well this one exists and they\u0027re both continuous on the following rectangle,"},{"Start":"00:51.440 ","End":"00:53.404","Text":"which is the upper half plane,"},{"Start":"00:53.404 ","End":"00:56.940","Text":"which can be described as follows."},{"Start":"00:56.960 ","End":"01:03.855","Text":"Let\u0027s choose a closed rectangle D inside R, which contains (0,1)."},{"Start":"01:03.855 ","End":"01:07.475","Text":"The point which the solution goes through."},{"Start":"01:07.475 ","End":"01:11.510","Text":"We can describe it as absolute value of x is less than or equal"},{"Start":"01:11.510 ","End":"01:16.445","Text":"to a and absolute value of y minus 1 is less than or equal to b."},{"Start":"01:16.445 ","End":"01:18.810","Text":"The picture, this is (0,1)."},{"Start":"01:18.810 ","End":"01:22.115","Text":"What we\u0027re describing is the following rectangle."},{"Start":"01:22.115 ","End":"01:25.310","Text":"There are restrictions on a and b."},{"Start":"01:25.310 ","End":"01:28.685","Text":"D must be inside the upper half plane."},{"Start":"01:28.685 ","End":"01:30.920","Text":"Now inside this rectangle,"},{"Start":"01:30.920 ","End":"01:35.780","Text":"we have the absolute value of f(x,y) and I\u0027m looking here,"},{"Start":"01:35.780 ","End":"01:38.705","Text":"the minus disappears and y is positive."},{"Start":"01:38.705 ","End":"01:41.320","Text":"Just take the absolute value of the x,"},{"Start":"01:41.320 ","End":"01:43.550","Text":"so it\u0027s the absolute value of x over y."},{"Start":"01:43.550 ","End":"01:48.830","Text":"Now, y is above the line, 1 minus b."},{"Start":"01:48.830 ","End":"01:52.190","Text":"If y is bigger or equal to 1 minus b,"},{"Start":"01:52.190 ","End":"01:55.910","Text":"then when we put it on the denominator,"},{"Start":"01:55.910 ","End":"02:00.095","Text":"it\u0027s going to be less than or equal to 1/1 minus b."},{"Start":"02:00.095 ","End":"02:04.945","Text":"An absolute value of x is less than or equal to a and have it here."},{"Start":"02:04.945 ","End":"02:10.650","Text":"We\u0027ll let this be our M, a/1 minus b."},{"Start":"02:11.800 ","End":"02:17.890","Text":"The absolute value of f with respect to y is the following."},{"Start":"02:17.890 ","End":"02:26.030","Text":"We can also estimate that similarly as a/(1 minus b)^2, less than or equal to,"},{"Start":"02:26.030 ","End":"02:30.230","Text":"and that\u0027s our other bound K. The theorem"},{"Start":"02:30.230 ","End":"02:35.195","Text":"guarantees a solution on the interval minus Alpha to Alpha,"},{"Start":"02:35.195 ","End":"02:42.605","Text":"where Alpha is the minimum of a and b/M and the M is this."},{"Start":"02:42.605 ","End":"02:50.075","Text":"Rewrite this, invert the fraction and put it on the top so we get b(1 minus b)/a."},{"Start":"02:50.075 ","End":"02:53.345","Text":"Now the question said find an interval so,"},{"Start":"02:53.345 ","End":"02:59.215","Text":"just arbitrarily choose a and b within the bounds here,"},{"Start":"02:59.215 ","End":"03:02.195","Text":"a positive and b between 0 and 1."},{"Start":"03:02.195 ","End":"03:05.555","Text":"For example, a=1 and b=1/2."},{"Start":"03:05.555 ","End":"03:12.630","Text":"What we get is that Alpha is the minimum of 1 and 1/2 times 1 minus 1/2 over 1 is 1/4,"},{"Start":"03:12.630 ","End":"03:21.505","Text":"and the minimum is a 1/4 and so the problem has unique solution on minus 1/4 to 1/4."},{"Start":"03:21.505 ","End":"03:26.180","Text":"Now in part b, we want to find the largest such interval."},{"Start":"03:26.180 ","End":"03:27.710","Text":"What I mean is,"},{"Start":"03:27.710 ","End":"03:30.470","Text":"we\u0027re going to play around with a and b to try and get"},{"Start":"03:30.470 ","End":"03:33.880","Text":"the best possible combination on Alpha."},{"Start":"03:33.880 ","End":"03:41.435","Text":"We\u0027re going to show in a moment that the maximum of the numerator here,"},{"Start":"03:41.435 ","End":"03:45.995","Text":"b(1 minus b) is 1/4 and occurs when b=1/2."},{"Start":"03:45.995 ","End":"03:53.610","Text":"When b is a 1/2, b(1 minus b) is a 1/4."},{"Start":"03:53.610 ","End":"03:57.365","Text":"This becomes the minimum of a and 1/4a."},{"Start":"03:57.365 ","End":"04:03.365","Text":"Now, look at the graph of a and 1/4a plotted against a."},{"Start":"04:03.365 ","End":"04:06.145","Text":"From the graph you can see that,"},{"Start":"04:06.145 ","End":"04:12.535","Text":"the minimum of these two is going here and then along here."},{"Start":"04:12.535 ","End":"04:18.460","Text":"The maximum value of the minimum of the two is at this point."},{"Start":"04:18.460 ","End":"04:23.505","Text":"We can see algebraically that if a is strictly bigger than a 1/2,"},{"Start":"04:23.505 ","End":"04:25.950","Text":"then this is less than a 1/2 and if this is equal to a 1/2,"},{"Start":"04:25.950 ","End":"04:29.475","Text":"this is equal to a 1/2 and this is less than a 1/2, this is bigger."},{"Start":"04:29.475 ","End":"04:32.860","Text":"Anyway, it\u0027s clear from the picture."},{"Start":"04:32.860 ","End":"04:38.255","Text":"We\u0027ll take a to be a 1/2 and then we take b to be a 1/2 from here."},{"Start":"04:38.255 ","End":"04:43.420","Text":"Altogether, we get that the minimum of these 2 is less than or"},{"Start":"04:43.420 ","End":"04:49.895","Text":"equal to a half and equality when a=1/2."},{"Start":"04:49.895 ","End":"04:56.805","Text":"The maximum of Alpha is a 1/2 and occurs when a is a 1/2 and b is a 1/2 also."},{"Start":"04:56.805 ","End":"05:01.670","Text":"The largest interval that we get is minus"},{"Start":"05:01.670 ","End":"05:06.710","Text":"0.5 to 0.5 and that\u0027s bigger than what we got before,"},{"Start":"05:06.710 ","End":"05:10.565","Text":"which was from minus 0.25 to 0.25."},{"Start":"05:10.565 ","End":"05:15.390","Text":"But we can do still better and that\u0027s part c. In part c,"},{"Start":"05:15.390 ","End":"05:17.390","Text":"we\u0027ll find the larger interval, in fact,"},{"Start":"05:17.390 ","End":"05:21.560","Text":"we\u0027ll find the largest interval by using direct computation."},{"Start":"05:21.560 ","End":"05:24.200","Text":"Are you solving the differential equation?"},{"Start":"05:24.200 ","End":"05:31.815","Text":"Okay, here again is the initial value problem and two separation of variables,"},{"Start":"05:31.815 ","End":"05:33.720","Text":"dy/dx= minus x/y,"},{"Start":"05:33.720 ","End":"05:37.075","Text":"cross multiply ydy is minus xdx."},{"Start":"05:37.075 ","End":"05:39.935","Text":"Take the integral of both sides."},{"Start":"05:39.935 ","End":"05:43.750","Text":"1/2y^2= minus 1/2x^2 plus the constant,"},{"Start":"05:43.750 ","End":"05:47.835","Text":"but use the initial value when x is nil,"},{"Start":"05:47.835 ","End":"05:51.300","Text":"y=1 to get that C=1/2."},{"Start":"05:51.300 ","End":"05:55.650","Text":"Then we have the expression y^2 is minus x^2 plus 1,"},{"Start":"05:55.650 ","End":"05:58.500","Text":"after multiplying by 2 of course."},{"Start":"05:58.500 ","End":"06:06.375","Text":"This gives us that y is the square root of 1 minus x^2 because y is bigger than 0,"},{"Start":"06:06.375 ","End":"06:08.810","Text":"so we don\u0027t take the minus branch."},{"Start":"06:08.810 ","End":"06:14.330","Text":"This is defined for x between minus 1 and 1."},{"Start":"06:14.330 ","End":"06:20.655","Text":"But actually, x can\u0027t be minus 1 or 1 because in that case,"},{"Start":"06:20.655 ","End":"06:24.330","Text":"x^2 would be 1, square root of 1 minus 1 is 0,"},{"Start":"06:24.330 ","End":"06:29.180","Text":"but y is strictly bigger than 0 so we rule out the endpoints."},{"Start":"06:29.180 ","End":"06:30.904","Text":"This is the interval,"},{"Start":"06:30.904 ","End":"06:36.695","Text":"means you find a unique solution to this on the open interval minus 1 to 1,"},{"Start":"06:36.695 ","End":"06:40.145","Text":"and certainly bigger than the minus 1/2 to 1/2."},{"Start":"06:40.145 ","End":"06:44.670","Text":"That concludes part c and this clip."}],"Thumbnail":null,"ID":26142},{"Watched":false,"Name":"Exercise 8","Duration":"3m 30s","ChapterTopicVideoID":25326,"CourseChapterTopicPlaylistID":4226,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.824","Text":"In this exercise, we have an initial value problem."},{"Start":"00:03.824 ","End":"00:09.150","Text":"This part is the differential equation and the initial value,"},{"Start":"00:09.150 ","End":"00:10.620","Text":"the initial condition is this."},{"Start":"00:10.620 ","End":"00:13.800","Text":"It\u0027s general, x_0 and y_0 are general."},{"Start":"00:13.800 ","End":"00:19.095","Text":"We have to show that this has a unique solution on the whole real line."},{"Start":"00:19.095 ","End":"00:24.180","Text":"As usual, we let the right-hand side of this be our f(x,y)."},{"Start":"00:24.180 ","End":"00:26.280","Text":"Sorry, there\u0027s no minus here."},{"Start":"00:26.280 ","End":"00:33.920","Text":"Anyway, that\u0027s f. Obviously f is continuous and it\u0027s differentiable in both x and y."},{"Start":"00:33.920 ","End":"00:38.210","Text":"But particularly the derivative with respect to y is cosine y,"},{"Start":"00:38.210 ","End":"00:40.295","Text":"which is also continuous,"},{"Start":"00:40.295 ","End":"00:42.935","Text":"both of them are continuous in the whole plane."},{"Start":"00:42.935 ","End":"00:49.210","Text":"Now let\u0027s define a closed rectangle D. I\u0027ll show you a picture of this."},{"Start":"00:49.210 ","End":"00:51.805","Text":"Here we are x_0, y_0,"},{"Start":"00:51.805 ","End":"00:56.310","Text":"and then plus or minus b for y and plus or minus a for x."},{"Start":"00:56.310 ","End":"00:59.900","Text":"Let me get this rectangle with the border."},{"Start":"00:59.900 ","End":"01:02.060","Text":"While we\u0027re on this page,"},{"Start":"01:02.060 ","End":"01:05.075","Text":"this is a reminder of the theorem that we\u0027re going to use."},{"Start":"01:05.075 ","End":"01:06.740","Text":"You can just pause and look at it."},{"Start":"01:06.740 ","End":"01:08.320","Text":"It\u0027s just to remind you."},{"Start":"01:08.320 ","End":"01:11.089","Text":"What we want to do is find upper bounds"},{"Start":"01:11.089 ","End":"01:14.735","Text":"for absolute value of f and absolute value of f_y."},{"Start":"01:14.735 ","End":"01:18.810","Text":"This we\u0027ll call m and this we\u0027ll call k the maxima."},{"Start":"01:18.820 ","End":"01:23.210","Text":"First, f, f(x,y) is absolute value of x plus sin"},{"Start":"01:23.210 ","End":"01:27.995","Text":"y less than or equal to by triangle inequality this."},{"Start":"01:27.995 ","End":"01:33.780","Text":"We can write x as x minus x_0 plus x_0 plus sine y."},{"Start":"01:33.780 ","End":"01:37.580","Text":"This is less than or equal to again triangle inequality."},{"Start":"01:37.580 ","End":"01:41.680","Text":"Then we know that x minus x_0 in absolute value is less than or equal to a."},{"Start":"01:41.680 ","End":"01:46.570","Text":"We have this, but this is now a constant doesn\u0027t depend on x."},{"Start":"01:46.570 ","End":"01:54.230","Text":"We can call this m. Absolute value of f(x,y) is less than or equal to M, where M is this."},{"Start":"01:54.230 ","End":"01:56.570","Text":"Now for the other one,"},{"Start":"01:56.570 ","End":"01:59.310","Text":"f_y, this is equal to cosine y."},{"Start":"01:59.310 ","End":"02:03.515","Text":"We know that cosine is bounded by the constant one."},{"Start":"02:03.515 ","End":"02:05.945","Text":"We can take our k=1,"},{"Start":"02:05.945 ","End":"02:09.460","Text":"it\u0027s going to have M and K. By"},{"Start":"02:09.460 ","End":"02:13.340","Text":"the version of the existence and uniqueness theorem that I just showed you a moment ago,"},{"Start":"02:13.340 ","End":"02:16.550","Text":"we know that the problem has a unique solution"},{"Start":"02:16.550 ","End":"02:20.870","Text":"on this interval plus or minus Alpha from x_0,"},{"Start":"02:20.870 ","End":"02:24.335","Text":"where Alpha is the minimum of a and b over M,"},{"Start":"02:24.335 ","End":"02:25.700","Text":"M we got from here,"},{"Start":"02:25.700 ","End":"02:28.135","Text":"so it\u0027s the minimum of these."},{"Start":"02:28.135 ","End":"02:32.255","Text":"Now what remains to show is that Alpha can be arbitrarily large."},{"Start":"02:32.255 ","End":"02:34.940","Text":"When we\u0027ve shown that we\u0027re practically done."},{"Start":"02:34.940 ","End":"02:38.035","Text":"Let Beta be bigger than 0,"},{"Start":"02:38.035 ","End":"02:40.365","Text":"Beta can be as large as you want."},{"Start":"02:40.365 ","End":"02:43.835","Text":"I\u0027m going to show you that we can make Alpha equal to Beta,"},{"Start":"02:43.835 ","End":"02:45.725","Text":"meaning Alpha can be arbitrarily large."},{"Start":"02:45.725 ","End":"02:48.920","Text":"Choose a not Alpha don\u0027t confuse the two."},{"Start":"02:48.920 ","End":"02:56.450","Text":"Let a equal Beta and b is equal to this constant M times Beta."},{"Start":"02:56.450 ","End":"03:00.645","Text":"Alpha is the minimum of a,"},{"Start":"03:00.645 ","End":"03:03.265","Text":"which is Beta, and b/M,"},{"Start":"03:03.265 ","End":"03:05.330","Text":"which is this over this,"},{"Start":"03:05.330 ","End":"03:06.550","Text":"which is also Beta."},{"Start":"03:06.550 ","End":"03:08.915","Text":"The minimum of Beta and Beta is Beta."},{"Start":"03:08.915 ","End":"03:11.600","Text":"Alpha is equal to Beta,"},{"Start":"03:11.600 ","End":"03:14.455","Text":"but Beta was arbitrarily large."},{"Start":"03:14.455 ","End":"03:17.030","Text":"Any Beta bigger than naught,"},{"Start":"03:17.030 ","End":"03:19.040","Text":"we have a solution on this interval."},{"Start":"03:19.040 ","End":"03:20.990","Text":"We take the union of all those,"},{"Start":"03:20.990 ","End":"03:23.480","Text":"it covers the whole real line."},{"Start":"03:23.480 ","End":"03:28.175","Text":"Any point can be covered by one of these intervals if you make Beta large enough."},{"Start":"03:28.175 ","End":"03:31.050","Text":"That concludes the proof."}],"Thumbnail":null,"ID":26143}],"ID":4226},{"Name":"Bernoulli Equations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Bernoulli Equations","Duration":"3m 3s","ChapterTopicVideoID":7613,"CourseChapterTopicPlaylistID":216828,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.076","Text":"This exercise is not really an exercise."},{"Start":"00:03.076 ","End":"00:09.960","Text":"It\u0027s actually a tutorial on what is a Bernoulli equation and introduction."},{"Start":"00:09.960 ","End":"00:13.815","Text":"What is Bernoulli\u0027s equation and explain how to solve it."},{"Start":"00:13.815 ","End":"00:16.694","Text":"That\u0027s in the context of ordinary differential equations."},{"Start":"00:16.694 ","End":"00:19.245","Text":"First of all, I\u0027ll tell you what it is."},{"Start":"00:19.245 ","End":"00:21.480","Text":"Here\u0027s what the Bernoulli\u0027s equation is."},{"Start":"00:21.480 ","End":"00:24.930","Text":"It\u0027s of the form y\u0027 plus p(x),"},{"Start":"00:24.930 ","End":"00:29.460","Text":"some function of x times y, plus q(x)."},{"Start":"00:29.460 ","End":"00:32.130","Text":"If we didn\u0027t have this extra bit, y^n,"},{"Start":"00:32.130 ","End":"00:33.449","Text":"it would be linear,"},{"Start":"00:33.449 ","End":"00:35.820","Text":"but we do have y^n."},{"Start":"00:35.820 ","End":"00:37.400","Text":"n is an integer,"},{"Start":"00:37.400 ","End":"00:41.540","Text":"but there are a couple of restrictions on n. n is not going to be 0."},{"Start":"00:41.540 ","End":"00:45.310","Text":"If n is 0, then this thing disappears and it\u0027s a linear equation."},{"Start":"00:45.310 ","End":"00:47.450","Text":"We don\u0027t need Bernoulli\u0027s equation for that."},{"Start":"00:47.450 ","End":"00:48.895","Text":"If n is 1,"},{"Start":"00:48.895 ","End":"00:51.285","Text":"and here we have y and here we have y,"},{"Start":"00:51.285 ","End":"00:56.390","Text":"so we can just combine and put p minus q on the left-hand side,"},{"Start":"00:56.390 ","End":"00:59.765","Text":"and that will actually turn out to be separable."},{"Start":"00:59.765 ","End":"01:03.335","Text":"Let\u0027s just take it that n is not 0 or 1."},{"Start":"01:03.335 ","End":"01:06.230","Text":"That answers what it is."},{"Start":"01:06.230 ","End":"01:09.635","Text":"Now, the recipe on how to solve it."},{"Start":"01:09.635 ","End":"01:11.555","Text":"I\u0027m going to give you the steps,"},{"Start":"01:11.555 ","End":"01:15.110","Text":"but this will really only makes sense when you get into the examples."},{"Start":"01:15.110 ","End":"01:17.820","Text":"Still I want you to have an outline,"},{"Start":"01:17.820 ","End":"01:19.715","Text":"the main steps that we do,"},{"Start":"01:19.715 ","End":"01:21.530","Text":"and together with the exercises,"},{"Start":"01:21.530 ","End":"01:23.195","Text":"you\u0027ll be able to follow it."},{"Start":"01:23.195 ","End":"01:25.205","Text":"What we do is a substitution,"},{"Start":"01:25.205 ","End":"01:29.030","Text":"v equals y to the power of 1 minus n,"},{"Start":"01:29.030 ","End":"01:30.860","Text":"where n is the n from here."},{"Start":"01:30.860 ","End":"01:34.100","Text":"Now, it\u0027s guaranteed you don\u0027t have to do any work,"},{"Start":"01:34.100 ","End":"01:35.810","Text":"that after the substitution,"},{"Start":"01:35.810 ","End":"01:41.450","Text":"what you\u0027re left with is a differential equation in v, which is linear."},{"Start":"01:41.450 ","End":"01:46.865","Text":"We just take our a(x) and our b(x) from the original equation,"},{"Start":"01:46.865 ","End":"01:51.325","Text":"we just take p and multiply it by 1 minus n,"},{"Start":"01:51.325 ","End":"01:52.670","Text":"which is by the way,"},{"Start":"01:52.670 ","End":"01:55.910","Text":"not 0 because n is not equal to 1."},{"Start":"01:55.910 ","End":"02:00.830","Text":"Also we take the q from here and multiply it by 1 minus n,"},{"Start":"02:00.830 ","End":"02:04.100","Text":"and that we take this differential equation in v."},{"Start":"02:04.100 ","End":"02:07.985","Text":"I\u0027m going to solve it for v and at the end we will do a reverse substitution."},{"Start":"02:07.985 ","End":"02:16.140","Text":"Anyway, the next step is to solve this linear equation in v,"},{"Start":"02:16.140 ","End":"02:19.100","Text":"not the original equation because that\u0027s the whole point."},{"Start":"02:19.100 ","End":"02:21.454","Text":"For example, we could use the formula."},{"Start":"02:21.454 ","End":"02:23.420","Text":"If I use the formula with a and b,"},{"Start":"02:23.420 ","End":"02:25.895","Text":"then we get the v is equal to this."},{"Start":"02:25.895 ","End":"02:29.257","Text":"I didn\u0027t say it, but A, it\u0027s understood,"},{"Start":"02:29.257 ","End":"02:33.160","Text":"is the integral of little a in this formula."},{"Start":"02:33.160 ","End":"02:35.000","Text":"There\u0027s 2 integrations to be done,"},{"Start":"02:35.000 ","End":"02:37.865","Text":"one to find big A and then this other integral."},{"Start":"02:37.865 ","End":"02:39.589","Text":"Anyway, that\u0027s linear equations."},{"Start":"02:39.589 ","End":"02:40.910","Text":"That was Step 2."},{"Start":"02:40.910 ","End":"02:46.078","Text":"Instead of v, I put y to the power of 1 minus n wherever v was,"},{"Start":"02:46.078 ","End":"02:50.210","Text":"and then we try to extract y if possible."},{"Start":"02:50.210 ","End":"02:53.930","Text":"Basically, y would equal v to the power of 1"},{"Start":"02:53.930 ","End":"02:57.710","Text":"over 1 minus n. We\u0027ll see this in the examples."},{"Start":"02:57.710 ","End":"03:00.560","Text":"I just wanted to outline the 3 main steps."},{"Start":"03:00.560 ","End":"03:04.470","Text":"We\u0027re done here and onto the exercises."}],"Thumbnail":null,"ID":26229},{"Watched":false,"Name":"Exercise 1","Duration":"6m 31s","ChapterTopicVideoID":7616,"CourseChapterTopicPlaylistID":216828,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.905","Text":"In this exercise, we\u0027re asked to solve this differential equation."},{"Start":"00:04.905 ","End":"00:08.280","Text":"I happen to know it\u0027s a Bernoulli equation."},{"Start":"00:08.280 ","End":"00:11.295","Text":"We\u0027re in the chapter on Bernoulli equations."},{"Start":"00:11.295 ","End":"00:13.395","Text":"I brought with me the steps,"},{"Start":"00:13.395 ","End":"00:16.605","Text":"the essence of how we solve Bernoulli equations."},{"Start":"00:16.605 ","End":"00:22.670","Text":"First of all, we bring it into the standard form where we have y\u0027, and so on."},{"Start":"00:22.670 ","End":"00:25.915","Text":"I want to divide by x^2."},{"Start":"00:25.915 ","End":"00:27.840","Text":"We have just y\u0027."},{"Start":"00:27.840 ","End":"00:33.890","Text":"Also, I\u0027ll take this over to the other side because that\u0027s how this is."},{"Start":"00:33.890 ","End":"00:35.840","Text":"Here we are."},{"Start":"00:35.840 ","End":"00:37.685","Text":"The y^3 goes over,"},{"Start":"00:37.685 ","End":"00:41.275","Text":"we divide by x^2 so we get just y\u0027."},{"Start":"00:41.275 ","End":"00:42.580","Text":"Divide this by x^2,"},{"Start":"00:42.580 ","End":"00:44.000","Text":"we get 2 over x,"},{"Start":"00:44.000 ","End":"00:48.405","Text":"and over here, also 1 over x^2 times y^3."},{"Start":"00:48.405 ","End":"00:56.150","Text":"This indeed looks like this format where p is 2 over x and q is 1 over x^2."},{"Start":"00:56.150 ","End":"00:58.115","Text":"Because we divided by x^2,"},{"Start":"00:58.115 ","End":"01:01.765","Text":"I added the condition that x is not 0."},{"Start":"01:01.765 ","End":"01:05.345","Text":"Next is to make this substitution only."},{"Start":"01:05.345 ","End":"01:06.770","Text":"We just write it,"},{"Start":"01:06.770 ","End":"01:11.450","Text":"we don\u0027t actually need to substitute because in the theory after the substitution,"},{"Start":"01:11.450 ","End":"01:13.025","Text":"this is what we end up with."},{"Start":"01:13.025 ","End":"01:16.795","Text":"We only actually use this at the end when we substitute back."},{"Start":"01:16.795 ","End":"01:21.410","Text":"In our case, we identify n as being 3."},{"Start":"01:21.410 ","End":"01:26.240","Text":"That\u0027s this 3 here that corresponds to this n. If n is 3,"},{"Start":"01:26.240 ","End":"01:29.105","Text":"we need 1 minus n is negative 2."},{"Start":"01:29.105 ","End":"01:33.875","Text":"The substitution is v=y to the minus 2."},{"Start":"01:33.875 ","End":"01:36.260","Text":"As I said, we don\u0027t do anything with this yet,"},{"Start":"01:36.260 ","End":"01:38.150","Text":"just at the end."},{"Start":"01:38.150 ","End":"01:43.310","Text":"Now the substitution, and I look at this line here,"},{"Start":"01:43.310 ","End":"01:44.930","Text":"and to make it clear,"},{"Start":"01:44.930 ","End":"01:47.930","Text":"the 1 minus n that\u0027s here in red is here in red,"},{"Start":"01:47.930 ","End":"01:49.730","Text":"that\u0027s the minus 2."},{"Start":"01:49.730 ","End":"01:55.730","Text":"It\u0027s the original equation similar to anyway,"},{"Start":"01:55.730 ","End":"01:59.190","Text":"and instead of the q and the p,"},{"Start":"01:59.190 ","End":"02:03.500","Text":"we have 1 minus n times them and the y^n drops off."},{"Start":"02:03.500 ","End":"02:05.630","Text":"That\u0027s a way of remembering what you do,"},{"Start":"02:05.630 ","End":"02:09.950","Text":"and y is replaced by v. This is what we get."},{"Start":"02:09.950 ","End":"02:11.960","Text":"I didn\u0027t provide the proof for this,"},{"Start":"02:11.960 ","End":"02:16.490","Text":"but that\u0027s part of the theory that the substitution produces this."},{"Start":"02:16.490 ","End":"02:22.415","Text":"Now, we identify this as a linear equation in v,"},{"Start":"02:22.415 ","End":"02:26.180","Text":"v\u0027 plus some function of x v equal to another function of"},{"Start":"02:26.180 ","End":"02:31.910","Text":"x. I\u0027m going to use the theory for linear equations here."},{"Start":"02:31.910 ","End":"02:33.980","Text":"What we do is, first of all,"},{"Start":"02:33.980 ","End":"02:40.235","Text":"just identify that this is a(x) and this function we call b(x)."},{"Start":"02:40.235 ","End":"02:43.370","Text":"Though normally the template was with y,"},{"Start":"02:43.370 ","End":"02:52.730","Text":"but it works just as well if we have v\u0027 plus a(x)v = b(x),"},{"Start":"02:52.730 ","End":"02:55.970","Text":"the same formula that we used when we had y will work with"},{"Start":"02:55.970 ","End":"02:59.465","Text":"another letter v. Let\u0027s do this on another page."},{"Start":"02:59.465 ","End":"03:03.749","Text":"Here\u0027s the standard general formula we used for the linear equations,"},{"Start":"03:03.749 ","End":"03:05.728","Text":"the only difference is that here we have a v,"},{"Start":"03:05.728 ","End":"03:10.460","Text":"and then previously we used y. I didn\u0027t say what big A is,"},{"Start":"03:10.460 ","End":"03:15.140","Text":"but this is actually the integral of little a,"},{"Start":"03:15.140 ","End":"03:17.790","Text":"and little a if you look back,"},{"Start":"03:17.790 ","End":"03:19.860","Text":"it was minus 4 over x."},{"Start":"03:19.860 ","End":"03:24.170","Text":"We get minus 4 ln(x)."},{"Start":"03:24.170 ","End":"03:25.925","Text":"That\u0027s 1 integral."},{"Start":"03:25.925 ","End":"03:29.450","Text":"There\u0027s another integral we have to perform which is this."},{"Start":"03:29.450 ","End":"03:33.020","Text":"Though is 2 integrations we need to do for the linear."},{"Start":"03:33.020 ","End":"03:35.225","Text":"The second integral is this part."},{"Start":"03:35.225 ","End":"03:39.845","Text":"B we saw above was minus 2 over x^2."},{"Start":"03:39.845 ","End":"03:42.815","Text":"The a is what we get from the first part,"},{"Start":"03:42.815 ","End":"03:45.125","Text":"this is this here."},{"Start":"03:45.125 ","End":"03:51.590","Text":"We can simplify this if we use the formula e to the minus k ln"},{"Start":"03:51.590 ","End":"03:58.535","Text":"of something is 1 over that something to the power of k. We\u0027ve used before this formula,"},{"Start":"03:58.535 ","End":"04:03.330","Text":"and what we get then if I replace this with this"},{"Start":"04:03.330 ","End":"04:08.900","Text":"where k is 4 and the box is absolute value of x,"},{"Start":"04:08.900 ","End":"04:11.900","Text":"of course, absolute value to an even power,"},{"Start":"04:11.900 ","End":"04:17.540","Text":"I can just throw out the absolute value so it\u0027s x^4 and this combines with the x^2,"},{"Start":"04:17.540 ","End":"04:22.835","Text":"and that gives us this integral which is a straightforward integral."},{"Start":"04:22.835 ","End":"04:26.070","Text":"Let\u0027s see. Raise the exponent by 1,"},{"Start":"04:26.070 ","End":"04:29.285","Text":"that\u0027s negative 5 and divide by that exponent,"},{"Start":"04:29.285 ","End":"04:31.610","Text":"the minuses will cancel."},{"Start":"04:31.610 ","End":"04:33.425","Text":"Back to the formula,"},{"Start":"04:33.425 ","End":"04:36.560","Text":"this is the integral part."},{"Start":"04:36.560 ","End":"04:38.790","Text":"This integral, that\u0027s this."},{"Start":"04:38.790 ","End":"04:41.480","Text":"We already computed a(x) before,"},{"Start":"04:41.480 ","End":"04:43.055","Text":"that was the first integral."},{"Start":"04:43.055 ","End":"04:44.870","Text":"We marked this form with an asterisk,"},{"Start":"04:44.870 ","End":"04:46.875","Text":"and this was double asterisk."},{"Start":"04:46.875 ","End":"04:49.940","Text":"Now plug them in, I\u0027m not going to scroll back, take a look that,"},{"Start":"04:49.940 ","End":"04:56.060","Text":"we found that big A(x) was minus 4 ln absolute value of x."},{"Start":"04:56.060 ","End":"04:58.970","Text":"This integral, we still have it here,"},{"Start":"04:58.970 ","End":"05:00.680","Text":"the minuses cancel,"},{"Start":"05:00.680 ","End":"05:03.920","Text":"plus C. There was no need to add a constant for a,"},{"Start":"05:03.920 ","End":"05:07.115","Text":"this constant takes care of everything."},{"Start":"05:07.115 ","End":"05:13.430","Text":"Once again, I use the trick with the exponent of the Ln only this time,"},{"Start":"05:13.430 ","End":"05:14.810","Text":"previously it was a minus,"},{"Start":"05:14.810 ","End":"05:17.930","Text":"this time it\u0027s minus minus which is plus so it\u0027s on the numerator,"},{"Start":"05:17.930 ","End":"05:21.190","Text":"and we can throw out the absolute value if we want."},{"Start":"05:21.190 ","End":"05:23.290","Text":"Now, we\u0027ve found v. As I said,"},{"Start":"05:23.290 ","End":"05:25.045","Text":"this is just x^4."},{"Start":"05:25.045 ","End":"05:29.710","Text":"X^4 with 1 over x^5 is just 1 over x,"},{"Start":"05:29.710 ","End":"05:31.210","Text":"if I multiply it out,"},{"Start":"05:31.210 ","End":"05:36.880","Text":"and C gets multiplied by x^4 but that\u0027s v, we want y."},{"Start":"05:36.880 ","End":"05:41.980","Text":"If you remember, substitution was v=y to the minus 2,"},{"Start":"05:41.980 ","End":"05:51.100","Text":"I want to see the other way round what y is in terms of v. This is 1 over y^2."},{"Start":"05:51.100 ","End":"05:55.840","Text":"If you figure it, y comes out to be plus or minus 1 over the square root of"},{"Start":"05:55.840 ","End":"06:00.870","Text":"v. Now I can substitute v from here."},{"Start":"06:00.870 ","End":"06:03.375","Text":"After we plug in v as this,"},{"Start":"06:03.375 ","End":"06:07.205","Text":"we get that y is this expression plus or minus this."},{"Start":"06:07.205 ","End":"06:08.840","Text":"We could leave it at that."},{"Start":"06:08.840 ","End":"06:12.200","Text":"That\u0027s why explicitly in terms of x,"},{"Start":"06:12.200 ","End":"06:13.760","Text":"there\u0027s a plus, and a minus."},{"Start":"06:13.760 ","End":"06:15.260","Text":"We could simplify it."},{"Start":"06:15.260 ","End":"06:17.810","Text":"If you don\u0027t mind implicit functions,"},{"Start":"06:17.810 ","End":"06:19.729","Text":"you could square both sides."},{"Start":"06:19.729 ","End":"06:23.210","Text":"That would have the advantage that it gets rid of the plus or minus,"},{"Start":"06:23.210 ","End":"06:26.090","Text":"and it also would get rid of the square root."},{"Start":"06:26.090 ","End":"06:32.970","Text":"We could try 2 over 5x plus C x^4. That\u0027s an alternative."}],"Thumbnail":null,"ID":26230},{"Watched":false,"Name":"Exercise 2","Duration":"4m 46s","ChapterTopicVideoID":7614,"CourseChapterTopicPlaylistID":216828,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.285","Text":"Here, we have this differential equation to solve"},{"Start":"00:03.285 ","End":"00:07.583","Text":"and I can tell you right away that this is going to be a Bernoulli equation,"},{"Start":"00:07.583 ","End":"00:10.680","Text":"I\u0027ll just have to slightly rewrite it because"},{"Start":"00:10.680 ","End":"00:16.830","Text":"the Bernoulli equation looks like this and to remind you how we solve such a thing,"},{"Start":"00:16.830 ","End":"00:22.080","Text":"we substitute v=y^(1-n),"},{"Start":"00:22.080 ","End":"00:25.065","Text":"where n is this n here."},{"Start":"00:25.065 ","End":"00:26.850","Text":"After we do that,"},{"Start":"00:26.850 ","End":"00:31.890","Text":"we get a linear equation in v and we solve that,"},{"Start":"00:31.890 ","End":"00:35.025","Text":"at the end we substitute back from v to y."},{"Start":"00:35.025 ","End":"00:40.040","Text":"Let\u0027s begin. This is the original equation copied and then after I"},{"Start":"00:40.040 ","End":"00:45.260","Text":"move this over to the right-hand side and divide by x^2 plus 1,"},{"Start":"00:45.260 ","End":"00:47.450","Text":"then this is what we get,"},{"Start":"00:47.450 ","End":"00:54.725","Text":"which matches this, where this bit is p(x) and this bit here is q(x)."},{"Start":"00:54.725 ","End":"00:58.040","Text":"Most importantly, n=2,"},{"Start":"00:58.040 ","End":"01:00.055","Text":"we need the value of n,"},{"Start":"01:00.055 ","End":"01:02.930","Text":"and from n we can get 1 minus n,"},{"Start":"01:02.930 ","End":"01:08.210","Text":"which I marked in red the correspond with the red in the formula here."},{"Start":"01:08.210 ","End":"01:13.885","Text":"The substitution is v=y to the power of this 1 minus n,"},{"Start":"01:13.885 ","End":"01:18.231","Text":"so this is it and after we do that,"},{"Start":"01:18.231 ","End":"01:21.110","Text":"we get, and I\u0027m talking about this last line,"},{"Start":"01:21.110 ","End":"01:23.660","Text":"v\u0027 is wherever I see 1 minus n,"},{"Start":"01:23.660 ","End":"01:30.180","Text":"I put minus 1 and it\u0027s minus 1 times this and here we have minus"},{"Start":"01:30.180 ","End":"01:37.205","Text":"1 times this and this now becomes a linear equation in v,"},{"Start":"01:37.205 ","End":"01:41.450","Text":"the minus with minus cancels and this becomes of minus here."},{"Start":"01:41.450 ","End":"01:45.890","Text":"We\u0027ll just call this a and this b and then we\u0027re going to use the formula for"},{"Start":"01:45.890 ","End":"01:51.215","Text":"linear differential equations and to get the answer for what v is,"},{"Start":"01:51.215 ","End":"01:52.850","Text":"I\u0027ll do that on the next page."},{"Start":"01:52.850 ","End":"01:55.100","Text":"This is the formula in general,"},{"Start":"01:55.100 ","End":"01:56.720","Text":"we\u0027re used to y here,"},{"Start":"01:56.720 ","End":"02:00.755","Text":"but in our case it\u0027s v and there are 2 integrals to be done."},{"Start":"02:00.755 ","End":"02:04.545","Text":"We need A(x), and I should have mentioned,"},{"Start":"02:04.545 ","End":"02:09.160","Text":"you\u0027re supposed to know this is a big A is the integral of little a and"},{"Start":"02:09.160 ","End":"02:15.545","Text":"the computation is that since little a was 2x/ x^2 plus 1,"},{"Start":"02:15.545 ","End":"02:17.870","Text":"this is an easy integral because"},{"Start":"02:17.870 ","End":"02:22.490","Text":"the numerator is the derivative of the denominator and whenever that happens,"},{"Start":"02:22.490 ","End":"02:24.485","Text":"the answer is natural log."},{"Start":"02:24.485 ","End":"02:26.990","Text":"Technically, it should be absolute value here,"},{"Start":"02:26.990 ","End":"02:28.490","Text":"but since this thing is positive,"},{"Start":"02:28.490 ","End":"02:32.435","Text":"I don\u0027t need the absolute value and we also don\u0027t put the C here,"},{"Start":"02:32.435 ","End":"02:36.095","Text":"it\u0027s taken care of by the C over here."},{"Start":"02:36.095 ","End":"02:37.610","Text":"That was 1 integral,"},{"Start":"02:37.610 ","End":"02:39.950","Text":"that was the finding a,"},{"Start":"02:39.950 ","End":"02:44.390","Text":"this integral call that asterisk and the other"},{"Start":"02:44.390 ","End":"02:49.325","Text":"integral that we have to do is this bit here,"},{"Start":"02:49.325 ","End":"02:54.540","Text":"to the dx, we\u0027ll call that double asterisk and we\u0027ll compute this."},{"Start":"02:54.540 ","End":"02:56.540","Text":"Now there\u0027s always 2 integrals to perform."},{"Start":"02:56.540 ","End":"03:00.905","Text":"We do this linear using the formula and we need the,"},{"Start":"03:00.905 ","End":"03:04.610","Text":"this one has to be the first because we use this one here."},{"Start":"03:04.610 ","End":"03:08.350","Text":"What we get is the function little b,"},{"Start":"03:08.350 ","End":"03:10.130","Text":"I just copied it from above,"},{"Start":"03:10.130 ","End":"03:14.990","Text":"was minus 1/x^2 plus 1 then e to the power of"},{"Start":"03:14.990 ","End":"03:20.225","Text":"and then big A(x) is this and we use this formula often,"},{"Start":"03:20.225 ","End":"03:24.050","Text":"I\u0027m reminding you of it, e to the natural log of something is just that something."},{"Start":"03:24.050 ","End":"03:27.110","Text":"So over here, we have e to"},{"Start":"03:27.110 ","End":"03:31.370","Text":"the natural log of this thing is just this thing itself and even better,"},{"Start":"03:31.370 ","End":"03:33.110","Text":"this will now cancel."},{"Start":"03:33.110 ","End":"03:38.015","Text":"This over this, this whole thing comes out to be minus 1 and the integral of that"},{"Start":"03:38.015 ","End":"03:43.040","Text":"is minus x and now that I\u0027ve done the 2 integrals,"},{"Start":"03:43.040 ","End":"03:47.210","Text":"I can plug in this formula and we can get what v is,"},{"Start":"03:47.210 ","End":"03:52.745","Text":"it\u0027s e to the power of minus big A(x) so it\u0027s minus this."},{"Start":"03:52.745 ","End":"03:57.440","Text":"Then this integral, this double star was minus x plus,"},{"Start":"03:57.440 ","End":"03:59.750","Text":"this time we take the constant."},{"Start":"03:59.750 ","End":"04:03.545","Text":"Once again, we\u0027re going to use the properties of the exponent,"},{"Start":"04:03.545 ","End":"04:06.290","Text":"e to the power of the natural log of"},{"Start":"04:06.290 ","End":"04:09.470","Text":"something is that thing itself, but because it\u0027s a minus,"},{"Start":"04:09.470 ","End":"04:11.215","Text":"it\u0027s going to be 1 over,"},{"Start":"04:11.215 ","End":"04:13.680","Text":"so 1/x^2 plus 1,"},{"Start":"04:13.680 ","End":"04:15.660","Text":"and times the minus x plus C,"},{"Start":"04:15.660 ","End":"04:19.130","Text":"this gives us the answer for v. But don\u0027t forget,"},{"Start":"04:19.130 ","End":"04:21.950","Text":"don\u0027t stop here when you\u0027re solving an exercise because we need"},{"Start":"04:21.950 ","End":"04:25.700","Text":"to get y not v. Refer back,"},{"Start":"04:25.700 ","End":"04:30.890","Text":"you\u0027ll see that we substituted v=y to the minus 1, which is 1/y."},{"Start":"04:30.890 ","End":"04:35.420","Text":"If you reverse it, that means that y is 1/v."},{"Start":"04:35.420 ","End":"04:38.340","Text":"If I want 1/v and I have a fraction,"},{"Start":"04:38.340 ","End":"04:41.030","Text":"all I have to do is reverse numerator and"},{"Start":"04:41.030 ","End":"04:47.440","Text":"denominator and that gives us that y is this and that\u0027s the answer."}],"Thumbnail":null,"ID":26231},{"Watched":false,"Name":"Exercise 3","Duration":"3m 56s","ChapterTopicVideoID":7617,"CourseChapterTopicPlaylistID":216828,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"Here we have a differential equation to solve,"},{"Start":"00:03.240 ","End":"00:06.755","Text":"and I\u0027ll show you that it\u0027s actually a Bernoulli equation,"},{"Start":"00:06.755 ","End":"00:09.510","Text":"and I have brought in the details for Bernoulli equation,"},{"Start":"00:09.510 ","End":"00:11.190","Text":"I won\u0027t go over them again."},{"Start":"00:11.190 ","End":"00:15.705","Text":"Just have to slightly rewrite this to get it to look in this form,"},{"Start":"00:15.705 ","End":"00:18.660","Text":"y prime instead of dy by dx,"},{"Start":"00:18.660 ","End":"00:20.625","Text":"and now we divide by x."},{"Start":"00:20.625 ","End":"00:21.990","Text":"This is what we get."},{"Start":"00:21.990 ","End":"00:24.890","Text":"The x\u0027s appears, we get an x in the denominator here,"},{"Start":"00:24.890 ","End":"00:26.490","Text":"and one of the x\u0027s here"},{"Start":"00:26.490 ","End":"00:30.720","Text":"disappears because I have to add the restriction x not equal to 0."},{"Start":"00:30.720 ","End":"00:36.225","Text":"Now this is in this form where n is equal to one-half,"},{"Start":"00:36.225 ","End":"00:37.800","Text":"and if n is one-half,"},{"Start":"00:37.800 ","End":"00:39.090","Text":"then 1 minus n,"},{"Start":"00:39.090 ","End":"00:40.770","Text":"which is what we want,"},{"Start":"00:40.770 ","End":"00:43.185","Text":"is equal to one-half,"},{"Start":"00:43.185 ","End":"00:48.560","Text":"so we let v equal y^one -half."},{"Start":"00:48.560 ","End":"00:53.930","Text":"Now if I just substitute what\u0027s in this line here,"},{"Start":"00:53.930 ","End":"01:00.660","Text":"then this is what we get because our P was minus 2 over x and the Q was equal to just x,"},{"Start":"01:00.660 ","End":"01:05.000","Text":"and this simplifies the half with the 2 cancels."},{"Start":"01:05.000 ","End":"01:08.840","Text":"This, which is a linear equation,"},{"Start":"01:08.840 ","End":"01:10.685","Text":"v as a function of x,"},{"Start":"01:10.685 ","End":"01:12.230","Text":"where this is a of x,"},{"Start":"01:12.230 ","End":"01:13.525","Text":"this is b of x."},{"Start":"01:13.525 ","End":"01:18.290","Text":"We\u0027re going to use the formula for linear differential equation."},{"Start":"01:18.290 ","End":"01:21.320","Text":"The first order, this is the formula for"},{"Start":"01:21.320 ","End":"01:25.145","Text":"the solution for a linear equation when V is a function of x,"},{"Start":"01:25.145 ","End":"01:28.685","Text":"and as usual, A is the integral of a."},{"Start":"01:28.685 ","End":"01:30.874","Text":"There are two integrals to solve."},{"Start":"01:30.874 ","End":"01:35.495","Text":"The first integral is this a itself which is here,"},{"Start":"01:35.495 ","End":"01:37.190","Text":"and that\u0027s we call the asterisk,"},{"Start":"01:37.190 ","End":"01:42.245","Text":"and the other integral is this bit here called that double asterisk."},{"Start":"01:42.245 ","End":"01:46.010","Text":"But we need to do this one first because this one uses A."},{"Start":"01:46.010 ","End":"01:48.500","Text":"A integral a,"},{"Start":"01:48.500 ","End":"01:51.650","Text":"if you look at the a was minus 1 over x,"},{"Start":"01:51.650 ","End":"01:56.610","Text":"so the integral is minus natural log of x."},{"Start":"01:56.610 ","End":"02:00.260","Text":"That\u0027s just normally natural log of absolute value of x."},{"Start":"02:00.260 ","End":"02:04.410","Text":"But I\u0027m going to restrict x to be positive for convenience,"},{"Start":"02:04.410 ","End":"02:07.010","Text":"in any event x can\u0027t be 0 as we saw,"},{"Start":"02:07.010 ","End":"02:08.810","Text":"let\u0027s just consider a positive x,"},{"Start":"02:08.810 ","End":"02:11.115","Text":"so that\u0027s the A,"},{"Start":"02:11.115 ","End":"02:16.185","Text":"and now we can substitute A here from here,"},{"Start":"02:16.185 ","End":"02:21.215","Text":"and so second integral becomes b of x was one-half x."},{"Start":"02:21.215 ","End":"02:26.460","Text":"You can look back and see e to the power of now a was this."},{"Start":"02:26.460 ","End":"02:28.080","Text":"Then we use this formula,"},{"Start":"02:28.080 ","End":"02:32.210","Text":"e^ minus natural log of something is just one over that something."},{"Start":"02:32.210 ","End":"02:36.830","Text":"In our case, this bit here just comes out to be 1 over x,"},{"Start":"02:36.830 ","End":"02:38.900","Text":"so this is the integral."},{"Start":"02:38.900 ","End":"02:42.195","Text":"We get, the x and the x cancel."},{"Start":"02:42.195 ","End":"02:46.265","Text":"All I\u0027m left with is integral of the constant one-half,"},{"Start":"02:46.265 ","End":"02:48.200","Text":"and that\u0027s one-half x."},{"Start":"02:48.200 ","End":"02:50.905","Text":"We don\u0027t put a constant in at this stage."},{"Start":"02:50.905 ","End":"02:53.720","Text":"Now we have to plug in both the integrals,"},{"Start":"02:53.720 ","End":"02:58.880","Text":"the A and this expression here that we found here"},{"Start":"02:58.880 ","End":"03:05.045","Text":"and here we plug those in here and here, e^minus."},{"Start":"03:05.045 ","End":"03:08.255","Text":"Now this is A of x this bit,"},{"Start":"03:08.255 ","End":"03:14.500","Text":"and the integral came out to be one-half x,"},{"Start":"03:14.500 ","End":"03:17.535","Text":"which was this, but we add plus C,"},{"Start":"03:17.535 ","End":"03:20.510","Text":"and so again, using rules of exponents,"},{"Start":"03:20.510 ","End":"03:21.950","Text":"this comes out to be,"},{"Start":"03:21.950 ","End":"03:23.570","Text":"the minus minus cancels,"},{"Start":"03:23.570 ","End":"03:26.930","Text":"e^natural log of x is just x itself,"},{"Start":"03:26.930 ","End":"03:30.590","Text":"so this bit here is our answer for v,"},{"Start":"03:30.590 ","End":"03:33.695","Text":"but we need to go back and find y."},{"Start":"03:33.695 ","End":"03:40.130","Text":"Now the substitution was the v equals y^one-half, so reversing it,"},{"Start":"03:40.130 ","End":"03:44.960","Text":"y equals v squared and the v squared was"},{"Start":"03:44.960 ","End":"03:49.940","Text":"just v this is v. If I just square everything separately,"},{"Start":"03:49.940 ","End":"03:52.475","Text":"x squared and then this thing squared,"},{"Start":"03:52.475 ","End":"03:53.960","Text":"that will give me y,"},{"Start":"03:53.960 ","End":"03:56.370","Text":"and that\u0027s the answer."}],"Thumbnail":null,"ID":26232},{"Watched":false,"Name":"Exercise 4","Duration":"5m 41s","ChapterTopicVideoID":7612,"CourseChapterTopicPlaylistID":216828,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.430","Text":"In this exercise, we have an initial value problem,"},{"Start":"00:03.430 ","End":"00:07.920","Text":"that\u0027s a differential equation with an initial condition."},{"Start":"00:07.920 ","End":"00:13.395","Text":"The differential equation is a Bernoulli equation."},{"Start":"00:13.395 ","End":"00:17.430","Text":"We can see this is the definition of a Bernoulli equation."},{"Start":"00:17.430 ","End":"00:21.125","Text":"This bit here with the minus is my p,"},{"Start":"00:21.125 ","End":"00:25.340","Text":"and the q is minus x^3 and the n is 2."},{"Start":"00:25.340 ","End":"00:29.660","Text":"In this case we know what to do, we make a substitution of v,"},{"Start":"00:29.660 ","End":"00:34.565","Text":"and solve the linear equation in v that in the end we substitute back."},{"Start":"00:34.565 ","End":"00:37.250","Text":"I need this,1 minus n,"},{"Start":"00:37.250 ","End":"00:39.540","Text":"since n is 2,"},{"Start":"00:39.980 ","End":"00:45.645","Text":"1 minus is negative 1 so the substitution we make is v is y to the power of negative 1."},{"Start":"00:45.645 ","End":"00:50.630","Text":"Then I need to rewrite this with our p and q."},{"Start":"00:50.630 ","End":"00:52.415","Text":"This is what we get,"},{"Start":"00:52.415 ","End":"00:55.070","Text":"remembering that 1 minus n is minus 1,"},{"Start":"00:55.070 ","End":"00:57.520","Text":"it\u0027s all marked in red so you can\u0027t go wrong."},{"Start":"00:57.520 ","End":"00:59.570","Text":"Then the minus simplification,"},{"Start":"00:59.570 ","End":"01:03.829","Text":"the minus with the minus cancels so it does here also,"},{"Start":"01:03.829 ","End":"01:05.585","Text":"this is what we end up with,"},{"Start":"01:05.585 ","End":"01:08.900","Text":"which is a linear differential equation,"},{"Start":"01:08.900 ","End":"01:10.730","Text":"v as a function of x."},{"Start":"01:10.730 ","End":"01:13.780","Text":"This function we call a and this one we call b,"},{"Start":"01:13.780 ","End":"01:16.204","Text":"then we plug in to the famous formula."},{"Start":"01:16.204 ","End":"01:18.040","Text":"I\u0027ll do that on the next page,"},{"Start":"01:18.040 ","End":"01:20.240","Text":"and this is the formula,"},{"Start":"01:20.240 ","End":"01:21.740","Text":"usually it\u0027s for the y,"},{"Start":"01:21.740 ","End":"01:27.425","Text":"but in our case we have a v. Remember that A is the integral of a."},{"Start":"01:27.425 ","End":"01:29.720","Text":"Basically we have two rules to do."},{"Start":"01:29.720 ","End":"01:33.815","Text":"We have to compute A and then we have the integral also over here,"},{"Start":"01:33.815 ","End":"01:35.960","Text":"this one I\u0027ll call asterisk,"},{"Start":"01:35.960 ","End":"01:39.725","Text":"and this integral double Asterisk and here they are."},{"Start":"01:39.725 ","End":"01:41.650","Text":"A is the integral of a,"},{"Start":"01:41.650 ","End":"01:43.800","Text":"a if you look back,"},{"Start":"01:43.800 ","End":"01:46.510","Text":"was this function here, 1 over x plus 5x^4."},{"Start":"01:46.700 ","End":"01:51.370","Text":"So we get the natural log of x plus x^5."},{"Start":"01:51.380 ","End":"01:55.340","Text":"Let\u0027s just assume that x is bigger than 0,"},{"Start":"01:55.340 ","End":"02:00.995","Text":"so we don\u0027t need the absolute value and we\u0027ll just restrict our x to positive numbers."},{"Start":"02:00.995 ","End":"02:02.870","Text":"Once we found A,"},{"Start":"02:02.870 ","End":"02:05.045","Text":"we can substitute that in here,"},{"Start":"02:05.045 ","End":"02:10.575","Text":"and b(x) was x^3 so we have this expression."},{"Start":"02:10.575 ","End":"02:13.955","Text":"I want to simplify this and use some rules of logs."},{"Start":"02:13.955 ","End":"02:16.650","Text":"One rule of logarithms is that if you have a plus in"},{"Start":"02:16.650 ","End":"02:19.500","Text":"the exponent it converts to a product."},{"Start":"02:19.500 ","End":"02:24.945","Text":"The other thing we can do is that e to the power of natural Iog(x) is just x."},{"Start":"02:24.945 ","End":"02:31.010","Text":"The integral we need to compute x with x^3 is x^4 is this one."},{"Start":"02:31.010 ","End":"02:37.140","Text":"Now, the derivative of x^5 is 5x^4 and if I had a 5 here,"},{"Start":"02:37.140 ","End":"02:39.545","Text":"I\u0027ll better off because then I\u0027d have a template of"},{"Start":"02:39.545 ","End":"02:42.155","Text":"e to the something and that something derivative,"},{"Start":"02:42.155 ","End":"02:44.559","Text":"but if I needed 5 here no problem,"},{"Start":"02:44.559 ","End":"02:45.980","Text":"just write a 5 here,"},{"Start":"02:45.980 ","End":"02:49.940","Text":"but to compensate, I write a 1/5 in front."},{"Start":"02:49.940 ","End":"02:54.105","Text":"Now this integral is just the 1/5 from here,"},{"Start":"02:54.105 ","End":"02:56.085","Text":"and this is e^x^5."},{"Start":"02:56.085 ","End":"02:59.810","Text":"I\u0027ll say again that if we have"},{"Start":"02:59.810 ","End":"03:04.270","Text":"the derivative of something and then e to the power of something,"},{"Start":"03:04.270 ","End":"03:09.290","Text":"then this integral is just e to the power of that something."},{"Start":"03:09.290 ","End":"03:12.230","Text":"Because you can see by differentiating this from"},{"Start":"03:12.230 ","End":"03:15.965","Text":"the chain rule that we get e to the something and then the inner derivative."},{"Start":"03:15.965 ","End":"03:17.780","Text":"Anyway, we\u0027ve seen this sort of thing before,"},{"Start":"03:17.780 ","End":"03:20.285","Text":"we don\u0027t need the constant at this stage,"},{"Start":"03:20.285 ","End":"03:23.810","Text":"so that gives us one of the integrals."},{"Start":"03:23.810 ","End":"03:25.380","Text":"This was the double asterisk,"},{"Start":"03:25.380 ","End":"03:30.195","Text":"this was this part here and we also figured this out."},{"Start":"03:30.195 ","End":"03:33.980","Text":"Just substituting what we got just off screen at"},{"Start":"03:33.980 ","End":"03:38.490","Text":"the moment but A(x) was once written here, and this here,"},{"Start":"03:40.360 ","End":"03:46.620","Text":"which I wrote as e^x^5 over 5 plus C. Well,"},{"Start":"03:46.620 ","End":"03:48.930","Text":"I slightly adopted it instead of C,"},{"Start":"03:48.930 ","End":"03:53.165","Text":"it could\u0027ve been C over 5 because C over 5 is a general constant,"},{"Start":"03:53.165 ","End":"03:58.460","Text":"just as much as C. The reason I did that is that then I can write it a bit more neatly."},{"Start":"03:58.460 ","End":"04:01.195","Text":"I can put the C on the numerator here,"},{"Start":"04:01.195 ","End":"04:05.390","Text":"and then we\u0027ve got the answer for what v equals,"},{"Start":"04:05.390 ","End":"04:10.130","Text":"just combining this into the denominator but we\u0027re far from done."},{"Start":"04:10.130 ","End":"04:12.620","Text":"For one thing, we have to get back to y"},{"Start":"04:12.620 ","End":"04:15.275","Text":"and then there\u0027s the matter of the initial condition,"},{"Start":"04:15.275 ","End":"04:17.665","Text":"there\u0027s still some more work to do."},{"Start":"04:17.665 ","End":"04:21.705","Text":"If you look back, you\u0027ll see that v was y to the minus 1,"},{"Start":"04:21.705 ","End":"04:24.885","Text":"which is 1 over y and if v is 1 over y,"},{"Start":"04:24.885 ","End":"04:29.900","Text":"then y is 1 over v. So all I have to do now is reverse this,"},{"Start":"04:29.900 ","End":"04:34.165","Text":"the top and the bottom reciprocal, and that gives me y."},{"Start":"04:34.165 ","End":"04:38.720","Text":"That\u0027s the general answer for y but we had an initial condition."},{"Start":"04:38.720 ","End":"04:46.590","Text":"We had y(1) is 2.5, which means that when x is 1, y is 2.5."},{"Start":"04:46.590 ","End":"04:50.315","Text":"So we substitute that here and rewriting,"},{"Start":"04:50.315 ","End":"04:51.620","Text":"let\u0027s not use decimal,"},{"Start":"04:51.620 ","End":"04:53.360","Text":"I\u0027ll use a fraction,"},{"Start":"04:53.360 ","End":"05:01.595","Text":"5 over 2 is 2 and 1/2 so 5 over 2 or just substituting x equals 1, 1^5 is 1."},{"Start":"05:01.595 ","End":"05:07.520","Text":"So this is 5 times 1 times e. This is just 5e over e plus C. Now I"},{"Start":"05:07.520 ","End":"05:13.965","Text":"can divide both sides by 5 so I\u0027ve got 1/2 is e over this."},{"Start":"05:13.965 ","End":"05:18.750","Text":"Now cross-multiply, e plus C equals 2e,"},{"Start":"05:18.750 ","End":"05:21.330","Text":"and that gives us our constant C,"},{"Start":"05:21.330 ","End":"05:24.990","Text":"which happens to be equal to the number e. What we"},{"Start":"05:24.990 ","End":"05:28.915","Text":"will have to do is plug in the value of C over here,"},{"Start":"05:28.915 ","End":"05:30.240","Text":"and as we do that,"},{"Start":"05:30.240 ","End":"05:34.220","Text":"this is what we get and this is the final answer"},{"Start":"05:34.220 ","End":"05:38.675","Text":"for y the differential equation with the initial condition."},{"Start":"05:38.675 ","End":"05:41.850","Text":"This is the answer and we\u0027re done."}],"Thumbnail":null,"ID":26233},{"Watched":false,"Name":"Exercise 5","Duration":"4m 42s","ChapterTopicVideoID":7615,"CourseChapterTopicPlaylistID":216828,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.585","Text":"Here we have a differential equation to solve,"},{"Start":"00:03.585 ","End":"00:08.710","Text":"it\u0027s slightly different is that for a change instead of y we have z,"},{"Start":"00:08.710 ","End":"00:10.905","Text":"so if you want to think of it,"},{"Start":"00:10.905 ","End":"00:14.310","Text":"just replace z with y and you have the standard letters,"},{"Start":"00:14.310 ","End":"00:16.200","Text":"but it shouldn\u0027t make any difference,"},{"Start":"00:16.200 ","End":"00:19.530","Text":"the practice if we use z."},{"Start":"00:19.530 ","End":"00:21.300","Text":"It is a Bernoulli equation,"},{"Start":"00:21.300 ","End":"00:25.410","Text":"which is why I put this box in with the summary of the Bernoulli equation."},{"Start":"00:25.410 ","End":"00:30.135","Text":"It\u0027s written for variable y but it will work just as well if we have z."},{"Start":"00:30.135 ","End":"00:34.350","Text":"The p(x) is minus cotangent (x),"},{"Start":"00:34.350 ","End":"00:37.335","Text":"q(x) is 1 over sine x,"},{"Start":"00:37.335 ","End":"00:40.385","Text":"and the n will be 3."},{"Start":"00:40.385 ","End":"00:44.030","Text":"The domain is restricted from 0 to Pi,"},{"Start":"00:44.030 ","End":"00:47.720","Text":"and when x is strictly between 0 and 180 degrees,"},{"Start":"00:47.720 ","End":"00:50.735","Text":"the sine of x will never be 0, so we\u0027re okay."},{"Start":"00:50.735 ","End":"00:55.250","Text":"Also, the cotangent of x is cosine over sine,"},{"Start":"00:55.250 ","End":"01:00.080","Text":"it\u0027s also going to work from 0-180 degrees."},{"Start":"01:00.080 ","End":"01:03.025","Text":"Let me just copy this down here."},{"Start":"01:03.025 ","End":"01:06.510","Text":"Now as I said, the n is 3,"},{"Start":"01:06.510 ","End":"01:08.505","Text":"I need 1 minus n,"},{"Start":"01:08.505 ","End":"01:12.975","Text":"1 minus n is 1 minus 3, is minus 2."},{"Start":"01:12.975 ","End":"01:17.400","Text":"The substitution is V = not y but z,"},{"Start":"01:17.400 ","End":"01:19.100","Text":"to the minus 2."},{"Start":"01:19.100 ","End":"01:22.770","Text":"Then we substitute in this last equation,"},{"Start":"01:22.770 ","End":"01:26.405","Text":"this is what we\u0027re guaranteed to get if we do this substitution,"},{"Start":"01:26.405 ","End":"01:30.395","Text":"and we get a linear equation in v,"},{"Start":"01:30.395 ","End":"01:33.480","Text":"v\u0027 minus with the minus cancels,"},{"Start":"01:33.480 ","End":"01:35.145","Text":"it\u0027s the 2 cotangent,"},{"Start":"01:35.145 ","End":"01:39.089","Text":"and I just put the 2 into the numerator here."},{"Start":"01:39.089 ","End":"01:46.265","Text":"This is like my a(x) and b(x) in the standard format for linear differential equation."},{"Start":"01:46.265 ","End":"01:48.920","Text":"Then I\u0027ll continue on the next page,"},{"Start":"01:48.920 ","End":"01:50.600","Text":"we\u0027ll do this with the formula."},{"Start":"01:50.600 ","End":"01:54.020","Text":"This is the formula for the solution of a linear equation,"},{"Start":"01:54.020 ","End":"02:01.160","Text":"but in our case we have v and capital A is just the integral of little a,"},{"Start":"02:01.160 ","End":"02:02.990","Text":"and let\u0027s compute that for us."},{"Start":"02:02.990 ","End":"02:06.990","Text":"The asterisk is just to note that there are 2 integrals to solve,"},{"Start":"02:06.990 ","End":"02:12.115","Text":"this is 1 integral and the other integral is this bit here"},{"Start":"02:12.115 ","End":"02:17.950","Text":"that\u0027s like the double asterisk and a is the single asterisks 2 integral."},{"Start":"02:17.950 ","End":"02:20.370","Text":"The first one, integral of a,"},{"Start":"02:20.370 ","End":"02:21.720","Text":"a if you look back,"},{"Start":"02:21.720 ","End":"02:24.795","Text":"is 2 cotangent of x,"},{"Start":"02:24.795 ","End":"02:27.205","Text":"and to take out front."},{"Start":"02:27.205 ","End":"02:29.230","Text":"The integral of cotangent,"},{"Start":"02:29.230 ","End":"02:34.710","Text":"happens to be natural log of absolute value of sine x,"},{"Start":"02:34.710 ","End":"02:38.950","Text":"actually don\u0027t need the absolute value, but doesn\u0027t hurt."},{"Start":"02:38.950 ","End":"02:40.720","Text":"That\u0027s 1 integral,"},{"Start":"02:40.720 ","End":"02:42.955","Text":"we don\u0027t need the constant at this stage."},{"Start":"02:42.955 ","End":"02:44.795","Text":"Now let\u0027s do the other integral,"},{"Start":"02:44.795 ","End":"02:47.355","Text":"that\u0027s this part over here,"},{"Start":"02:47.355 ","End":"02:48.930","Text":"b(x), if you look back,"},{"Start":"02:48.930 ","End":"02:51.135","Text":"was minus 2 over sine x."},{"Start":"02:51.135 ","End":"02:53.930","Text":"Big A, we\u0027ve got from here,"},{"Start":"02:53.930 ","End":"02:55.565","Text":"we plug that into here."},{"Start":"02:55.565 ","End":"02:59.030","Text":"Then remember the rules of exponents."},{"Start":"02:59.030 ","End":"03:08.060","Text":"One of the rules is what\u0027s here we have it with k equals 2 and the box is sine x."},{"Start":"03:08.060 ","End":"03:10.940","Text":"That\u0027s simplifies this bit."},{"Start":"03:10.940 ","End":"03:14.620","Text":"Then we can also cancel with sine x."},{"Start":"03:14.620 ","End":"03:18.570","Text":"All we\u0027re left with is minus 2 sine x,"},{"Start":"03:18.570 ","End":"03:20.085","Text":"the integral of that,"},{"Start":"03:20.085 ","End":"03:23.300","Text":"and that gives us 2 cosine x."},{"Start":"03:23.300 ","End":"03:29.149","Text":"Now all I have to do is substitute in that bit of screen now that formula for the linear."},{"Start":"03:29.149 ","End":"03:30.925","Text":"We have the 2 integrals,"},{"Start":"03:30.925 ","End":"03:33.345","Text":"the formula says V equals this,"},{"Start":"03:33.345 ","End":"03:35.660","Text":"capital A, we have over here,"},{"Start":"03:35.660 ","End":"03:38.330","Text":"this part is over here."},{"Start":"03:38.330 ","End":"03:40.985","Text":"If I make the substitution,"},{"Start":"03:40.985 ","End":"03:42.565","Text":"this is what we get,"},{"Start":"03:42.565 ","End":"03:45.915","Text":"A is that, B is this."},{"Start":"03:45.915 ","End":"03:49.505","Text":"Again, we\u0027re going to simplify using this rule for"},{"Start":"03:49.505 ","End":"03:54.620","Text":"the exponents with the logarithms with k being 2 here."},{"Start":"03:54.620 ","End":"03:56.540","Text":"All of this boils down to this,"},{"Start":"03:56.540 ","End":"04:02.635","Text":"but that\u0027s just v. We\u0027re not done because we need to get back to y,"},{"Start":"04:02.635 ","End":"04:05.210","Text":"only in our case it\u0027s not y it\u0027s z."},{"Start":"04:05.210 ","End":"04:07.655","Text":"We have to be different in this exercise."},{"Start":"04:07.655 ","End":"04:10.300","Text":"V was z to the minus 2,"},{"Start":"04:10.300 ","End":"04:14.670","Text":"so if you just compute that v is 1 over z squared,"},{"Start":"04:14.670 ","End":"04:16.110","Text":"so z squared is 1 over V,"},{"Start":"04:16.110 ","End":"04:18.650","Text":"so z is plus or minus 1 over"},{"Start":"04:18.650 ","End":"04:22.600","Text":"the square root of v. Now just have to put instead of v this here,"},{"Start":"04:22.600 ","End":"04:24.470","Text":"but I also did 1 extra step."},{"Start":"04:24.470 ","End":"04:26.135","Text":"Instead of having the reciprocal,"},{"Start":"04:26.135 ","End":"04:28.790","Text":"I just switched the top and the bottom."},{"Start":"04:28.790 ","End":"04:33.300","Text":"Whoops, it looks like I forgot the 2,"},{"Start":"04:33.300 ","End":"04:36.212","Text":"well, let me just stick it in here, a 2."},{"Start":"04:36.212 ","End":"04:38.285","Text":"Sorry about that typo."},{"Start":"04:38.285 ","End":"04:40.354","Text":"Anyway, this is the answer."},{"Start":"04:40.354 ","End":"04:42.690","Text":"We\u0027re done."}],"Thumbnail":null,"ID":26234}],"ID":216828},{"Name":"Clairaut Equation","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"9m 42s","ChapterTopicVideoID":28318,"CourseChapterTopicPlaylistID":280701,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.040","Text":"Hi. In this video,"},{"Start":"00:02.040 ","End":"00:05.865","Text":"we\u0027re looking at the differential equation,"},{"Start":"00:05.865 ","End":"00:13.315","Text":"y equals x times dy by dx minus the cosine of dy by dx,"},{"Start":"00:13.315 ","End":"00:17.655","Text":"and we\u0027re being asked here to derive the family of solutions"},{"Start":"00:17.655 ","End":"00:23.310","Text":"and the singular solution that solves this differential equation here."},{"Start":"00:23.310 ","End":"00:27.390","Text":"This differential equation is a special type of"},{"Start":"00:27.390 ","End":"00:31.200","Text":"differential equation and it has the name of,"},{"Start":"00:31.200 ","End":"00:33.795","Text":"it\u0027s called a Clairaut\u0027s equation."},{"Start":"00:33.795 ","End":"00:37.140","Text":"Why this is worth mentioning is because if you see"},{"Start":"00:37.140 ","End":"00:42.120","Text":"a differential equation in the future that matches this form exactly,"},{"Start":"00:42.120 ","End":"00:45.740","Text":"then you can apply the method that\u0027s used in this video"},{"Start":"00:45.740 ","End":"00:51.440","Text":"today almost exactly to solve that differential equation as well."},{"Start":"00:51.440 ","End":"00:59.030","Text":"A Clairaut\u0027s equation is one that can be expressed in the form y is equal"},{"Start":"00:59.030 ","End":"01:06.450","Text":"to x multiplied by some function Psi of dy by dx,"},{"Start":"01:06.450 ","End":"01:09.075","Text":"so here we\u0027re just replacing dy by dx with"},{"Start":"01:09.075 ","End":"01:12.390","Text":"y dash just because it\u0027s a bit easier to write,"},{"Start":"01:12.390 ","End":"01:18.780","Text":"and then we have plus another function Phi of y dash."},{"Start":"01:18.780 ","End":"01:22.519","Text":"You can solve C immediately the things that match up."},{"Start":"01:22.519 ","End":"01:25.860","Text":"Of course, the y\u0027s here match up,"},{"Start":"01:25.860 ","End":"01:29.715","Text":"and then we\u0027ve got the x\u0027s here that match up."},{"Start":"01:29.715 ","End":"01:37.395","Text":"Then here, our function Psi of y\u0027 is just dy by dx and then our function"},{"Start":"01:37.395 ","End":"01:47.350","Text":"Phi of y\u0027 is just this cosine dy by dx or cosine of y\u0027."},{"Start":"01:47.350 ","End":"01:50.600","Text":"Anything that has this exact form is known as"},{"Start":"01:50.600 ","End":"01:56.360","Text":"a Clairaut\u0027s equation and then we\u0027re going to see how we\u0027re going to solve this now."},{"Start":"01:56.360 ","End":"02:02.585","Text":"Let\u0027s just rewrite our differential equation here in a bit of a simpler form."},{"Start":"02:02.585 ","End":"02:09.080","Text":"We\u0027re going to have it as y\u0027 is equal to x,"},{"Start":"02:09.080 ","End":"02:13.650","Text":"y\u0027 minus cosine and"},{"Start":"02:13.650 ","End":"02:18.950","Text":"then in here remember we just have y dash because we just have cosine of dy by dx."},{"Start":"02:18.950 ","End":"02:21.695","Text":"Now the first step in solving"},{"Start":"02:21.695 ","End":"02:28.130","Text":"a Clairaut\u0027s equation is to just differentiate the whole thing with respect to x."},{"Start":"02:28.130 ","End":"02:31.085","Text":"It might seem like a bit of an arbitrary step,"},{"Start":"02:31.085 ","End":"02:32.180","Text":"but when we do it,"},{"Start":"02:32.180 ","End":"02:35.990","Text":"then hopefully it will appear a bit more obvious as to why we do it."},{"Start":"02:35.990 ","End":"02:43.220","Text":"Sorry, this should just be a y remember,"},{"Start":"02:43.220 ","End":"02:46.400","Text":"so if we differentiate everything with respect to x,"},{"Start":"02:46.400 ","End":"02:51.530","Text":"then the left-hand side is just going to be a y\u0027 and the right-hand side,"},{"Start":"02:51.530 ","End":"02:53.105","Text":"this first bit here,"},{"Start":"02:53.105 ","End":"02:55.400","Text":"we\u0027re going to have to use the product rule,"},{"Start":"02:55.400 ","End":"02:59.360","Text":"so we are to do x times y\u0027\u0027,"},{"Start":"02:59.360 ","End":"03:00.770","Text":"so that\u0027s this first bit,"},{"Start":"03:00.770 ","End":"03:03.200","Text":"multiplied by the derivative of y\u0027,"},{"Start":"03:03.200 ","End":"03:09.890","Text":"and then we\u0027ve got plus y\u0027 multiplied by the derivative of x which is just 1."},{"Start":"03:09.890 ","End":"03:11.240","Text":"Then the final bit,"},{"Start":"03:11.240 ","End":"03:15.140","Text":"which might seem a little bit more complicated,"},{"Start":"03:15.140 ","End":"03:17.225","Text":"we just use the chain rule here,"},{"Start":"03:17.225 ","End":"03:21.720","Text":"and then we just get plus sine of y\u0027."},{"Start":"03:21.720 ","End":"03:26.585","Text":"Then remember we have to multiply by the derivative of what\u0027s in the inside,"},{"Start":"03:26.585 ","End":"03:29.885","Text":"so we just get y\u0027\u0027 here."},{"Start":"03:29.885 ","End":"03:32.750","Text":"Let\u0027s just make a note of what we did here,"},{"Start":"03:32.750 ","End":"03:38.989","Text":"so we differentiate it with respect to x."},{"Start":"03:38.989 ","End":"03:43.250","Text":"Now, you can see already that these y primes"},{"Start":"03:43.250 ","End":"03:48.980","Text":"will cancel and then so what we\u0027re just left with here is x,"},{"Start":"03:48.980 ","End":"03:56.900","Text":"y\u0027\u0027 plus sine of y\u0027 multiplied by"},{"Start":"03:56.900 ","End":"04:06.170","Text":"y\u0027\u0027 is equal to 0 because remember we\u0027ve just removed this y\u0027 from the other side."},{"Start":"04:06.170 ","End":"04:11.150","Text":"Now if we factor out a y\u0027\u0027,"},{"Start":"04:11.150 ","End":"04:12.954","Text":"then we get y\u0027\u0027,"},{"Start":"04:12.954 ","End":"04:18.770","Text":"and then in this bracket it\u0027s x plus sine of y\u0027,"},{"Start":"04:18.770 ","End":"04:24.965","Text":"and then remember big bracket is equal to 0."},{"Start":"04:24.965 ","End":"04:27.665","Text":"We have 2 options here."},{"Start":"04:27.665 ","End":"04:31.820","Text":"We can either have it so that y\u0027\u0027 is equal to"},{"Start":"04:31.820 ","End":"04:38.725","Text":"0 or we can have it that x plus sine of y\u0027 is equal to 0."},{"Start":"04:38.725 ","End":"04:42.985","Text":"Let\u0027s explore what happens in both of these 2 cases."},{"Start":"04:42.985 ","End":"04:44.710","Text":"In this first case,"},{"Start":"04:44.710 ","End":"04:48.430","Text":"which might be the most simple of the 2 cases,"},{"Start":"04:48.430 ","End":"04:51.850","Text":"we have y\u0027\u0027 is equal to 0."},{"Start":"04:51.850 ","End":"05:02.725","Text":"This is the same as d^2y over dx^2 is equal to 0."},{"Start":"05:02.725 ","End":"05:06.760","Text":"What we can do here is to find y on its own,"},{"Start":"05:06.760 ","End":"05:11.605","Text":"then we can just integrate both sides with respect to x twice."},{"Start":"05:11.605 ","End":"05:14.125","Text":"On integrating the first time,"},{"Start":"05:14.125 ","End":"05:19.510","Text":"we just get dy by dx is equal to,"},{"Start":"05:19.510 ","End":"05:21.835","Text":"let\u0027s just say some constant A."},{"Start":"05:21.835 ","End":"05:27.640","Text":"Then integrating this again gives us y(x) is equal"},{"Start":"05:27.640 ","End":"05:31.750","Text":"to Ax plus B and maybe we"},{"Start":"05:31.750 ","End":"05:36.505","Text":"should call this y1 of x because we can see that we\u0027ve got 2 cases here."},{"Start":"05:36.505 ","End":"05:40.840","Text":"This is the first solution to this differential equation."},{"Start":"05:40.840 ","End":"05:44.305","Text":"Now let\u0027s explore what happens in Case 2."},{"Start":"05:44.305 ","End":"05:50.020","Text":"Here we have x plus sine of y\u0027 is equal to 0."},{"Start":"05:50.020 ","End":"05:54.850","Text":"Let\u0027s bring this sine of y\u0027 or this x to the other side,"},{"Start":"05:54.850 ","End":"05:57.485","Text":"so then we get sine of,"},{"Start":"05:57.485 ","End":"06:00.955","Text":"and then we\u0027re going to write the dy by dx now,"},{"Start":"06:00.955 ","End":"06:04.745","Text":"is equal to minus x."},{"Start":"06:04.745 ","End":"06:07.670","Text":"Now how did we get dy by dx on its own?"},{"Start":"06:07.670 ","End":"06:11.195","Text":"Well, we have to take the arcsine of both sides."},{"Start":"06:11.195 ","End":"06:20.210","Text":"That gives us the dy by dx is equal to arcsine of minus x."},{"Start":"06:20.210 ","End":"06:25.655","Text":"Now, the interesting property about the function arcsine is that it\u0027s an odd function,"},{"Start":"06:25.655 ","End":"06:34.020","Text":"so that means the arcsine of minus x is equal to minus arcsine of x."},{"Start":"06:34.020 ","End":"06:35.660","Text":"Now this isn\u0027t a necessary step,"},{"Start":"06:35.660 ","End":"06:39.175","Text":"it\u0027s just usually what we do for convenience."},{"Start":"06:39.175 ","End":"06:42.755","Text":"Now, if we want to find y on its own,"},{"Start":"06:42.755 ","End":"06:46.580","Text":"then you can see this is a separable differential equation."},{"Start":"06:46.580 ","End":"06:49.189","Text":"So if we bring the dx to the other side,"},{"Start":"06:49.189 ","End":"06:54.870","Text":"then what we get here is we get dy is equal to"},{"Start":"06:54.870 ","End":"07:05.400","Text":"minus arcsine of x dx and then we can just integrate both sides."},{"Start":"07:05.400 ","End":"07:08.600","Text":"Let\u0027s just tidy this up a little bit."},{"Start":"07:08.600 ","End":"07:11.540","Text":"Now if we integrate the left-hand side,"},{"Start":"07:11.540 ","End":"07:13.895","Text":"then that\u0027s just going to give us our y,"},{"Start":"07:13.895 ","End":"07:16.355","Text":"and remember which is a function of x,"},{"Start":"07:16.355 ","End":"07:19.205","Text":"and then integrating the right-hand side."},{"Start":"07:19.205 ","End":"07:20.990","Text":"Well, this is a bit of a tricky one,"},{"Start":"07:20.990 ","End":"07:24.919","Text":"so you may need to re-consult your integrals."},{"Start":"07:24.919 ","End":"07:29.930","Text":"But the integral of arcsine of x is equal to"},{"Start":"07:29.930 ","End":"07:34.305","Text":"x arcsine of x"},{"Start":"07:34.305 ","End":"07:41.130","Text":"plus and then that\u0027s going to be 1 minus x^2,"},{"Start":"07:41.130 ","End":"07:43.140","Text":"square root it,"},{"Start":"07:43.140 ","End":"07:45.830","Text":"and then we just have some constant function,"},{"Start":"07:45.830 ","End":"07:50.180","Text":"we might want to call it C here because we\u0027ve used A and B already."},{"Start":"07:50.180 ","End":"07:53.150","Text":"This is our other solutions,"},{"Start":"07:53.150 ","End":"07:57.770","Text":"so we will call this y_2(x) and that is just equal"},{"Start":"07:57.770 ","End":"08:02.750","Text":"to minus and then x arcsine of x plus the square root of"},{"Start":"08:02.750 ","End":"08:07.705","Text":"1 minus x^2 plus C. But remember the question asked"},{"Start":"08:07.705 ","End":"08:13.915","Text":"us to find the singular solution and the family of solutions."},{"Start":"08:13.915 ","End":"08:18.845","Text":"How do we determine which is the singular one and which is the family?"},{"Start":"08:18.845 ","End":"08:24.350","Text":"Well, a singular solution is one in which there\u0027s"},{"Start":"08:24.350 ","End":"08:33.035","Text":"an interval of values of x where the function is not defined."},{"Start":"08:33.035 ","End":"08:35.870","Text":"We can see in y_1(x),"},{"Start":"08:35.870 ","End":"08:40.205","Text":"no matter what our choices are or choice we take for x,"},{"Start":"08:40.205 ","End":"08:43.025","Text":"this function will be defined,"},{"Start":"08:43.025 ","End":"08:45.589","Text":"so that can\u0027t be our singular solution."},{"Start":"08:45.589 ","End":"08:48.530","Text":"But if we look at our y_2(x),"},{"Start":"08:48.530 ","End":"08:53.345","Text":"we can see that this is not defined for modulus of"},{"Start":"08:53.345 ","End":"09:00.545","Text":"x is greater than 1 because we have this square roots of 1 minus x^2."},{"Start":"09:00.545 ","End":"09:03.575","Text":"So if the modulus of x is greater than 1,"},{"Start":"09:03.575 ","End":"09:08.255","Text":"then this is going to give us a complex or imaginary number here."},{"Start":"09:08.255 ","End":"09:14.270","Text":"Our y_2(x) is our singular solution and"},{"Start":"09:14.270 ","End":"09:21.065","Text":"this therefore implies that our y_1(x) is our family of solutions."},{"Start":"09:21.065 ","End":"09:27.290","Text":"We\u0027ve just solved this specific case of a Clairaut\u0027s differential equation and remember,"},{"Start":"09:27.290 ","End":"09:33.000","Text":"if you see 1 of these in future that matches the form that we discussed earlier,"},{"Start":"09:33.000 ","End":"09:34.615","Text":"then this method here,"},{"Start":"09:34.615 ","End":"09:40.565","Text":"where we differentiate and then solve can be applied directly as well to that."},{"Start":"09:40.565 ","End":"09:43.260","Text":"Okay, thank you very much."}],"Thumbnail":null,"ID":29553}],"ID":280701}]

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