Nonhomogeneous Equation around Regular Point
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Homogeneous Equation around Regular-Singular Point
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[{"Name":"Nonhomogeneous Equation around Regular Point","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Nonhomogeneous Equation around Regular Point","Duration":"7m 45s","ChapterTopicVideoID":7817,"CourseChapterTopicPlaylistID":4243,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this clip we\u0027ll be learning how to solve second order linear"},{"Start":"00:03.630 ","End":"00:07.050","Text":"non-homogeneous ordinary differential equations and"},{"Start":"00:07.050 ","End":"00:10.170","Text":"we\u0027re going to be solving them with power series"},{"Start":"00:10.170 ","End":"00:14.175","Text":"around a regular point which will shortly be defined."},{"Start":"00:14.175 ","End":"00:17.040","Text":"I suggest you review power series,"},{"Start":"00:17.040 ","End":"00:22.005","Text":"especially Taylor and Maclaurin series before you continue."},{"Start":"00:22.005 ","End":"00:23.760","Text":"The setup is this,"},{"Start":"00:23.760 ","End":"00:28.800","Text":"we have an ordinary differential equation of this form as written,"},{"Start":"00:28.800 ","End":"00:34.740","Text":"and we assume that x equals naught is irregular point shortly to be defined."},{"Start":"00:34.740 ","End":"00:42.290","Text":"We also assume that p and q are polynomials or possibly rational functions."},{"Start":"00:42.290 ","End":"00:46.025","Text":"Rational as you recall means the quotient of two polynomials."},{"Start":"00:46.025 ","End":"00:47.660","Text":"We can actually generalize this,"},{"Start":"00:47.660 ","End":"00:51.455","Text":"but for the moment we\u0027ll focus on these kinds of p and q."},{"Start":"00:51.455 ","End":"00:55.970","Text":"What we\u0027ll be doing is to give a recipe algorithm for solving"},{"Start":"00:55.970 ","End":"01:00.890","Text":"this equation by developing the solution as a power series around x"},{"Start":"01:00.890 ","End":"01:03.170","Text":"equals naught which is also"},{"Start":"01:03.170 ","End":"01:07.670","Text":"a Maclaurin series and then after that we\u0027re going to generalize from x"},{"Start":"01:07.670 ","End":"01:10.090","Text":"equals naught to any irregular point x equals"},{"Start":"01:10.090 ","End":"01:15.200","Text":"some x_0 and that\u0027s why I asked you to review Taylor series."},{"Start":"01:15.200 ","End":"01:16.820","Text":"I mentioned regular points,"},{"Start":"01:16.820 ","End":"01:18.295","Text":"it\u0027s time to define them."},{"Start":"01:18.295 ","End":"01:23.615","Text":"A point x=x_0 is called a regular point of the equation above,"},{"Start":"01:23.615 ","End":"01:25.130","Text":"if both p,"},{"Start":"01:25.130 ","End":"01:28.950","Text":"and q are defined at that point."},{"Start":"01:28.950 ","End":"01:31.665","Text":"I\u0027ll give a couple of examples,"},{"Start":"01:31.665 ","End":"01:38.683","Text":"x equals naught is a regular point of this differential equation because p(x),"},{"Start":"01:38.683 ","End":"01:41.420","Text":"and q(x) are defined everywhere in particular at"},{"Start":"01:41.420 ","End":"01:44.570","Text":"x equals naught and now I\u0027m going to give an example"},{"Start":"01:44.570 ","End":"01:47.750","Text":"that something is not x equals naught is not"},{"Start":"01:47.750 ","End":"01:51.545","Text":"a regular point of this differential equation."},{"Start":"01:51.545 ","End":"01:53.480","Text":"Notice the difference, it looks similar,"},{"Start":"01:53.480 ","End":"01:56.720","Text":"but look at this 1 over x for p and"},{"Start":"01:56.720 ","End":"02:01.100","Text":"p(x) isn\u0027t defined at x equals naught so not a regular point."},{"Start":"02:01.100 ","End":"02:03.755","Text":"Now we come to the algorithm,"},{"Start":"02:03.755 ","End":"02:07.744","Text":"meaning how to solve such an equation with power series."},{"Start":"02:07.744 ","End":"02:12.080","Text":"Before I get into it I just wanted to tell you that it probably won\u0027t make sense."},{"Start":"02:12.080 ","End":"02:15.505","Text":"It\u0027s abstract, but after you\u0027ve done"},{"Start":"02:15.505 ","End":"02:20.660","Text":"one example or more you can come back to this and it will make a lot more sense."},{"Start":"02:20.660 ","End":"02:24.295","Text":"I\u0027m just going to give a general outline of how we solve it."},{"Start":"02:24.295 ","End":"02:26.270","Text":"Hopefully you understand some of it,"},{"Start":"02:26.270 ","End":"02:28.850","Text":"I\u0027ll get a general idea and there examples and"},{"Start":"02:28.850 ","End":"02:31.715","Text":"there\u0027ll be plenty of those will make it a lot clearer."},{"Start":"02:31.715 ","End":"02:33.785","Text":"We talked about power series,"},{"Start":"02:33.785 ","End":"02:38.550","Text":"so we\u0027ll take our function y and write it as a power series."},{"Start":"02:38.550 ","End":"02:42.370","Text":"I\u0027ve written a lot of terms from 0-5 and then dot,"},{"Start":"02:42.370 ","End":"02:44.450","Text":"dot, dot meaning etc."},{"Start":"02:44.450 ","End":"02:51.005","Text":"and I\u0027ve written the last 4 terms so of course in practice you wouldn\u0027t write so many."},{"Start":"02:51.005 ","End":"02:54.395","Text":"I just get the general idea you could even write it in Sigma form,"},{"Start":"02:54.395 ","End":"03:01.805","Text":"possibly Sigma from naught to infinity of a_n x^n."},{"Start":"03:01.805 ","End":"03:05.600","Text":"We substitute this, but it\u0027s a second-order differential equations."},{"Start":"03:05.600 ","End":"03:08.990","Text":"We\u0027re going to have to substitute y\u0027 and y\u0027\u0027."},{"Start":"03:08.990 ","End":"03:11.790","Text":"Here\u0027s y\u0027 we just differentiate each term,"},{"Start":"03:11.790 ","End":"03:14.085","Text":"a_1x gives us a_1,"},{"Start":"03:14.085 ","End":"03:17.355","Text":"a_2x^2 gives 2a_2x and so on."},{"Start":"03:17.355 ","End":"03:23.510","Text":"The last term which deliberately wrote up to n plus 2,"},{"Start":"03:23.510 ","End":"03:26.345","Text":"and actually there should be continuous infinitely."},{"Start":"03:26.345 ","End":"03:31.705","Text":"Then we\u0027ll get, the last term will be n plus 2x^n plus 1,"},{"Start":"03:31.705 ","End":"03:36.395","Text":"again, it continues and the second derivative will make it x^n,"},{"Start":"03:36.395 ","End":"03:37.766","Text":"here is the last time anyway."},{"Start":"03:37.766 ","End":"03:41.450","Text":"Just look this over and see that you agree with this."},{"Start":"03:41.450 ","End":"03:46.600","Text":"Now we substitute these y\u0027\u0027 in the ODE"},{"Start":"03:46.600 ","End":"03:52.475","Text":"and when you\u0027ve seen first example it already will be clearer what I\u0027m talking about."},{"Start":"03:52.475 ","End":"03:56.420","Text":"The next step will be to compare coefficients of x^n"},{"Start":"03:56.420 ","End":"04:01.175","Text":"for an arbitrary n on both sides of the equation and then"},{"Start":"04:01.175 ","End":"04:05.570","Text":"what you get is a recursive formula for a_n which typically"},{"Start":"04:05.570 ","End":"04:11.000","Text":"doesn\u0027t hold for all n but from a certain n onwards from n=2 or 3 onwards."},{"Start":"04:11.000 ","End":"04:16.150","Text":"We still have to take care of the head of the series."},{"Start":"04:16.150 ","End":"04:19.370","Text":"You do the first few coefficients manually,"},{"Start":"04:19.370 ","End":"04:22.475","Text":"typically a_0 or a_1 maybe a_2,"},{"Start":"04:22.475 ","End":"04:25.955","Text":"again you\u0027ll see this in the examples."},{"Start":"04:25.955 ","End":"04:30.410","Text":"You could return to this tutorial after you\u0027ve done one or"},{"Start":"04:30.410 ","End":"04:34.925","Text":"more of the solved exercises and have some remarks."},{"Start":"04:34.925 ","End":"04:37.580","Text":"The first remark is that if the function r,"},{"Start":"04:37.580 ","End":"04:42.805","Text":"that\u0027s the one on the right-hand side of the OD is not a polynomial,"},{"Start":"04:42.805 ","End":"04:46.850","Text":"you replace it by its Maclaurin series and there\u0027s"},{"Start":"04:46.850 ","End":"04:52.115","Text":"a table or list of Maclaurin series or there should be at the end of the exercise book."},{"Start":"04:52.115 ","End":"04:55.700","Text":"There\u0027s an example of this in the solved exercises should"},{"Start":"04:55.700 ","End":"04:59.495","Text":"be number 6 unless someone\u0027s changed the numbering."},{"Start":"04:59.495 ","End":"05:01.615","Text":"Go take a look at that."},{"Start":"05:01.615 ","End":"05:06.350","Text":"Now sometimes we\u0027re given the initial conditions and that means that we\u0027re"},{"Start":"05:06.350 ","End":"05:11.075","Text":"given y of naught and y\u0027 of naught that\u0027s typically,"},{"Start":"05:11.075 ","End":"05:14.570","Text":"and in that case if you just substitute in the power series,"},{"Start":"05:14.570 ","End":"05:17.180","Text":"you see that y naught is just a naught and"},{"Start":"05:17.180 ","End":"05:20.555","Text":"y\u0027 of naught is a_1 so we get a naught and a_1."},{"Start":"05:20.555 ","End":"05:24.800","Text":"Then we can use recursive definition to get the others."},{"Start":"05:24.800 ","End":"05:26.869","Text":"But if we\u0027re not given them explicitly,"},{"Start":"05:26.869 ","End":"05:31.940","Text":"we can still express the other coefficients in terms of a naught and a_1,"},{"Start":"05:31.940 ","End":"05:36.440","Text":"which allow like parameters like c_1 and c_2 or something."},{"Start":"05:36.440 ","End":"05:41.420","Text":"Now as I mentioned, we don\u0027t always have a power series around x equals naught,"},{"Start":"05:41.420 ","End":"05:45.680","Text":"could be a general point x_0 and then what you do is you"},{"Start":"05:45.680 ","End":"05:50.195","Text":"make a substitution to let t=x minus x_0."},{"Start":"05:50.195 ","End":"05:51.830","Text":"Or if you look at it the other way,"},{"Start":"05:51.830 ","End":"05:56.750","Text":"x=t plus x_0 and you also replace y, y\u0027,"},{"Start":"05:56.750 ","End":"06:02.790","Text":"y\u0027\u0027 as functions of t by replacing x with this,"},{"Start":"06:02.790 ","End":"06:08.840","Text":"so you get them as functions of t. Then you solve the differential equation for y(t),"},{"Start":"06:08.840 ","End":"06:15.830","Text":"and it\u0027s now around t equals naught and at the end you replace t with x minus x_0."},{"Start":"06:15.830 ","End":"06:21.125","Text":"That\u0027s examples of these should be numbers 8 and 9, an extra mark."},{"Start":"06:21.125 ","End":"06:25.700","Text":"Now, here we talked about order to differential equations,"},{"Start":"06:25.700 ","End":"06:29.990","Text":"but it\u0027s very similar to extend to higher orders."},{"Start":"06:29.990 ","End":"06:35.195","Text":"Just note slight difference that whereas earlier here we had that"},{"Start":"06:35.195 ","End":"06:40.630","Text":"a naught is just y of naught and a_1 is y\u0027 of naught."},{"Start":"06:40.630 ","End":"06:42.395","Text":"When we get to higher powers,"},{"Start":"06:42.395 ","End":"06:50.165","Text":"the factorial creeps in y\u0027\u0027 is 2 a_2, 2 factorial."},{"Start":"06:50.165 ","End":"06:54.915","Text":"Next we get 3 times 2 3 factorial a_2,"},{"Start":"06:54.915 ","End":"06:59.240","Text":"y\u0027\u0027\u0027\u0027 is 4 factorial a_4 and so on."},{"Start":"06:59.240 ","End":"07:04.085","Text":"The nth derivative of naught is n factorial times a_n."},{"Start":"07:04.085 ","End":"07:08.295","Text":"Finally, this is going to be a pretty technical remark."},{"Start":"07:08.295 ","End":"07:15.035","Text":"We assume that p and q where polynomials or possibly quotients of polynomials."},{"Start":"07:15.035 ","End":"07:18.470","Text":"But actually it can be extended to the case where"},{"Start":"07:18.470 ","End":"07:22.280","Text":"p and q are what is called analytic at x equals naught."},{"Start":"07:22.280 ","End":"07:25.185","Text":"Don\u0027t worry if this is a bit abstract."},{"Start":"07:25.185 ","End":"07:28.370","Text":"Anyway, you don\u0027t usually get this condition,"},{"Start":"07:28.370 ","End":"07:29.660","Text":"but I have to mention it."},{"Start":"07:29.660 ","End":"07:33.500","Text":"Analytic means in the case of a function f that it\u0027s"},{"Start":"07:33.500 ","End":"07:39.160","Text":"Maclaurin series converges in some interval containing x_0."},{"Start":"07:39.160 ","End":"07:46.060","Text":"There you have it. The examples will really be your main source of learning. That\u0027s it."}],"Thumbnail":null,"ID":7918},{"Watched":false,"Name":"Exercise 1","Duration":"9m 5s","ChapterTopicVideoID":7818,"CourseChapterTopicPlaylistID":4243,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.560","Text":"In this exercise, we\u0027re given this differential equation to solve."},{"Start":"00:04.560 ","End":"00:08.190","Text":"It\u0027s second-order linear and it\u0027s non-homogeneous"},{"Start":"00:08.190 ","End":"00:12.465","Text":"and we\u0027re given initial conditions at x=0."},{"Start":"00:12.465 ","End":"00:15.690","Text":"We\u0027re going to use power series to solve it."},{"Start":"00:15.690 ","End":"00:19.490","Text":"I\u0027ll just note that the functions that we call P and Q,"},{"Start":"00:19.490 ","End":"00:22.870","Text":"this and this are defined at x=0."},{"Start":"00:22.870 ","End":"00:24.780","Text":"Naught is a regular point."},{"Start":"00:24.780 ","End":"00:26.730","Text":"Now we\u0027re going to substitute y as"},{"Start":"00:26.730 ","End":"00:30.990","Text":"a power series and actually this continues indefinitely."},{"Start":"00:30.990 ","End":"00:36.785","Text":"y\u0027 as the derivative of this and y\u0027\u0027, the second derivative."},{"Start":"00:36.785 ","End":"00:41.625","Text":"I\u0027m going to plug them all into this differential equation."},{"Start":"00:41.625 ","End":"00:46.310","Text":"Start with y\u0027\u0027. I\u0027ll just copy it from here."},{"Start":"00:46.310 ","End":"00:51.575","Text":"But what we\u0027re gonna be doing is looking for the coefficient of x^n."},{"Start":"00:51.575 ","End":"00:54.740","Text":"It\u0027s not a mistake to take these,"},{"Start":"00:54.740 ","End":"00:58.745","Text":"but there\u0027s no need to take all these because I\u0027m going to be,"},{"Start":"00:58.745 ","End":"01:01.280","Text":"as I said, focusing on x^n."},{"Start":"01:01.280 ","End":"01:04.220","Text":"Similarly, when I take the term with y\u0027,"},{"Start":"01:04.220 ","End":"01:06.485","Text":"this one to x^2 y\u0027,"},{"Start":"01:06.485 ","End":"01:11.110","Text":"then I didn\u0027t bother with all these,"},{"Start":"01:11.110 ","End":"01:17.490","Text":"I just went up to x to the n-2 because I\u0027m going to be multiplying by the x^2."},{"Start":"01:17.490 ","End":"01:19.010","Text":"Yeah, here it is, x^2."},{"Start":"01:19.010 ","End":"01:22.025","Text":"That\u0027s going to bump this exponent up to x^n."},{"Start":"01:22.025 ","End":"01:27.075","Text":"That\u0027s the one I\u0027m really focusing on besides the first few, okay?"},{"Start":"01:27.075 ","End":"01:33.425","Text":"Lastly, we have this 4x times y. That\u0027s here."},{"Start":"01:33.425 ","End":"01:39.710","Text":"This time, I just went to up to n minus 1."},{"Start":"01:39.710 ","End":"01:42.190","Text":"I didn\u0027t need these."},{"Start":"01:42.190 ","End":"01:43.700","Text":"It wouldn\u0027t be a mistake,"},{"Start":"01:43.700 ","End":"01:45.725","Text":"but it would just make it more cumbersome."},{"Start":"01:45.725 ","End":"01:50.870","Text":"Because the x to the n minus 1 with the x is also going to give me x^n."},{"Start":"01:50.870 ","End":"01:52.810","Text":"x^n is what I\u0027m collecting."},{"Start":"01:52.810 ","End":"01:54.740","Text":"Of course, on the right-hand side,"},{"Start":"01:54.740 ","End":"01:58.010","Text":"I have to write the x^2 plus 2x plus 2, that\u0027s here."},{"Start":"01:58.010 ","End":"02:00.144","Text":"This is what we have now."},{"Start":"02:00.144 ","End":"02:03.605","Text":"That was step 1 where we substitute."},{"Start":"02:03.605 ","End":"02:06.710","Text":"Next we want to rearrange these collect like terms."},{"Start":"02:06.710 ","End":"02:11.720","Text":"On the left, we\u0027re going to have terms times 1 or constant terms then times x,"},{"Start":"02:11.720 ","End":"02:16.925","Text":"x^2, x^3, and so on up to x^n and the same on the right."},{"Start":"02:16.925 ","End":"02:19.850","Text":"We just put this in increasing order of exponents."},{"Start":"02:19.850 ","End":"02:21.170","Text":"Since we have initial conditions,"},{"Start":"02:21.170 ","End":"02:24.320","Text":"remember that y(0) is a_0,"},{"Start":"02:24.320 ","End":"02:27.050","Text":"but that was given to be 3 and y\u0027(0),"},{"Start":"02:27.050 ","End":"02:29.710","Text":"which is a_1, was given to be 12."},{"Start":"02:29.710 ","End":"02:31.955","Text":"Now we start comparing coefficients."},{"Start":"02:31.955 ","End":"02:37.580","Text":"The constant term, like coefficients of 1 or x^0 gives"},{"Start":"02:37.580 ","End":"02:43.520","Text":"us that this 2a_2 is equal to the constant term here, which is 2."},{"Start":"02:43.520 ","End":"02:46.235","Text":"That gives us that a_2 is 1."},{"Start":"02:46.235 ","End":"02:50.090","Text":"Just let me highlight what I did when I compared the coefficients here."},{"Start":"02:50.090 ","End":"02:55.295","Text":"I compared this on the left to this on the right."},{"Start":"02:55.295 ","End":"02:57.800","Text":"You can probably guess what I\u0027m going to write next."},{"Start":"02:57.800 ","End":"02:59.210","Text":"I\u0027m going to be comparing this,"},{"Start":"02:59.210 ","End":"03:02.425","Text":"the coefficient of x^1 with this,"},{"Start":"03:02.425 ","End":"03:09.715","Text":"which is what I just wrote here now and this gives us that a_3, in terms of a_0."},{"Start":"03:09.715 ","End":"03:12.255","Text":"We already have a_0 and a_1."},{"Start":"03:12.255 ","End":"03:15.435","Text":"From here I get a_3 is this,"},{"Start":"03:15.435 ","End":"03:17.490","Text":"a_0 is 3,"},{"Start":"03:17.490 ","End":"03:22.530","Text":"so this gives 1/3 minus 2, which is minus 5/3."},{"Start":"03:22.530 ","End":"03:25.640","Text":"Next and then to compare coefficients of x^2."},{"Start":"03:25.640 ","End":"03:28.885","Text":"I\u0027m going to equate this to this."},{"Start":"03:28.885 ","End":"03:30.710","Text":"This is what we get,"},{"Start":"03:30.710 ","End":"03:33.710","Text":"but we already have a_1 so we can find a_4."},{"Start":"03:33.710 ","End":"03:35.735","Text":"Here\u0027s is the computation,"},{"Start":"03:35.735 ","End":"03:38.435","Text":"and this is getting a bit tedious."},{"Start":"03:38.435 ","End":"03:42.035","Text":"Let\u0027s then jump onto the general term."},{"Start":"03:42.035 ","End":"03:43.340","Text":"You know what? Let\u0027s do one more."},{"Start":"03:43.340 ","End":"03:44.825","Text":"Let\u0027s do the x^3 term."},{"Start":"03:44.825 ","End":"03:47.750","Text":"It\u0027s fairly typical because I wanted to do one where you have 0 here,"},{"Start":"03:47.750 ","End":"03:54.790","Text":"because I can continue this 0 x^3 plus and up to 0, x^n."},{"Start":"03:54.790 ","End":"03:56.580","Text":"Everything else now is going to be compared to 0."},{"Start":"03:56.580 ","End":"04:03.230","Text":"I just take this one, so we get that 20a_5 is equal to 0."},{"Start":"04:03.230 ","End":"04:05.165","Text":"That gives us a_5 is 0."},{"Start":"04:05.165 ","End":"04:08.570","Text":"Now let\u0027s jump to the x^n."},{"Start":"04:08.570 ","End":"04:11.420","Text":"What I\u0027m going to say is this whole thing."},{"Start":"04:11.420 ","End":"04:15.500","Text":"The coefficient of x^n is going to be equal to 0."},{"Start":"04:15.500 ","End":"04:17.645","Text":"Here I just wrote that."},{"Start":"04:17.645 ","End":"04:23.460","Text":"Note that this works only for n bigger or equal to 3,"},{"Start":"04:23.460 ","End":"04:29.890","Text":"because we took care of the case where n is x^0 x^1 x^2 manually here,"},{"Start":"04:29.890 ","End":"04:35.405","Text":"then the 0 start from x^3 onwards."},{"Start":"04:35.405 ","End":"04:39.790","Text":"Like this would also be color it also."},{"Start":"04:39.790 ","End":"04:44.790","Text":"This one also. We get equals 0 from x^3 onwards."},{"Start":"04:44.790 ","End":"04:48.150","Text":"If we extract a_n plus"},{"Start":"04:48.150 ","End":"04:52.130","Text":"2 from this equation and bring everything else to the right-hand side,"},{"Start":"04:52.130 ","End":"04:57.140","Text":"we get a_n plus 2 in terms of n and in terms of a_n minus 1."},{"Start":"04:57.140 ","End":"04:59.735","Text":"This is the recursion formula."},{"Start":"04:59.735 ","End":"05:01.340","Text":"Just want to show you that something works out."},{"Start":"05:01.340 ","End":"05:04.850","Text":"You don\u0027t have to do this, but I wanted to I was just wondering if n=3,"},{"Start":"05:04.850 ","End":"05:06.905","Text":"which is covered here and in here."},{"Start":"05:06.905 ","End":"05:08.540","Text":"If it\u0027s consistent, well,"},{"Start":"05:08.540 ","End":"05:11.490","Text":"if we put n=3, what do we get?"},{"Start":"05:11.490 ","End":"05:13.890","Text":"We get a_3 plus 2,"},{"Start":"05:13.890 ","End":"05:18.945","Text":"which is 5 equals twice 3 minus 6 over,"},{"Start":"05:18.945 ","End":"05:25.360","Text":"doesn\u0027t matter times a then 3 minus 1 is 2."},{"Start":"05:25.360 ","End":"05:29.224","Text":"It doesn\u0027t matter because 2 times 3 minus 6 is 0."},{"Start":"05:29.224 ","End":"05:31.910","Text":"We get 0, which is what we got here."},{"Start":"05:31.910 ","End":"05:36.560","Text":"Let\u0027s just check that we didn\u0027t even have to do this line from x^3 onwards."},{"Start":"05:36.560 ","End":"05:38.900","Text":"We have this formula which gives this."},{"Start":"05:38.900 ","End":"05:41.495","Text":"Anyway, we have this recursive equation,"},{"Start":"05:41.495 ","End":"05:46.340","Text":"which is not the same as an explicit equation in closed form."},{"Start":"05:46.340 ","End":"05:49.730","Text":"We can compute any a_n we want."},{"Start":"05:49.730 ","End":"05:54.665","Text":"I mean, we did it up to A5 and you could substitute what n=4 would give us what?"},{"Start":"05:54.665 ","End":"05:56.660","Text":"a_6 equals and so on."},{"Start":"05:56.660 ","End":"05:58.745","Text":"We can compute as many a_n as we want,"},{"Start":"05:58.745 ","End":"06:00.740","Text":"even if we don\u0027t have a closed formula."},{"Start":"06:00.740 ","End":"06:04.865","Text":"Now another thing is that there\u0027s an alternative way of writing this."},{"Start":"06:04.865 ","End":"06:08.240","Text":"You might want to say, what is a_n equal or not?"},{"Start":"06:08.240 ","End":"06:10.930","Text":"This plus 2 looks a bit messy."},{"Start":"06:10.930 ","End":"06:16.460","Text":"If you take this formula and everywhere substitute instead of n,"},{"Start":"06:16.460 ","End":"06:22.310","Text":"you substitute this, write it as an arrow like this n minus 2."},{"Start":"06:22.310 ","End":"06:28.580","Text":"Then here we\u0027ll get n minus 2 plus 2 is a_n here twice n minus 2 minus 6 is this."},{"Start":"06:28.580 ","End":"06:35.550","Text":"n minus 2 plus 1 is n minus 1 and plus minus 2 plus 2 is n. Here,"},{"Start":"06:35.550 ","End":"06:38.805","Text":"n minus 2 minus 1 a_n minus 3."},{"Start":"06:38.805 ","End":"06:41.670","Text":"It would be from n minus 2 big or equal to 3,"},{"Start":"06:41.670 ","End":"06:43.485","Text":"which is n big or equal to 5."},{"Start":"06:43.485 ","End":"06:45.140","Text":"Not saying you have to do this,"},{"Start":"06:45.140 ","End":"06:49.445","Text":"but you could have written the recursive formula this way."},{"Start":"06:49.445 ","End":"06:53.570","Text":"I also want to write what y is in terms of the a_n,"},{"Start":"06:53.570 ","End":"06:55.605","Text":"the ones that we\u0027ve found already."},{"Start":"06:55.605 ","End":"06:57.814","Text":"See if I can see a few more."},{"Start":"06:57.814 ","End":"06:59.615","Text":"If I scroll a little bit."},{"Start":"06:59.615 ","End":"07:03.610","Text":"Yeah, I\u0027ve got a_0 is 3,"},{"Start":"07:03.610 ","End":"07:05.730","Text":"then we have a_1 x,"},{"Start":"07:05.730 ","End":"07:07.290","Text":"which is 12 x."},{"Start":"07:07.290 ","End":"07:08.910","Text":"Then we have a_2,"},{"Start":"07:08.910 ","End":"07:11.535","Text":"which is 1 x^2."},{"Start":"07:11.535 ","End":"07:12.920","Text":"Look at these numbers here,"},{"Start":"07:12.920 ","End":"07:16.220","Text":"minus 5/3 minus 23 over 12."},{"Start":"07:16.220 ","End":"07:20.450","Text":"In general, a_n which we don\u0027t know explicitly,"},{"Start":"07:20.450 ","End":"07:23.375","Text":"but we have a recursion formula for it."},{"Start":"07:23.375 ","End":"07:26.405","Text":"This is what you might write that this y equals,"},{"Start":"07:26.405 ","End":"07:31.280","Text":"so-and-so where we have this recursion formula or"},{"Start":"07:31.280 ","End":"07:36.140","Text":"alternatively this recursion formula with its limitation, I\u0027m not done."},{"Start":"07:36.140 ","End":"07:38.075","Text":"I want to say a couple of more things."},{"Start":"07:38.075 ","End":"07:47.990","Text":"My recommendation is not to do the part where we calculate the small coefficients first."},{"Start":"07:47.990 ","End":"07:49.220","Text":"What we had, well,"},{"Start":"07:49.220 ","End":"07:50.450","Text":"if I go back, we had,"},{"Start":"07:50.450 ","End":"07:53.810","Text":"I know a_0, a_1, a_2, and so on."},{"Start":"07:53.810 ","End":"07:56.465","Text":"Then we got the recursion formula."},{"Start":"07:56.465 ","End":"08:03.030","Text":"My advice is to actually start with this case here, with the x^n."},{"Start":"08:03.030 ","End":"08:05.910","Text":"Start with this, get to this."},{"Start":"08:05.910 ","End":"08:10.040","Text":"Then it\u0027s possible that this works for some of these cases."},{"Start":"08:10.040 ","End":"08:13.070","Text":"Maybe we\u0027re doing some extra work here that we don\u0027t need to."},{"Start":"08:13.070 ","End":"08:17.880","Text":"In this case, this only applies from big or equal to 3."},{"Start":"08:17.880 ","End":"08:24.985","Text":"We had to compute the case where n is 0,1 and 2 separately."},{"Start":"08:24.985 ","End":"08:29.450","Text":"Sometimes you can save calculations if you do this first."},{"Start":"08:29.450 ","End":"08:34.235","Text":"The final thing I want to note is that this expression does have a domain of definition."},{"Start":"08:34.235 ","End":"08:36.170","Text":"Well, denominator can\u0027t be 0,"},{"Start":"08:36.170 ","End":"08:38.340","Text":"but that\u0027s not usually an issue."},{"Start":"08:38.340 ","End":"08:43.869","Text":"For one thing, the a series only begins from index 0."},{"Start":"08:43.869 ","End":"08:47.155","Text":"You couldn\u0027t put, for example, n=0 here,"},{"Start":"08:47.155 ","End":"08:49.040","Text":"but if you put n=0,"},{"Start":"08:49.040 ","End":"08:52.155","Text":"that would give a_n minus 1 which doesn\u0027t exist."},{"Start":"08:52.155 ","End":"08:56.705","Text":"This might give you an extra restriction on the ends that you can substitute."},{"Start":"08:56.705 ","End":"08:59.810","Text":"Anyway, I\u0027ve said enough, maybe too much."},{"Start":"08:59.810 ","End":"09:06.660","Text":"Go ahead and do more exercises and you\u0027ll soon catch on how to do these."}],"Thumbnail":null,"ID":7919},{"Watched":false,"Name":"Exercise 2","Duration":"5m 29s","ChapterTopicVideoID":7819,"CourseChapterTopicPlaylistID":4243,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"Here we have this second-order differential equation to solve."},{"Start":"00:03.990 ","End":"00:07.470","Text":"It\u0027s homogeneous this time and we have"},{"Start":"00:07.470 ","End":"00:11.235","Text":"initial conditions and we\u0027re going to solve it with power series."},{"Start":"00:11.235 ","End":"00:14.250","Text":"Just note that x=0 is a regular point."},{"Start":"00:14.250 ","End":"00:17.850","Text":"Well, the P is missing and the q is minus x,"},{"Start":"00:17.850 ","End":"00:21.194","Text":"but it\u0027s defined at x=0, so no problem."},{"Start":"00:21.194 ","End":"00:24.240","Text":"Here we have y as a power series in x,"},{"Start":"00:24.240 ","End":"00:26.385","Text":"we have y\u0027 and we have y\u0027\u0027."},{"Start":"00:26.385 ","End":"00:29.895","Text":"In this case, we don\u0027t need y\u0027 because it\u0027s absent here."},{"Start":"00:29.895 ","End":"00:33.535","Text":"But, we\u0027re going to substitute y and y\u0027\u0027 in here,"},{"Start":"00:33.535 ","End":"00:35.670","Text":"so this is y\u0027\u0027,"},{"Start":"00:35.670 ","End":"00:37.560","Text":"it\u0027s what we need here,"},{"Start":"00:37.560 ","End":"00:39.660","Text":"but we don\u0027t need all of these terms."},{"Start":"00:39.660 ","End":"00:41.150","Text":"I\u0027ve put it in way too many."},{"Start":"00:41.150 ","End":"00:43.550","Text":"In fact, we can just ignore these,"},{"Start":"00:43.550 ","End":"00:47.225","Text":"and the reason for that is we\u0027re going to be collecting the x^n,"},{"Start":"00:47.225 ","End":"00:50.885","Text":"so we just need the x^n term."},{"Start":"00:50.885 ","End":"00:52.700","Text":"Next we want, well,"},{"Start":"00:52.700 ","End":"00:53.990","Text":"there\u0027s no y\u0027,"},{"Start":"00:53.990 ","End":"00:58.775","Text":"but we need x times y. X times y"},{"Start":"00:58.775 ","End":"01:04.310","Text":"will need to go up to the a_n-1 to get x^n,"},{"Start":"01:04.310 ","End":"01:05.540","Text":"and here we are."},{"Start":"01:05.540 ","End":"01:10.550","Text":"Like I said, we just need to go up to here because x times,"},{"Start":"01:10.550 ","End":"01:12.395","Text":"this is going to give us x^n,"},{"Start":"01:12.395 ","End":"01:14.800","Text":"and then this is equal to 0,"},{"Start":"01:14.800 ","End":"01:17.395","Text":"which is just what I copied from the equation."},{"Start":"01:17.395 ","End":"01:20.605","Text":"Next, we want to do a bit of simplification."},{"Start":"01:20.605 ","End":"01:22.104","Text":"Let me just get some space,"},{"Start":"01:22.104 ","End":"01:26.755","Text":"collect like terms together ordered in increasing powers of x,"},{"Start":"01:26.755 ","End":"01:29.300","Text":"like x^0, which is 1,"},{"Start":"01:29.300 ","End":"01:31.710","Text":"x^1 which is x, x^2,"},{"Start":"01:31.710 ","End":"01:34.715","Text":"x^3, x^4, and so on up to x^n."},{"Start":"01:34.715 ","End":"01:37.120","Text":"Now in a moment we\u0027ll compare coefficients,"},{"Start":"01:37.120 ","End":"01:40.780","Text":"but this thing starts from a_2 and we\u0027ll need"},{"Start":"01:40.780 ","End":"01:44.770","Text":"a_0 and a_1 for computing like this and this."},{"Start":"01:44.770 ","End":"01:48.690","Text":"Remember we had initial conditions that a_0 was 1."},{"Start":"01:48.690 ","End":"01:51.300","Text":"Well, y(0) is a_0 is 1,"},{"Start":"01:51.300 ","End":"01:53.235","Text":"and y\u0027(0) is a_1,"},{"Start":"01:53.235 ","End":"01:56.810","Text":"which is 2 so now we can start comparing."},{"Start":"01:56.810 ","End":"02:01.490","Text":"But my recommendation is not to start at the head of the series,"},{"Start":"02:01.490 ","End":"02:05.315","Text":"but to take the general term with x^n in it."},{"Start":"02:05.315 ","End":"02:11.330","Text":"What we get is that this is equal to 0 and that\u0027s what\u0027s over here."},{"Start":"02:11.330 ","End":"02:13.325","Text":"I\u0027ll just highlight it so you see what I mean."},{"Start":"02:13.325 ","End":"02:15.110","Text":"This is 0,"},{"Start":"02:15.110 ","End":"02:18.080","Text":"it\u0027s got 0 in all the powers of x."},{"Start":"02:18.080 ","End":"02:20.985","Text":"That thing, which is this,"},{"Start":"02:20.985 ","End":"02:23.415","Text":"is equal to 0."},{"Start":"02:23.415 ","End":"02:26.790","Text":"What we get is that a_n+2,"},{"Start":"02:26.790 ","End":"02:29.240","Text":"if we put everything else on the other side of the equation,"},{"Start":"02:29.240 ","End":"02:30.515","Text":"is equal to this."},{"Start":"02:30.515 ","End":"02:35.960","Text":"Now it\u0027s important to know from which n onwards this recursion formula applies."},{"Start":"02:35.960 ","End":"02:38.660","Text":"It\u0027s always n, big or equal to something."},{"Start":"02:38.660 ","End":"02:43.730","Text":"n can\u0027t be 0, because then we\u0027d get an index of minus 1."},{"Start":"02:43.730 ","End":"02:47.460","Text":"From this point alone, it has to be from at least 1."},{"Start":"02:47.460 ","End":"02:54.155","Text":"Now, if you just look at this part,"},{"Start":"02:54.155 ","End":"02:57.215","Text":"if n equals 1, we get 2 times 3,"},{"Start":"02:57.215 ","End":"03:02.015","Text":"which is 6a_3-a_0, which is this."},{"Start":"03:02.015 ","End":"03:05.030","Text":"We can see that from here onwards it"},{"Start":"03:05.030 ","End":"03:08.510","Text":"works so it\u0027s only of an exception for the first term."},{"Start":"03:08.510 ","End":"03:10.550","Text":"I\u0027m going to have to handle that separately."},{"Start":"03:10.550 ","End":"03:14.390","Text":"Now all the coefficients here are 0 because we have 0 on the right."},{"Start":"03:14.390 ","End":"03:17.585","Text":"We\u0027ve dealt with this general term which applies from here onwards."},{"Start":"03:17.585 ","End":"03:19.220","Text":"We still need to this."},{"Start":"03:19.220 ","End":"03:21.800","Text":"This is going to equal 0."},{"Start":"03:21.800 ","End":"03:24.710","Text":"That gives us that a_2 is naught."},{"Start":"03:24.710 ","End":"03:30.020","Text":"Now look, we already figured out a_0, a_1 and a_2."},{"Start":"03:30.020 ","End":"03:32.420","Text":"We have this, we have this,"},{"Start":"03:32.420 ","End":"03:34.010","Text":"and we have this."},{"Start":"03:34.010 ","End":"03:36.050","Text":"I mean, this 1 will help us compute a_3,"},{"Start":"03:36.050 ","End":"03:37.670","Text":"this will help us compute a_4,"},{"Start":"03:37.670 ","End":"03:39.260","Text":"this will give us a_5."},{"Start":"03:39.260 ","End":"03:40.640","Text":"Then we\u0027ll have a_3 already."},{"Start":"03:40.640 ","End":"03:42.845","Text":"We can compute a_6 and so on."},{"Start":"03:42.845 ","End":"03:46.400","Text":"Each time, the 1 we have to compute will be in terms of ones we\u0027ve"},{"Start":"03:46.400 ","End":"03:50.165","Text":"computed already and this is the recursion formula."},{"Start":"03:50.165 ","End":"03:51.605","Text":"Let\u0027s do another 1."},{"Start":"03:51.605 ","End":"03:54.960","Text":"If I put n equals 1 in here,"},{"Start":"03:54.960 ","End":"04:02.040","Text":"then we get that a_3 is equal to a_0 because 1-1 is naught."},{"Start":"04:02.040 ","End":"04:05.665","Text":"Here we have this 2 times 3, so it\u0027s 1/6."},{"Start":"04:05.665 ","End":"04:08.300","Text":"Then just get some space."},{"Start":"04:08.300 ","End":"04:13.730","Text":"Let\u0027s do another 1 and put n equals 2 here so 2+2 is 4,"},{"Start":"04:13.730 ","End":"04:18.515","Text":"and that gives us a_4 in terms of a_1, 2-1."},{"Start":"04:18.515 ","End":"04:19.850","Text":"Here, if n is 2,"},{"Start":"04:19.850 ","End":"04:21.175","Text":"3 times 4,"},{"Start":"04:21.175 ","End":"04:24.660","Text":"which is a_1 was 2,"},{"Start":"04:24.660 ","End":"04:27.735","Text":"2 over 12 is 1/6."},{"Start":"04:27.735 ","End":"04:30.885","Text":"We\u0027ve computed a few terms so we can say that y is,"},{"Start":"04:30.885 ","End":"04:32.535","Text":"a_0 was 1,"},{"Start":"04:32.535 ","End":"04:36.315","Text":"and a_1 was 2,"},{"Start":"04:36.315 ","End":"04:37.740","Text":"a_3, well,"},{"Start":"04:37.740 ","End":"04:39.565","Text":"we can see it here is a 1/6."},{"Start":"04:39.565 ","End":"04:43.310","Text":"I was missing x^2 term because a_2 is 0."},{"Start":"04:43.310 ","End":"04:45.755","Text":"Then a_4 was also a 1/6,"},{"Start":"04:45.755 ","End":"04:48.260","Text":"and so on and so on up to a_n,"},{"Start":"04:48.260 ","End":"04:52.685","Text":"where a_n is given by an adaptation of this,"},{"Start":"04:52.685 ","End":"04:57.800","Text":"where we just replace n+2 by n or if you like,"},{"Start":"04:57.800 ","End":"05:01.310","Text":"we replace n by n-2."},{"Start":"05:01.310 ","End":"05:03.530","Text":"Perhaps that\u0027s a better way of saying it."},{"Start":"05:03.530 ","End":"05:07.100","Text":"Then n-2+1 is n-1,"},{"Start":"05:07.100 ","End":"05:09.370","Text":"n-2+2 is n,"},{"Start":"05:09.370 ","End":"05:12.900","Text":"n-2-1 is n-3, and so on."},{"Start":"05:12.900 ","End":"05:17.460","Text":"n+2, big or equal to 1 becomes n big or equal to 3."},{"Start":"05:17.460 ","End":"05:20.780","Text":"We have the first few to start off with,"},{"Start":"05:20.780 ","End":"05:22.685","Text":"and then we have the recursion formula."},{"Start":"05:22.685 ","End":"05:24.755","Text":"This is how we would leave the answer."},{"Start":"05:24.755 ","End":"05:29.730","Text":"These 2 define it completely and we\u0027re done."}],"Thumbnail":null,"ID":7920},{"Watched":false,"Name":"Exercise 3","Duration":"5m 32s","ChapterTopicVideoID":7866,"CourseChapterTopicPlaylistID":4243,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"We have here this differential equation to solve."},{"Start":"00:02.760 ","End":"00:08.760","Text":"Then we\u0027re going to do it by expanding y as a power series around x equals note."},{"Start":"00:08.760 ","End":"00:12.300","Text":"The first we have to check that x equals note as irregular point."},{"Start":"00:12.300 ","End":"00:14.370","Text":"This is not in standard format."},{"Start":"00:14.370 ","End":"00:16.380","Text":"Y double-prime has to be on its own."},{"Start":"00:16.380 ","End":"00:25.725","Text":"I can rewrite this as by dividing by 1-x^2 and then plus 2/1-x^2."},{"Start":"00:25.725 ","End":"00:30.975","Text":"If I got the y prime here and then y here equals 0."},{"Start":"00:30.975 ","End":"00:33.165","Text":"This bit is P, this bit is Q."},{"Start":"00:33.165 ","End":"00:36.000","Text":"Now, those problems when x is plus or minus 1,"},{"Start":"00:36.000 ","End":"00:38.745","Text":"but around 0, there\u0027s absolutely no problems."},{"Start":"00:38.745 ","End":"00:43.000","Text":"X equals notice irregular points so we can continue."},{"Start":"00:43.000 ","End":"00:48.920","Text":"Expand the 1-x ^2 times y double prime to 2 separate terms."},{"Start":"00:48.920 ","End":"00:52.410","Text":"Then I display, we\u0027ve seen this before."},{"Start":"00:52.410 ","End":"00:58.400","Text":"Y is the power series y-prime differentiate each term y double-prime,"},{"Start":"00:58.400 ","End":"01:02.390","Text":"and we want to substitute these in this equation."},{"Start":"01:02.390 ","End":"01:06.770","Text":"We have four terms here and the 0 on the right that starts with the first one,"},{"Start":"01:06.770 ","End":"01:10.070","Text":"y double-prime, essentially just copy it from here,"},{"Start":"01:10.070 ","End":"01:12.470","Text":"but remove redundant terms."},{"Start":"01:12.470 ","End":"01:14.725","Text":"We\u0027re going to be comparing x^n."},{"Start":"01:14.725 ","End":"01:19.100","Text":"I don\u0027t need these that swallowed up in the dot, dot, dot."},{"Start":"01:19.100 ","End":"01:22.685","Text":"Next we want x ^2, y double-prime."},{"Start":"01:22.685 ","End":"01:25.595","Text":"Notice if I multiply this thing by x^2,"},{"Start":"01:25.595 ","End":"01:32.720","Text":"then the x^n-2 term is going to become x^n."},{"Start":"01:32.720 ","End":"01:36.800","Text":"This time, since this is where I want to get up to,"},{"Start":"01:36.800 ","End":"01:41.660","Text":"I\u0027m not going to use these and you\u0027re going to swallow it up in the dot,"},{"Start":"01:41.660 ","End":"01:43.250","Text":"dot, dot or the ellipsis."},{"Start":"01:43.250 ","End":"01:45.665","Text":"I guess I don\u0027t need this one either."},{"Start":"01:45.665 ","End":"01:49.445","Text":"We get minus x^2 times this and this,"},{"Start":"01:49.445 ","End":"01:52.340","Text":"and that\u0027s going to give us also an x^n term."},{"Start":"01:52.340 ","End":"01:54.050","Text":"Now the next one,"},{"Start":"01:54.050 ","End":"01:56.600","Text":"it\u0027s 2x times y prime."},{"Start":"01:56.600 ","End":"02:02.195","Text":"The XY prime will mean that the x^n term is going to come from this."},{"Start":"02:02.195 ","End":"02:05.385","Text":"I don\u0027t need to copy these."},{"Start":"02:05.385 ","End":"02:10.080","Text":"We get the minus 2x and that\u0027s the y-prime."},{"Start":"02:10.080 ","End":"02:13.725","Text":"Finally, we need a plus 2y."},{"Start":"02:13.725 ","End":"02:16.670","Text":"This one will be as is,"},{"Start":"02:16.670 ","End":"02:20.540","Text":"but we only need to go up to x to the n. We won\u0027t need this and here we are."},{"Start":"02:20.540 ","End":"02:22.700","Text":"I\u0027m actually, we don\u0027t even really need this,"},{"Start":"02:22.700 ","End":"02:25.370","Text":"doesn\u0027t matter if you leave it in and it\u0027s equal to 0,"},{"Start":"02:25.370 ","End":"02:27.230","Text":"of course, that\u0027s the right-hand side."},{"Start":"02:27.230 ","End":"02:30.005","Text":"It\u0027s clear a bit of room."},{"Start":"02:30.005 ","End":"02:32.990","Text":"Next we collect together all the like terms."},{"Start":"02:32.990 ","End":"02:38.575","Text":"For example, the constant term would be 2 a2 plus 2a naught and so on."},{"Start":"02:38.575 ","End":"02:41.000","Text":"This term is the term with x^n,"},{"Start":"02:41.000 ","End":"02:47.010","Text":"which is essentially the summation of 4 different things they\u0027ll gather together here."},{"Start":"02:47.010 ","End":"02:51.995","Text":"We compare this to 0 and do a bit of algebra here."},{"Start":"02:51.995 ","End":"02:55.040","Text":"I\u0027ve got a_ n plus 2, but all these contain a_ n. Let\u0027s take"},{"Start":"02:55.040 ","End":"02:58.745","Text":"minus a_ n out of the brackets just in these last three terms."},{"Start":"02:58.745 ","End":"03:02.510","Text":"What I\u0027m left with is n times n minus 1 is"},{"Start":"03:02.510 ","End":"03:10.040","Text":"n^2-n. Then I have plus 2n because I took minus out and minus 2."},{"Start":"03:10.040 ","End":"03:16.635","Text":"That comes out this bit here just becomes n^2-3n+2."},{"Start":"03:16.635 ","End":"03:22.465","Text":"This can be factorized into n-1, n+2."},{"Start":"03:22.465 ","End":"03:24.000","Text":"This is not minus 3,"},{"Start":"03:24.000 ","End":"03:25.815","Text":"this is plus 1 fixed."},{"Start":"03:25.815 ","End":"03:28.680","Text":"Now, n is at least 0,"},{"Start":"03:28.680 ","End":"03:30.080","Text":"m+ 2 is not 0."},{"Start":"03:30.080 ","End":"03:33.505","Text":"I can cancel out the n+2 from both sides."},{"Start":"03:33.505 ","End":"03:35.960","Text":"Extract a_ n plus 2 and we get"},{"Start":"03:35.960 ","End":"03:41.840","Text":"this recursion formula and it actually holds for all n. For example,"},{"Start":"03:41.840 ","End":"03:48.105","Text":"if we put n equals 0, we get that a_2 is minus a_1,"},{"Start":"03:48.105 ","End":"03:50.175","Text":"which we would get from here anyway."},{"Start":"03:50.175 ","End":"03:52.065","Text":"If you put n equals 1,"},{"Start":"03:52.065 ","End":"03:54.615","Text":"you\u0027d get that a_3 is 0,"},{"Start":"03:54.615 ","End":"03:56.790","Text":"which we would get from 6, a_3=0."},{"Start":"03:56.790 ","End":"03:59.400","Text":"It just works for all n from 0 onwards."},{"Start":"03:59.400 ","End":"04:05.415","Text":"This formula gives me from a_2 on what\u0027s it can\u0027t give me a_0 and a_1."},{"Start":"04:05.415 ","End":"04:07.370","Text":"In fact, in this exercise,"},{"Start":"04:07.370 ","End":"04:12.979","Text":"since we have no initial conditions these a_0 a_1 are going to remain"},{"Start":"04:12.979 ","End":"04:19.385","Text":"unknown parameters just like we had c_1 and c_2 and other differential equations."},{"Start":"04:19.385 ","End":"04:23.120","Text":"As we said, we get a_2 is minus a naught from here."},{"Start":"04:23.120 ","End":"04:25.130","Text":"From here we get a3 is 0,"},{"Start":"04:25.130 ","End":"04:29.885","Text":"and everything else from here on would becomes from the recursive formula."},{"Start":"04:29.885 ","End":"04:36.305","Text":"For example, a_4 by plugging in n=2 and we get 2-1/2+1."},{"Start":"04:36.305 ","End":"04:41.340","Text":"The third, a_2 and a_2 from here is minus a_0."},{"Start":"04:41.340 ","End":"04:44.330","Text":"I\u0027ve got a_4 in terms of a_0. We keep going."},{"Start":"04:44.330 ","End":"04:48.590","Text":"We can always put everything in terms of a_1 and a_0."},{"Start":"04:48.590 ","End":"04:51.215","Text":"Here I\u0027ve written out some of the terms."},{"Start":"04:51.215 ","End":"04:56.000","Text":"Similar effect a_0 appears everywhere except for this coefficient where I"},{"Start":"04:56.000 ","End":"05:00.935","Text":"have an a_1 because it skips to each time and once I have something depending on a_0,"},{"Start":"05:00.935 ","End":"05:02.810","Text":"a_0 will keep perpetuating itself."},{"Start":"05:02.810 ","End":"05:05.110","Text":"Anyway, that\u0027s just maybe of interest, maybe not."},{"Start":"05:05.110 ","End":"05:08.290","Text":"Finally, what we want is a recursion formula for a_"},{"Start":"05:08.290 ","End":"05:11.570","Text":"n. We prefer it that way that they have a_ n plus 2."},{"Start":"05:11.570 ","End":"05:18.065","Text":"If you replace everywhere n you replace it by n-2,"},{"Start":"05:18.065 ","End":"05:20.555","Text":"you\u0027ll see that what you get is"},{"Start":"05:20.555 ","End":"05:25.279","Text":"this recursion formula and basically you could say this is the answer."},{"Start":"05:25.279 ","End":"05:33.030","Text":"This is the recursion formula and a_ 0 and a_ 1 arbitrary constants and we\u0027re done."}],"Thumbnail":null,"ID":7926},{"Watched":false,"Name":"Exercise 4","Duration":"6m 23s","ChapterTopicVideoID":7867,"CourseChapterTopicPlaylistID":4243,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.710","Text":"Here we have a differential equation, linear non-homogeneous."},{"Start":"00:04.710 ","End":"00:07.305","Text":"We want to find y as a function of x,"},{"Start":"00:07.305 ","End":"00:13.350","Text":"and we\u0027re going to do it by treating y as an expanded power series around x equals note."},{"Start":"00:13.350 ","End":"00:16.815","Text":"I have to note that x equals note is a regular point,"},{"Start":"00:16.815 ","End":"00:20.820","Text":"and I have to divide by x^2 plus 4 throughout to get it in"},{"Start":"00:20.820 ","End":"00:24.750","Text":"standard form but since x^2 plus 4 is always positive,"},{"Start":"00:24.750 ","End":"00:28.905","Text":"it\u0027s even at least 4 all the functions in particular,"},{"Start":"00:28.905 ","End":"00:30.240","Text":"the P and Q."},{"Start":"00:30.240 ","End":"00:32.250","Text":"Well, P is 0, Q is x,"},{"Start":"00:32.250 ","End":"00:33.915","Text":"will be x over this,"},{"Start":"00:33.915 ","End":"00:35.880","Text":"is defined at x=0,"},{"Start":"00:35.880 ","End":"00:39.720","Text":"so x=0 is a regular point."},{"Start":"00:39.720 ","End":"00:43.425","Text":"Now, let\u0027s write the power series expansion."},{"Start":"00:43.425 ","End":"00:45.510","Text":"Here\u0027s the power series expansion."},{"Start":"00:45.510 ","End":"00:48.350","Text":"We\u0027ve been using this in every exercise,"},{"Start":"00:48.350 ","End":"00:50.150","Text":"so you should be familiar with it."},{"Start":"00:50.150 ","End":"00:54.480","Text":"Also expanded this to 2 separate terms,"},{"Start":"00:54.480 ","End":"00:57.030","Text":"x^2 separately and 4 separately."},{"Start":"00:57.030 ","End":"01:00.505","Text":"Now, we\u0027re going to substitute y here,"},{"Start":"01:00.505 ","End":"01:04.010","Text":"y\u0027\u0027 here, and y\u0027 is missing."},{"Start":"01:04.010 ","End":"01:05.780","Text":"We will take the terms 1 at a time."},{"Start":"01:05.780 ","End":"01:07.580","Text":"There\u0027s 1, 2, 3 of them here."},{"Start":"01:07.580 ","End":"01:10.880","Text":"The first one, x^2 y\u0027\u0027,"},{"Start":"01:10.880 ","End":"01:17.705","Text":"take x^2 multiplied by y\u0027\u0027 but allow me to take everything."},{"Start":"01:17.705 ","End":"01:24.080","Text":"I could stop here because I\u0027m looking for the coefficient of x^n."},{"Start":"01:24.080 ","End":"01:30.110","Text":"This x^2 with this x^n minus 2 will give it to me but also don\u0027t need this."},{"Start":"01:30.110 ","End":"01:32.000","Text":"That was this term."},{"Start":"01:32.000 ","End":"01:34.640","Text":"Now the next one 4y\u0027\u0027,"},{"Start":"01:34.640 ","End":"01:38.490","Text":"so it\u0027s 4 and I just copy the y\u0027\u0027."},{"Start":"01:38.490 ","End":"01:41.205","Text":"This time I do go up to x^n,"},{"Start":"01:41.205 ","End":"01:47.515","Text":"but in this case the terms that I don\u0027t need that are just clutter I call them."},{"Start":"01:47.515 ","End":"01:50.300","Text":"It\u0027s there, it\u0027s swallowed up in the dot,"},{"Start":"01:50.300 ","End":"01:52.130","Text":"dot, dot if you like."},{"Start":"01:52.130 ","End":"01:54.860","Text":"Now, let\u0027s do the third term,"},{"Start":"01:54.860 ","End":"01:56.885","Text":"the x times y."},{"Start":"01:56.885 ","End":"01:59.030","Text":"We need the x^n,"},{"Start":"01:59.030 ","End":"02:02.425","Text":"so obviously we\u0027re going to stop here."},{"Start":"02:02.425 ","End":"02:04.575","Text":"This is what I did in fact."},{"Start":"02:04.575 ","End":"02:07.450","Text":"It\u0027s like these are not necessary,"},{"Start":"02:07.450 ","End":"02:09.580","Text":"they\u0027re part of the dot, dot, dot now."},{"Start":"02:09.580 ","End":"02:13.330","Text":"Then I\u0027ll get x^n when I multiply it by the x."},{"Start":"02:13.330 ","End":"02:16.060","Text":"Then from here we collect like terms."},{"Start":"02:16.060 ","End":"02:18.640","Text":"Terms just constants without x,"},{"Start":"02:18.640 ","End":"02:21.150","Text":"terms multiplied by x,"},{"Start":"02:21.150 ","End":"02:27.460","Text":"the x^2 terms, the x^3 terms and so on but we would like the n term."},{"Start":"02:27.460 ","End":"02:29.690","Text":"Let me just take some space here."},{"Start":"02:29.690 ","End":"02:32.245","Text":"This is the x^n term,"},{"Start":"02:32.245 ","End":"02:33.790","Text":"which we can see."},{"Start":"02:33.790 ","End":"02:38.020","Text":"We got this x^2 times x^n minus 2,"},{"Start":"02:38.020 ","End":"02:40.375","Text":"so we have this bit,"},{"Start":"02:40.375 ","End":"02:45.055","Text":"and here we\u0027re going to get for x^n, this."},{"Start":"02:45.055 ","End":"02:46.910","Text":"Here for x^n,"},{"Start":"02:46.910 ","End":"02:48.180","Text":"we\u0027re going to get x times this,"},{"Start":"02:48.180 ","End":"02:49.950","Text":"so this is this."},{"Start":"02:49.950 ","End":"02:52.185","Text":"Put all these pieces together,"},{"Start":"02:52.185 ","End":"02:54.480","Text":"and this goes with the 4 of course,"},{"Start":"02:54.480 ","End":"02:56.550","Text":"and we\u0027ll get this."},{"Start":"02:56.550 ","End":"03:00.390","Text":"On the right-hand side the equation was x plus 2,"},{"Start":"03:00.390 ","End":"03:01.960","Text":"you can go back and check."},{"Start":"03:01.960 ","End":"03:03.430","Text":"Now the right-hand side,"},{"Start":"03:03.430 ","End":"03:07.030","Text":"I could write as 2 times 1,"},{"Start":"03:07.030 ","End":"03:08.815","Text":"or that\u0027s x to the note,"},{"Start":"03:08.815 ","End":"03:11.755","Text":"plus x is 1 times x^1,"},{"Start":"03:11.755 ","End":"03:16.480","Text":"and then I\u0027d get 0x^2 plus 0x^3."},{"Start":"03:16.480 ","End":"03:21.550","Text":"Everything else would be 0 after the first 2 terms."},{"Start":"03:21.550 ","End":"03:25.345","Text":"Now assuming that n is bigger or equal to 2,"},{"Start":"03:25.345 ","End":"03:27.200","Text":"I mean from 2 onwards,"},{"Start":"03:27.200 ","End":"03:29.280","Text":"the coefficient is 0."},{"Start":"03:29.280 ","End":"03:35.090","Text":"So all of this is 0 as long as n is bigger or equal to 2."},{"Start":"03:35.090 ","End":"03:37.220","Text":"Now we get this recursion formula."},{"Start":"03:37.220 ","End":"03:41.210","Text":"We just stopped to isolate this an plus 2 by taking the rest of"},{"Start":"03:41.210 ","End":"03:46.070","Text":"the stuff to the other side and then dividing by the coefficient of an plus 2,"},{"Start":"03:46.070 ","End":"03:50.930","Text":"so we get this and this holds for n bigger or equal to 2."},{"Start":"03:50.930 ","End":"03:57.350","Text":"Now, note that the smallest value of n that I can put in here is n equals 2."},{"Start":"03:57.350 ","End":"03:59.510","Text":"If I put n equals 2 here,"},{"Start":"03:59.510 ","End":"04:05.040","Text":"I\u0027ll get a4 in terms of a1 and a2."},{"Start":"04:05.040 ","End":"04:09.965","Text":"The first term that I can get from this formula is a4,"},{"Start":"04:09.965 ","End":"04:13.129","Text":"then a5 and so on onwards."},{"Start":"04:13.129 ","End":"04:17.210","Text":"Also, we know that we\u0027re not given initial conditions,"},{"Start":"04:17.210 ","End":"04:21.620","Text":"so a note and a1 are unknown constants."},{"Start":"04:21.620 ","End":"04:22.820","Text":"We treat them like parameters,"},{"Start":"04:22.820 ","End":"04:25.760","Text":"like the c1 and c2 in other equations."},{"Start":"04:25.760 ","End":"04:32.450","Text":"What we\u0027re missing in the gap is what is a2 and what is a3."},{"Start":"04:32.450 ","End":"04:34.925","Text":"I\u0027m going to need to scroll back a minute."},{"Start":"04:34.925 ","End":"04:38.195","Text":"8a2 is got to equal 2,"},{"Start":"04:38.195 ","End":"04:45.040","Text":"and this, which is the coefficient of x^1 has got to equal 1."},{"Start":"04:45.040 ","End":"04:48.950","Text":"We got 8a2=2,"},{"Start":"04:48.950 ","End":"04:51.700","Text":"and that gives us a2 is a quarter."},{"Start":"04:51.700 ","End":"04:57.870","Text":"We also got 24a3 plus a note is 1."},{"Start":"04:57.870 ","End":"05:00.070","Text":"That gives us a3,"},{"Start":"05:00.070 ","End":"05:02.315","Text":"but it depends on a note."},{"Start":"05:02.315 ","End":"05:04.039","Text":"Like I said earlier,"},{"Start":"05:04.039 ","End":"05:08.750","Text":"a note and a1 going to be our general constants,"},{"Start":"05:08.750 ","End":"05:12.170","Text":"and everything will be expressed in terms of 2 constants."},{"Start":"05:12.170 ","End":"05:14.920","Text":"Let\u0027s compute another one. Let\u0027s do a4."},{"Start":"05:14.920 ","End":"05:19.875","Text":"If we put n equals 2 here we get a4 equals,"},{"Start":"05:19.875 ","End":"05:21.330","Text":"here we have a1,"},{"Start":"05:21.330 ","End":"05:23.010","Text":"here we have a2,"},{"Start":"05:23.010 ","End":"05:24.755","Text":"this is the expression."},{"Start":"05:24.755 ","End":"05:27.075","Text":"This comes out to this."},{"Start":"05:27.075 ","End":"05:30.840","Text":"But we know a2 is a quarter,"},{"Start":"05:30.840 ","End":"05:32.615","Text":"and this is what we get."},{"Start":"05:32.615 ","End":"05:37.715","Text":"Now we can write the general shape of the function y as a power series."},{"Start":"05:37.715 ","End":"05:42.035","Text":"I have all the coefficients up to the power of x^4."},{"Start":"05:42.035 ","End":"05:44.090","Text":"Then the general term an,"},{"Start":"05:44.090 ","End":"05:45.650","Text":"we don\u0027t have an explicit formula,"},{"Start":"05:45.650 ","End":"05:47.855","Text":"but we have a recursive formula,"},{"Start":"05:47.855 ","End":"05:50.055","Text":"which was this,"},{"Start":"05:50.055 ","End":"05:53.930","Text":"but it would be nicer to write it differently."},{"Start":"05:53.930 ","End":"05:55.790","Text":"We can write it this way,"},{"Start":"05:55.790 ","End":"05:58.625","Text":"which I got from this by replacing,"},{"Start":"05:58.625 ","End":"06:01.040","Text":"I want to make n plus 2 equal n,"},{"Start":"06:01.040 ","End":"06:06.540","Text":"so I replace n everywhere by n minus 2."},{"Start":"06:06.540 ","End":"06:09.540","Text":"That will give me here an, for example,"},{"Start":"06:09.540 ","End":"06:12.430","Text":"n bigger/equal to 2 will become n bigger/equal to 4,"},{"Start":"06:12.430 ","End":"06:13.775","Text":"and replacing throughout,"},{"Start":"06:13.775 ","End":"06:15.950","Text":"we get this recursion formula."},{"Start":"06:15.950 ","End":"06:23.970","Text":"This together with this is our power series and we\u0027re done."}],"Thumbnail":null,"ID":7927},{"Watched":false,"Name":"Exercise 5","Duration":"6m 21s","ChapterTopicVideoID":7820,"CourseChapterTopicPlaylistID":4243,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.159","Text":"Here, we have a differential equation,"},{"Start":"00:02.159 ","End":"00:05.730","Text":"it\u0027s linear and it\u0027s actually homogeneous."},{"Start":"00:05.730 ","End":"00:07.680","Text":"The coefficient here is 1,"},{"Start":"00:07.680 ","End":"00:12.300","Text":"and I note that x equals naught is irregular point because"},{"Start":"00:12.300 ","End":"00:17.595","Text":"both these coefficients of y prime and y are defined at x equals naught."},{"Start":"00:17.595 ","End":"00:19.350","Text":"We\u0027re going to do it with power series,"},{"Start":"00:19.350 ","End":"00:21.525","Text":"of course, that\u0027s the chapter we\u0027re in."},{"Start":"00:21.525 ","End":"00:25.665","Text":"We\u0027re going to expand around x equals naught with power series."},{"Start":"00:25.665 ","End":"00:28.290","Text":"We need to expand the brackets here and here,"},{"Start":"00:28.290 ","End":"00:31.095","Text":"and we actually get 5 terms,"},{"Start":"00:31.095 ","End":"00:33.600","Text":"and in each of them we\u0027re going to substitute y,"},{"Start":"00:33.600 ","End":"00:36.485","Text":"y prime or y double-prime as power series,"},{"Start":"00:36.485 ","End":"00:38.270","Text":"and here they are y, y prime,"},{"Start":"00:38.270 ","End":"00:40.549","Text":"and y double-prime you\u0027ve seen this before."},{"Start":"00:40.549 ","End":"00:43.160","Text":"The first of the 5 terms is y double-prime,"},{"Start":"00:43.160 ","End":"00:47.690","Text":"which I just copy from here but I don\u0027t need these,"},{"Start":"00:47.690 ","End":"00:50.240","Text":"they\u0027re swallowed up in the dot because I need"},{"Start":"00:50.240 ","End":"00:54.200","Text":"the coefficient of x^n in each of the 5 terms."},{"Start":"00:54.200 ","End":"00:57.275","Text":"The next one is xy prime."},{"Start":"00:57.275 ","End":"01:00.380","Text":"The x is going to increase the exponents by 1,"},{"Start":"01:00.380 ","End":"01:02.840","Text":"I stop at n minus 1,"},{"Start":"01:02.840 ","End":"01:06.515","Text":"which means that I don\u0027t need these,"},{"Start":"01:06.515 ","End":"01:08.510","Text":"and I guess I don\u0027t need this either."},{"Start":"01:08.510 ","End":"01:13.055","Text":"They\u0027re all in the dot and we still have 3 more to go."},{"Start":"01:13.055 ","End":"01:16.770","Text":"Minus y prime, just minus this,"},{"Start":"01:16.770 ","End":"01:20.030","Text":"and this time there\u0027s no power of x here,"},{"Start":"01:20.030 ","End":"01:22.510","Text":"I need to go up to x^n,"},{"Start":"01:22.510 ","End":"01:24.375","Text":"and this is redundant."},{"Start":"01:24.375 ","End":"01:28.030","Text":"That\u0027s what we get. The fourth one is 2xy."},{"Start":"01:28.030 ","End":"01:31.040","Text":"Because of the x which increases the powers by 1,"},{"Start":"01:31.040 ","End":"01:34.015","Text":"we stop at n minus 1."},{"Start":"01:34.015 ","End":"01:37.140","Text":"These are the redundant ones,"},{"Start":"01:37.140 ","End":"01:41.940","Text":"and lastly we have the minus 3y and there\u0027s no x\u0027s here,"},{"Start":"01:41.940 ","End":"01:44.570","Text":"this time I need to go up to x^n."},{"Start":"01:44.570 ","End":"01:47.965","Text":"These are the redundant ones,"},{"Start":"01:47.965 ","End":"01:51.920","Text":"and now we need to collect like terms together,"},{"Start":"01:51.920 ","End":"01:54.050","Text":"I need to take constant terms, x terms,"},{"Start":"01:54.050 ","End":"01:55.370","Text":"x squared terms,"},{"Start":"01:55.370 ","End":"01:58.070","Text":"the constant terms from here,"},{"Start":"01:58.070 ","End":"01:59.750","Text":"here, and here,"},{"Start":"01:59.750 ","End":"02:02.525","Text":"and so on and we need the nth term,"},{"Start":"02:02.525 ","End":"02:06.155","Text":"this we get from here."},{"Start":"02:06.155 ","End":"02:09.455","Text":"That\u0027s x^n, here because of the x,"},{"Start":"02:09.455 ","End":"02:12.740","Text":"we stop here, this becomes x^n."},{"Start":"02:12.740 ","End":"02:17.265","Text":"Here we take this with the minus,"},{"Start":"02:17.265 ","End":"02:21.335","Text":"by the way, and here we have an x,"},{"Start":"02:21.335 ","End":"02:23.045","Text":"so we take up to here,"},{"Start":"02:23.045 ","End":"02:27.045","Text":"but we also have the 2 plus 2 times this,"},{"Start":"02:27.045 ","End":"02:31.320","Text":"and then minus 3 times this."},{"Start":"02:31.320 ","End":"02:33.680","Text":"Collecting all the highlighted stuff,"},{"Start":"02:33.680 ","End":"02:37.170","Text":"we get 1, 2,"},{"Start":"02:37.170 ","End":"02:39.105","Text":"3, 4,"},{"Start":"02:39.105 ","End":"02:42.540","Text":"5 bits and x^n and so on,"},{"Start":"02:42.540 ","End":"02:45.725","Text":"and the right-hand side of the ODE was 0."},{"Start":"02:45.725 ","End":"02:47.840","Text":"I\u0027d like you to note already that"},{"Start":"02:47.840 ","End":"02:52.430","Text":"this general term applies already from n bigger or equal to 1."},{"Start":"02:52.430 ","End":"02:57.440","Text":"It doesn\u0027t apply to n=0 because here we only took 3 of the terms not the full 5,"},{"Start":"02:57.440 ","End":"02:59.660","Text":"but from x onwards,"},{"Start":"02:59.660 ","End":"03:01.730","Text":"we did exactly the same things,"},{"Start":"03:01.730 ","End":"03:03.345","Text":"and we get this."},{"Start":"03:03.345 ","End":"03:05.600","Text":"This applies for n bigger or equal to 1."},{"Start":"03:05.600 ","End":"03:07.310","Text":"Just copy this."},{"Start":"03:07.310 ","End":"03:10.190","Text":"All the coefficients here of all the powers of"},{"Start":"03:10.190 ","End":"03:12.965","Text":"x are going to be 0 because on the right-hand side we have 0,"},{"Start":"03:12.965 ","End":"03:16.700","Text":"and if we look at the x^n coefficient, we make it 0,"},{"Start":"03:16.700 ","End":"03:22.205","Text":"we get this, which simplifies to this when we take each a with an index separately."},{"Start":"03:22.205 ","End":"03:25.915","Text":"What we want to isolate is the one with the largest index which is this,"},{"Start":"03:25.915 ","End":"03:31.205","Text":"and that gives us a_n plus 2 in terms of lower indices."},{"Start":"03:31.205 ","End":"03:35.270","Text":"We said before that the supplies for n bigger or equal to 1,"},{"Start":"03:35.270 ","End":"03:37.220","Text":"and another way to see this anyway,"},{"Start":"03:37.220 ","End":"03:39.260","Text":"is that if you put n=0,"},{"Start":"03:39.260 ","End":"03:41.160","Text":"you would get a negative 1,"},{"Start":"03:41.160 ","End":"03:44.655","Text":"which doesn\u0027t make sense because a starts from 0."},{"Start":"03:44.655 ","End":"03:46.800","Text":"We have n bigger or equal to 1."},{"Start":"03:46.800 ","End":"03:49.460","Text":"We take care of the case n equals 0 separately."},{"Start":"03:49.460 ","End":"03:53.705","Text":"We just look here the n=0 is this thing,"},{"Start":"03:53.705 ","End":"03:56.335","Text":"and that\u0027s got to equal 0,"},{"Start":"03:56.335 ","End":"03:58.425","Text":"and if this is 0,"},{"Start":"03:58.425 ","End":"04:01.980","Text":"then it gives us a_2 in terms of a_1 and a naught."},{"Start":"04:01.980 ","End":"04:03.915","Text":"We weren\u0027t given an initial condition,"},{"Start":"04:03.915 ","End":"04:06.750","Text":"a naught and a_1 are"},{"Start":"04:06.750 ","End":"04:10.355","Text":"our 2 arbitrary general constants"},{"Start":"04:10.355 ","End":"04:15.445","Text":"like c_1 and c_2 when we solve differential equations without initial conditions."},{"Start":"04:15.445 ","End":"04:18.810","Text":"Next, let\u0027s find a_3."},{"Start":"04:18.810 ","End":"04:22.155","Text":"To get a_3, we plug in n=1 here."},{"Start":"04:22.155 ","End":"04:25.765","Text":"Then we get a_1 plus 2 which is a_3,"},{"Start":"04:25.765 ","End":"04:27.080","Text":"is equal to, well,"},{"Start":"04:27.080 ","End":"04:28.865","Text":"just substitute everywhere,"},{"Start":"04:28.865 ","End":"04:30.740","Text":"and this is what we get."},{"Start":"04:30.740 ","End":"04:32.240","Text":"If you figure this out,"},{"Start":"04:32.240 ","End":"04:37.355","Text":"it comes out to be 1/2a_1 plus 1/6 a naught."},{"Start":"04:37.355 ","End":"04:42.635","Text":"Along the way, I substituted a_2 from the value here,"},{"Start":"04:42.635 ","End":"04:44.630","Text":"and like I mentioned before,"},{"Start":"04:44.630 ","End":"04:47.450","Text":"everything\u0027s going to be in terms of a_1 and a naught."},{"Start":"04:47.450 ","End":"04:49.800","Text":"Similarly, if you plug in n=2,"},{"Start":"04:49.800 ","End":"04:51.190","Text":"we get a_4,"},{"Start":"04:51.190 ","End":"04:55.005","Text":"but this will be a_3,"},{"Start":"04:55.005 ","End":"04:58.670","Text":"and we\u0027ll have to substitute it from here,"},{"Start":"04:58.670 ","End":"05:01.710","Text":"and again, we\u0027ll get in terms of a_1 and a naught."},{"Start":"05:01.710 ","End":"05:02.790","Text":"Well, a_1 doesn\u0027t appear,"},{"Start":"05:02.790 ","End":"05:04.425","Text":"but we\u0027ve got a_4."},{"Start":"05:04.425 ","End":"05:09.780","Text":"So far, we said that a naught and a_1,"},{"Start":"05:09.780 ","End":"05:11.970","Text":"these are unknowns,"},{"Start":"05:11.970 ","End":"05:15.995","Text":"we treat them as if we know them but we don\u0027t, they\u0027re parameters,"},{"Start":"05:15.995 ","End":"05:20.040","Text":"and then we got a_2,"},{"Start":"05:20.040 ","End":"05:22.800","Text":"we got a_3, we got a_4."},{"Start":"05:22.800 ","End":"05:26.795","Text":"Now we can see what the power series looks like."},{"Start":"05:26.795 ","End":"05:29.075","Text":"Just want to keep this formula."},{"Start":"05:29.075 ","End":"05:31.140","Text":"We have a naught plus a_1x,"},{"Start":"05:31.140 ","End":"05:32.940","Text":"and then all the rest of it,"},{"Start":"05:32.940 ","End":"05:37.185","Text":"we got a_2 we copied that from here to here,"},{"Start":"05:37.185 ","End":"05:39.525","Text":"a_3, we copied from here,"},{"Start":"05:39.525 ","End":"05:42.795","Text":"a_4, we copied from here,"},{"Start":"05:42.795 ","End":"05:44.780","Text":"and then we have the general term a_n,"},{"Start":"05:44.780 ","End":"05:49.325","Text":"which is governed by the recursion formula but this is not in a convenient form,"},{"Start":"05:49.325 ","End":"05:55.016","Text":"we like to know what a_n is equal to so what we do is,"},{"Start":"05:55.016 ","End":"05:57.110","Text":"like a substitution everywhere we see n,"},{"Start":"05:57.110 ","End":"05:59.975","Text":"we replace it with n minus 2,"},{"Start":"05:59.975 ","End":"06:02.210","Text":"and then n minus 2 plus 2 will be a_n,"},{"Start":"06:02.210 ","End":"06:03.920","Text":"and if you replace it everywhere,"},{"Start":"06:03.920 ","End":"06:07.340","Text":"then we will get this recursion formula."},{"Start":"06:07.340 ","End":"06:10.760","Text":"I\u0027ll leave you to check that and notice that this is from n bigger or equal to 3,"},{"Start":"06:10.760 ","End":"06:14.120","Text":"and I\u0027d say that this expression for"},{"Start":"06:14.120 ","End":"06:18.185","Text":"y together with recursion formula which tells us how to find a_n,"},{"Start":"06:18.185 ","End":"06:21.570","Text":"that\u0027s the answer and we\u0027re done."}],"Thumbnail":null,"ID":7921},{"Watched":false,"Name":"Exercise 6","Duration":"5m 3s","ChapterTopicVideoID":7821,"CourseChapterTopicPlaylistID":4243,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"Here, we have a differential equation to solve with"},{"Start":"00:02.820 ","End":"00:07.560","Text":"initial conditions and we\u0027re going to do it with power series."},{"Start":"00:07.560 ","End":"00:12.090","Text":"We\u0027re going to develop y as a power series around t=0."},{"Start":"00:12.090 ","End":"00:14.460","Text":"That\u0027s a little bit of a difference that of x,"},{"Start":"00:14.460 ","End":"00:16.905","Text":"we have t. It\u0027s just for variety\u0027s sake."},{"Start":"00:16.905 ","End":"00:20.040","Text":"We have to check that it\u0027s regular, which it is."},{"Start":"00:20.040 ","End":"00:25.160","Text":"When t is not, this co-efficient is defined and there is no coefficient of y\u0027."},{"Start":"00:25.160 ","End":"00:31.170","Text":"Anyway, we can write y as a power series in t. Here we have y,"},{"Start":"00:31.170 ","End":"00:36.499","Text":"y\u0027, and y\u0027\u0027 and these will be substituted on the left and we\u0027ll get a power series."},{"Start":"00:36.499 ","End":"00:38.990","Text":"Even when you multiply t by y,"},{"Start":"00:38.990 ","End":"00:42.770","Text":"it will still be a power series but what about the right-hand side?"},{"Start":"00:42.770 ","End":"00:45.125","Text":"The right-hand side is also different."},{"Start":"00:45.125 ","End":"00:46.730","Text":"We usually get a polynomial here."},{"Start":"00:46.730 ","End":"00:48.500","Text":"We have e^t+1."},{"Start":"00:48.500 ","End":"00:50.854","Text":"Well, this is where we\u0027ll use Maclaurin series."},{"Start":"00:50.854 ","End":"00:53.540","Text":"This is the Maclaurin series for e^t."},{"Start":"00:53.540 ","End":"00:56.542","Text":"We have e^t+1. No problem,"},{"Start":"00:56.542 ","End":"01:00.365","Text":"e^t+1 is e^t times e^1."},{"Start":"01:00.365 ","End":"01:02.930","Text":"It\u0027s just e times e^t,"},{"Start":"01:02.930 ","End":"01:05.990","Text":"so just multiply this everywhere by t,"},{"Start":"01:05.990 ","End":"01:09.050","Text":"and that will take care of the right-hand side."},{"Start":"01:09.050 ","End":"01:11.360","Text":"Now we have to start substituting."},{"Start":"01:11.360 ","End":"01:15.050","Text":"I start with copying y\u0027\u0027 from here."},{"Start":"01:15.050 ","End":"01:17.000","Text":"Some of the terms are redundant."},{"Start":"01:17.000 ","End":"01:20.420","Text":"I need to have t^n."},{"Start":"01:20.420 ","End":"01:23.270","Text":"I guess I don\u0027t really need these,"},{"Start":"01:23.270 ","End":"01:25.250","Text":"they\u0027re swallowed up in the dot, dot, dot."},{"Start":"01:25.250 ","End":"01:29.780","Text":"Next, I need t times y. I\u0027ll raise the exponents here by"},{"Start":"01:29.780 ","End":"01:34.495","Text":"1 and I\u0027m going to have to scroll and we\u0027ll lose the original equation. Never mind."},{"Start":"01:34.495 ","End":"01:36.000","Text":"Now, if we multiply this by t,"},{"Start":"01:36.000 ","End":"01:41.618","Text":"this becomes the t^n term so these are unnecessary,"},{"Start":"01:41.618 ","End":"01:43.475","Text":"and this is what we get."},{"Start":"01:43.475 ","End":"01:48.230","Text":"Now we need the right-hand side, which is e^t+1."},{"Start":"01:48.230 ","End":"01:50.540","Text":"This is what we get because like I said,"},{"Start":"01:50.540 ","End":"01:53.540","Text":"you just multiply e by e^t."},{"Start":"01:53.540 ","End":"01:58.295","Text":"Now we have to simplify and then compare coefficients."},{"Start":"01:58.295 ","End":"02:04.355","Text":"On the left, we collect together the various powers of t. The constant term is just this."},{"Start":"02:04.355 ","End":"02:07.250","Text":"The t term comes from here and here,"},{"Start":"02:07.250 ","End":"02:10.310","Text":"the t^2 term from here and here,"},{"Start":"02:10.310 ","End":"02:11.750","Text":"then from here and here,"},{"Start":"02:11.750 ","End":"02:16.715","Text":"and in the middle of the general term is t^n."},{"Start":"02:16.715 ","End":"02:20.510","Text":"We get that from here because this becomes t^n and here."},{"Start":"02:20.510 ","End":"02:23.690","Text":"Now, the only exception is this doesn\u0027t have a partner."},{"Start":"02:23.690 ","End":"02:26.240","Text":"I\u0027m mentioning this because there\u0027ll be a rule for these,"},{"Start":"02:26.240 ","End":"02:27.650","Text":"but it won\u0027t apply to this."},{"Start":"02:27.650 ","End":"02:28.835","Text":"On the right-hand side,"},{"Start":"02:28.835 ","End":"02:31.240","Text":"we just multiplied everything here by e,"},{"Start":"02:31.240 ","End":"02:32.525","Text":"just threw it in."},{"Start":"02:32.525 ","End":"02:35.660","Text":"The recursion rule comes from comparing the coefficient of"},{"Start":"02:35.660 ","End":"02:39.515","Text":"t^n on the left and on the right."},{"Start":"02:39.515 ","End":"02:43.535","Text":"On the right, it\u0027s e over n factorial,"},{"Start":"02:43.535 ","End":"02:48.140","Text":"and on the left, it\u0027s the sum of these two things here."},{"Start":"02:48.140 ","End":"02:53.910","Text":"This will apply to all n from 1 onwards,"},{"Start":"02:53.910 ","End":"02:56.209","Text":"and in any event you couldn\u0027t substitute"},{"Start":"02:56.209 ","End":"03:00.170","Text":"n=0 here because you\u0027d get a subscript negative 1, no."},{"Start":"03:00.170 ","End":"03:05.060","Text":"We take care separately of the case of n=0."},{"Start":"03:05.060 ","End":"03:06.665","Text":"Yeah, but that in a moment."},{"Start":"03:06.665 ","End":"03:09.380","Text":"Meanwhile, we\u0027ll take the a_n+2,"},{"Start":"03:09.380 ","End":"03:11.090","Text":"which is the largest index,"},{"Start":"03:11.090 ","End":"03:13.490","Text":"and write it in terms of the others,"},{"Start":"03:13.490 ","End":"03:17.780","Text":"just throw everything else to the right and divide by the coefficient."},{"Start":"03:17.780 ","End":"03:22.615","Text":"This is what we get. This is the recursion formula for a_n+2."},{"Start":"03:22.615 ","End":"03:26.380","Text":"Now, let\u0027s get to that matter of the a_0."},{"Start":"03:26.380 ","End":"03:30.050","Text":"The free coefficient which is this is this here."},{"Start":"03:30.050 ","End":"03:34.900","Text":"We get this equation and that gives us what a_2 is."},{"Start":"03:34.900 ","End":"03:41.915","Text":"Next, let\u0027s compare t. We have this on this side and on this side we also have e,"},{"Start":"03:41.915 ","End":"03:43.400","Text":"so we get this."},{"Start":"03:43.400 ","End":"03:49.400","Text":"The reason I got this was because a_0 is equal to 1,"},{"Start":"03:49.400 ","End":"03:51.440","Text":"that\u0027s from the initial conditions or while I\u0027m at it,"},{"Start":"03:51.440 ","End":"03:54.395","Text":"I\u0027ll remind you that a_1 is equal to 2."},{"Start":"03:54.395 ","End":"03:58.210","Text":"Let\u0027s do one more of these."},{"Start":"03:58.210 ","End":"04:01.310","Text":"If you compare the coefficients of t^2,"},{"Start":"04:01.310 ","End":"04:03.905","Text":"we get this on one side and this on the other."},{"Start":"04:03.905 ","End":"04:09.975","Text":"We get a_4 equals this because a_1 =2."},{"Start":"04:09.975 ","End":"04:15.080","Text":"We have the general shape of the power series for y as a function of t,"},{"Start":"04:15.080 ","End":"04:17.090","Text":"at least up to t^4,"},{"Start":"04:17.090 ","End":"04:21.050","Text":"we have the coefficients and for the general term,"},{"Start":"04:21.050 ","End":"04:22.400","Text":"we don\u0027t have an explicit formula,"},{"Start":"04:22.400 ","End":"04:24.265","Text":"but we have a recursive formula."},{"Start":"04:24.265 ","End":"04:28.700","Text":"We want to modify this though that it gives us what a_n equals."},{"Start":"04:28.700 ","End":"04:31.820","Text":"That\u0027s the standard trick of,"},{"Start":"04:31.820 ","End":"04:33.410","Text":"to get a_n here,"},{"Start":"04:33.410 ","End":"04:39.270","Text":"we\u0027d replace n by n-2 everywhere in this formula."},{"Start":"04:39.270 ","End":"04:45.590","Text":"Then n-2+2 would be a_n and we would get this formula."},{"Start":"04:45.590 ","End":"04:47.570","Text":"I guess I should have put it in a box."},{"Start":"04:47.570 ","End":"04:49.175","Text":"Maybe it\u0027s not too late."},{"Start":"04:49.175 ","End":"04:50.765","Text":"There we are."},{"Start":"04:50.765 ","End":"04:55.280","Text":"Really this together with this will give us the answer."},{"Start":"04:55.280 ","End":"04:56.570","Text":"We have the power series,"},{"Start":"04:56.570 ","End":"05:03.930","Text":"we have the recursion formula for computing as many a_n\u0027s as we want and we are done."}],"Thumbnail":null,"ID":7922},{"Watched":false,"Name":"Exercise 7","Duration":"9m 34s","ChapterTopicVideoID":7822,"CourseChapterTopicPlaylistID":4243,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.030","Text":"This exercise is actually one we\u0027ve solved already."},{"Start":"00:03.030 ","End":"00:04.980","Text":"If you look a couple of exercises back,"},{"Start":"00:04.980 ","End":"00:09.390","Text":"we had this with a slight difference that we had x in place of t there,"},{"Start":"00:09.390 ","End":"00:11.685","Text":"but that obviously makes no difference."},{"Start":"00:11.685 ","End":"00:16.210","Text":"The reason I\u0027m giving it again is as another way of doing it,"},{"Start":"00:16.210 ","End":"00:18.285","Text":"or organizing it using"},{"Start":"00:18.285 ","End":"00:23.925","Text":"the shorthand Sigma notation instead of the dot-dot-dot ellipsis notation."},{"Start":"00:23.925 ","End":"00:27.060","Text":"First, I\u0027ll open the brackets here and here,"},{"Start":"00:27.060 ","End":"00:30.195","Text":"so we get this as now 5 terms."},{"Start":"00:30.195 ","End":"00:31.815","Text":"I don\u0027t need this."},{"Start":"00:31.815 ","End":"00:33.877","Text":"We\u0027re still going to solve it with power series,"},{"Start":"00:33.877 ","End":"00:35.460","Text":"but using Sigma notation,"},{"Start":"00:35.460 ","End":"00:37.740","Text":"and I want to remind you of a couple of rules."},{"Start":"00:37.740 ","End":"00:43.730","Text":"If I have a sum going from sum number p to infinity of this power series,"},{"Start":"00:43.730 ","End":"00:47.195","Text":"I can change the starting point,"},{"Start":"00:47.195 ","End":"00:52.505","Text":"say I want to make it start by adding k. We can do that"},{"Start":"00:52.505 ","End":"00:55.340","Text":"provided that we subtract k in"},{"Start":"00:55.340 ","End":"00:59.615","Text":"these two places where we had n. To compensate for adding k,"},{"Start":"00:59.615 ","End":"01:03.215","Text":"we subtract k and it turns out to be the same sum."},{"Start":"01:03.215 ","End":"01:08.360","Text":"Similarly, if you want to make the starting point smaller by subtracting k,"},{"Start":"01:08.360 ","End":"01:11.330","Text":"then we do the opposite to n here, and here,"},{"Start":"01:11.330 ","End":"01:15.530","Text":"and we add k. Be aware of these two rules."},{"Start":"01:15.530 ","End":"01:18.895","Text":"Now when we write y as a power series in t,"},{"Start":"01:18.895 ","End":"01:20.670","Text":"around t equals Naught,"},{"Start":"01:20.670 ","End":"01:24.160","Text":"we don\u0027t have this dot-dot-dot and lots of terms."},{"Start":"01:24.160 ","End":"01:29.968","Text":"We have it nice, and compact in Sigma form where n goes from 0 to infinity,"},{"Start":"01:29.968 ","End":"01:33.215","Text":"and the general term is a_n t^n,"},{"Start":"01:33.215 ","End":"01:35.930","Text":"y\u0027 comes out to be this."},{"Start":"01:35.930 ","End":"01:38.960","Text":"But note that it starts from 1,"},{"Start":"01:38.960 ","End":"01:41.900","Text":"not from 0, and y\u0027\u0027,"},{"Start":"01:41.900 ","End":"01:44.610","Text":"which is this, starts from 2."},{"Start":"01:44.610 ","End":"01:46.920","Text":"This one starts from 0."},{"Start":"01:46.920 ","End":"01:49.760","Text":"At this point I want to emphasize this is optional,"},{"Start":"01:49.760 ","End":"01:52.070","Text":"if you don\u0027t like the Sigma notation,"},{"Start":"01:52.070 ","End":"01:54.905","Text":"let\u0027s go back to the old notation,"},{"Start":"01:54.905 ","End":"01:56.630","Text":"the ellipsis, the dot-dot-dot."},{"Start":"01:56.630 ","End":"01:58.700","Text":"This is shorter to write."},{"Start":"01:58.700 ","End":"02:00.605","Text":"It\u0027s neater, it\u0027s more precise,"},{"Start":"02:00.605 ","End":"02:03.395","Text":"but it\u0027s more error prone if you\u0027re not careful."},{"Start":"02:03.395 ","End":"02:05.248","Text":"If you\u0027re good with Sigmas,"},{"Start":"02:05.248 ","End":"02:07.370","Text":"and you\u0027ve seen them before, I suggest doing it this way."},{"Start":"02:07.370 ","End":"02:11.090","Text":"Otherwise, you might even skip this clip."},{"Start":"02:11.090 ","End":"02:15.110","Text":"Now, this is what I get if I substitute y, y\u0027,"},{"Start":"02:15.110 ","End":"02:18.815","Text":"and y\u0027\u0027 in here."},{"Start":"02:18.815 ","End":"02:21.770","Text":"y\u0027\u0027, y\u0027 here and here,"},{"Start":"02:21.770 ","End":"02:23.270","Text":"y here and here,"},{"Start":"02:23.270 ","End":"02:25.940","Text":"we get exactly this."},{"Start":"02:25.940 ","End":"02:29.555","Text":"Just check this is the same as this with these substitutions."},{"Start":"02:29.555 ","End":"02:33.275","Text":"Now, what I want to do is expand."},{"Start":"02:33.275 ","End":"02:35.105","Text":"I have a t here,"},{"Start":"02:35.105 ","End":"02:40.700","Text":"and I want to multiply this t inside by the t^n minus 1."},{"Start":"02:40.700 ","End":"02:43.460","Text":"Also here I have a t, anywhere,"},{"Start":"02:43.460 ","End":"02:44.870","Text":"I have powers of t,"},{"Start":"02:44.870 ","End":"02:46.760","Text":"I multiply them in."},{"Start":"02:46.760 ","End":"02:49.820","Text":"If I do that, then this is what we get."},{"Start":"02:49.820 ","End":"02:55.235","Text":"The difference is here that this t^n minus 1 became t^n."},{"Start":"02:55.235 ","End":"03:00.380","Text":"Here, this t^n became t^n plus 1."},{"Start":"03:00.380 ","End":"03:03.088","Text":"Now if I look at all these terms,"},{"Start":"03:03.088 ","End":"03:04.895","Text":"and look at the power of t,"},{"Start":"03:04.895 ","End":"03:06.490","Text":"it\u0027s out of sync."},{"Start":"03:06.490 ","End":"03:09.885","Text":"Here I have t^n minus 2, here t^n,"},{"Start":"03:09.885 ","End":"03:11.550","Text":"here t^n minus 1,"},{"Start":"03:11.550 ","End":"03:13.685","Text":"t^n plus 1, t^n."},{"Start":"03:13.685 ","End":"03:16.610","Text":"I\u0027d really like to have them all with t^n."},{"Start":"03:16.610 ","End":"03:20.419","Text":"That\u0027s where the rules I wrote earlier are going to come in useful."},{"Start":"03:20.419 ","End":"03:23.855","Text":"This is what I get and I\u0027ll explain it one term at a time."},{"Start":"03:23.855 ","End":"03:26.110","Text":"Now, in the first Sigma,"},{"Start":"03:26.110 ","End":"03:28.085","Text":"I subtracted 2 from here,"},{"Start":"03:28.085 ","End":"03:31.040","Text":"and to compensate, I have to add 2 here,"},{"Start":"03:31.040 ","End":"03:32.840","Text":"so n becomes n plus 2."},{"Start":"03:32.840 ","End":"03:35.900","Text":"I have to add 2 to n here and here."},{"Start":"03:35.900 ","End":"03:37.925","Text":"This is what I get."},{"Start":"03:37.925 ","End":"03:42.380","Text":"This, after the substitution becomes this,"},{"Start":"03:42.380 ","End":"03:44.905","Text":"when I subtract 2 add 2,"},{"Start":"03:44.905 ","End":"03:46.530","Text":"this is as is,"},{"Start":"03:46.530 ","End":"03:48.045","Text":"no need to comment."},{"Start":"03:48.045 ","End":"03:52.010","Text":"Here I have to subtract 1 here to get n equals 0,"},{"Start":"03:52.010 ","End":"03:56.705","Text":"and then I need to add 1. n becomes n plus 1 here and here,"},{"Start":"03:56.705 ","End":"04:03.515","Text":"and this n minus 1 becomes n. This term is this term."},{"Start":"04:03.515 ","End":"04:08.930","Text":"The next one, I want to subtract 1 here to get this t^n,"},{"Start":"04:08.930 ","End":"04:10.925","Text":"and I have to subtract 1 here also,"},{"Start":"04:10.925 ","End":"04:12.200","Text":"but here I add 1,"},{"Start":"04:12.200 ","End":"04:14.455","Text":"so I go from 0 to 1."},{"Start":"04:14.455 ","End":"04:18.560","Text":"This is the same as this,"},{"Start":"04:18.560 ","End":"04:19.895","Text":"and the last one also,"},{"Start":"04:19.895 ","End":"04:22.790","Text":"I leave as is. Now, it\u0027s good."},{"Start":"04:22.790 ","End":"04:25.340","Text":"I have everything in terms of t^n,"},{"Start":"04:25.340 ","End":"04:26.600","Text":"1, 2, 3,"},{"Start":"04:26.600 ","End":"04:28.265","Text":"4, 5 different terms."},{"Start":"04:28.265 ","End":"04:33.514","Text":"The only thing that\u0027s slightly out of sync now is the starting point, which here is 0."},{"Start":"04:33.514 ","End":"04:34.875","Text":"Here is 1,"},{"Start":"04:34.875 ","End":"04:38.625","Text":"here it\u0027s 0, here it\u0027s 1, and here it\u0027s 0."},{"Start":"04:38.625 ","End":"04:44.055","Text":"Certainly, the term from n equals 1 onwards belongs to all of them."},{"Start":"04:44.055 ","End":"04:48.670","Text":"I can collect the t^n coefficients,"},{"Start":"04:48.670 ","End":"04:52.985","Text":"add them up to get 0 because I have 0 t^n on the right."},{"Start":"04:52.985 ","End":"04:57.470","Text":"Collecting coefficients, I take this which is here,"},{"Start":"04:57.470 ","End":"05:00.215","Text":"this is here, and so on."},{"Start":"05:00.215 ","End":"05:02.360","Text":"We get this equation,"},{"Start":"05:02.360 ","End":"05:05.390","Text":"which is true when n is bigger or equal to 1,"},{"Start":"05:05.390 ","End":"05:06.905","Text":"because when n is 0,"},{"Start":"05:06.905 ","End":"05:09.065","Text":"here and here there is no term,"},{"Start":"05:09.065 ","End":"05:13.025","Text":"and we\u0027ll deal with the case of 0 later on."},{"Start":"05:13.025 ","End":"05:16.970","Text":"I\u0027ll get back to it if they were then bigger or equal to 1 meanwhile."},{"Start":"05:16.970 ","End":"05:21.380","Text":"Now, if I extract from here the largest index,"},{"Start":"05:21.380 ","End":"05:22.995","Text":"which is a_n plus 2,"},{"Start":"05:22.995 ","End":"05:25.160","Text":"not quite, I want to first collect like terms."},{"Start":"05:25.160 ","End":"05:30.095","Text":"The only like terms is this one with this one which gives me the n minus 3 here."},{"Start":"05:30.095 ","End":"05:32.180","Text":"Now I can extract,"},{"Start":"05:32.180 ","End":"05:34.040","Text":"this is just straightforward algebra,"},{"Start":"05:34.040 ","End":"05:35.563","Text":"bringing stuff to the other side,"},{"Start":"05:35.563 ","End":"05:37.010","Text":"and dividing by this coefficient,"},{"Start":"05:37.010 ","End":"05:43.645","Text":"we get a_n plus 2 recursively equal to a\u0027s with a lower index."},{"Start":"05:43.645 ","End":"05:47.210","Text":"Now, what I wrote here is back here."},{"Start":"05:47.210 ","End":"05:49.550","Text":"Remember we said we\u0027re going to take n equals 0"},{"Start":"05:49.550 ","End":"05:53.525","Text":"separately because it doesn\u0027t appear everywhere."},{"Start":"05:53.525 ","End":"05:55.790","Text":"Here we have n equals 0,"},{"Start":"05:55.790 ","End":"05:57.560","Text":"and when n is 0,"},{"Start":"05:57.560 ","End":"06:04.285","Text":"the coefficient is a_2 times 0 plus 2_0 plus 1 is 2, we get 2a_2."},{"Start":"06:04.285 ","End":"06:06.230","Text":"When n is 0 here,"},{"Start":"06:06.230 ","End":"06:09.140","Text":"we get minus and 0 plus 1 is 1."},{"Start":"06:09.140 ","End":"06:12.215","Text":"It\u0027s a_1 which is this."},{"Start":"06:12.215 ","End":"06:15.755","Text":"In this one where n is 0,"},{"Start":"06:15.755 ","End":"06:19.025","Text":"we get just minus 3a_0,"},{"Start":"06:19.025 ","End":"06:21.425","Text":"and all this is equal to 0."},{"Start":"06:21.425 ","End":"06:23.210","Text":"This is good for n big or equal to 1,"},{"Start":"06:23.210 ","End":"06:25.945","Text":"this is what happened when n was 0."},{"Start":"06:25.945 ","End":"06:28.905","Text":"We extract a_2,"},{"Start":"06:28.905 ","End":"06:31.570","Text":"that a_2 in terms of a_1 and a_0."},{"Start":"06:31.570 ","End":"06:35.405","Text":"Now, we\u0027re assuming, since we don\u0027t have initial conditions that"},{"Start":"06:35.405 ","End":"06:39.559","Text":"a_0 and a_1 are some general constants,"},{"Start":"06:39.559 ","End":"06:41.300","Text":"the parameters, we don\u0027t know what they are,"},{"Start":"06:41.300 ","End":"06:45.521","Text":"but we can express everything else in terms of a_0 and a_1,"},{"Start":"06:45.521 ","End":"06:51.800","Text":"just a bit like we used to have c_1 and c_2 is 2 parameters for second-order equations."},{"Start":"06:51.800 ","End":"06:54.500","Text":"We have a_0, and a_1 as if they\u0027re given."},{"Start":"06:54.500 ","End":"06:57.635","Text":"Now we have a_2 in terms of these,"},{"Start":"06:57.635 ","End":"07:00.030","Text":"and all the rest of them, a_3,"},{"Start":"07:00.030 ","End":"07:02.450","Text":"a_4 and so on can be gotten from here."},{"Start":"07:02.450 ","End":"07:04.930","Text":"For example, if I let n=1,"},{"Start":"07:04.930 ","End":"07:06.680","Text":"and this will be a_0, a_1,"},{"Start":"07:06.680 ","End":"07:09.800","Text":"a_2, and we\u0027ll use those to give a_3."},{"Start":"07:09.800 ","End":"07:11.675","Text":"In fact, let\u0027s do that."},{"Start":"07:11.675 ","End":"07:17.445","Text":"What we get if we put n is 1 is here,1 plus 2 is 3."},{"Start":"07:17.445 ","End":"07:19.530","Text":"This is a_2, this is a_1,"},{"Start":"07:19.530 ","End":"07:21.780","Text":"this is a_1 minus 1,"},{"Start":"07:21.780 ","End":"07:24.675","Text":"which is really a_0."},{"Start":"07:24.675 ","End":"07:27.020","Text":"Again, substituting n equals 1,"},{"Start":"07:27.020 ","End":"07:28.625","Text":"we get all these numbers."},{"Start":"07:28.625 ","End":"07:35.155","Text":"The last thing we could do is substitute a_2 we have over here,"},{"Start":"07:35.155 ","End":"07:38.910","Text":"so everything will now be in terms of a_1 and a_0,"},{"Start":"07:38.910 ","End":"07:41.075","Text":"and I\u0027ll show you what we get."},{"Start":"07:41.075 ","End":"07:43.175","Text":"It simplifies to this."},{"Start":"07:43.175 ","End":"07:44.825","Text":"Let me just do a brief summary,"},{"Start":"07:44.825 ","End":"07:46.355","Text":"a_0 and a_1,"},{"Start":"07:46.355 ","End":"07:48.260","Text":"we don\u0027t know, but they are general constants."},{"Start":"07:48.260 ","End":"07:49.670","Text":"It\u0027s as if we know them."},{"Start":"07:49.670 ","End":"07:54.035","Text":"Then we have a_2 in terms of the known concepts."},{"Start":"07:54.035 ","End":"07:56.770","Text":"Then we found a_3."},{"Start":"07:56.770 ","End":"07:59.860","Text":"Let\u0027s do one more, let\u0027s do a_4."},{"Start":"07:59.860 ","End":"08:02.075","Text":"This is what we get when we substitute."},{"Start":"08:02.075 ","End":"08:04.340","Text":"This of course is a_3."},{"Start":"08:04.340 ","End":"08:06.365","Text":"This is a_1,"},{"Start":"08:06.365 ","End":"08:11.020","Text":"and a_3 we substitute from here,"},{"Start":"08:11.020 ","End":"08:17.105","Text":"and a2 we substitute from over here."},{"Start":"08:17.105 ","End":"08:18.665","Text":"If we do all that,"},{"Start":"08:18.665 ","End":"08:24.180","Text":"we get that a_4 is 1/6a_0,"},{"Start":"08:24.180 ","End":"08:26.475","Text":"a1 doesn\u0027t explicitly appear here."},{"Start":"08:26.475 ","End":"08:28.400","Text":"We can keep going like that."},{"Start":"08:28.400 ","End":"08:30.830","Text":"We could do a_5 and so on."},{"Start":"08:30.830 ","End":"08:33.980","Text":"Essentially, this rule gives us how to perpetuate,"},{"Start":"08:33.980 ","End":"08:36.020","Text":"how to keep generating n\u0027s."},{"Start":"08:36.020 ","End":"08:40.160","Text":"Our series, if I drop the Sigma notation,"},{"Start":"08:40.160 ","End":"08:42.990","Text":"I can write the coefficients a naught,"},{"Start":"08:42.990 ","End":"08:44.685","Text":"a_1, a_2,"},{"Start":"08:44.685 ","End":"08:46.740","Text":"a_3, and a_4."},{"Start":"08:46.740 ","End":"08:51.260","Text":"Then as many as we want to compute and then the general term a_n t^n,"},{"Start":"08:51.260 ","End":"08:55.055","Text":"which is obtained by this recursive formula."},{"Start":"08:55.055 ","End":"09:01.070","Text":"But, the recursive formula would be more convenient if I had what is a_n."},{"Start":"09:01.070 ","End":"09:02.810","Text":"Again, we do a substitution."},{"Start":"09:02.810 ","End":"09:08.084","Text":"We replace n here by n minus 2 and we\u0027ve done this before."},{"Start":"09:08.084 ","End":"09:10.730","Text":"What we get is this."},{"Start":"09:10.730 ","End":"09:14.750","Text":"Note that we also substitute the n big,"},{"Start":"09:14.750 ","End":"09:18.090","Text":"or equal to 1 becomes now n big or equal to 3."},{"Start":"09:18.090 ","End":"09:23.675","Text":"This recursion rule, together with this general shape of y,"},{"Start":"09:23.675 ","End":"09:27.830","Text":"will be the recursive definition of the power series."},{"Start":"09:27.830 ","End":"09:29.870","Text":"If you go check the answer,"},{"Start":"09:29.870 ","End":"09:34.260","Text":"we did it two exercises ago. We\u0027re done."}],"Thumbnail":null,"ID":7923},{"Watched":false,"Name":"Exercise 8","Duration":"3m 4s","ChapterTopicVideoID":7823,"CourseChapterTopicPlaylistID":4243,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.580","Text":"In this exercise, we have to solve this differential equation and we have"},{"Start":"00:05.580 ","End":"00:08.790","Text":"the initial conditions and we\u0027re told that"},{"Start":"00:08.790 ","End":"00:12.225","Text":"we should develop the solution as a power series around x=1."},{"Start":"00:12.225 ","End":"00:15.885","Text":"Now we\u0027re in the chapter on power series and up to now,"},{"Start":"00:15.885 ","End":"00:19.770","Text":"we\u0027ve mostly or always seen around x equals 0."},{"Start":"00:19.770 ","End":"00:22.740","Text":"Now even if they haven\u0027t told us around x=1,"},{"Start":"00:22.740 ","End":"00:24.540","Text":"we would\u0027ve guessed that because look,"},{"Start":"00:24.540 ","End":"00:26.970","Text":"the value of y and y\u0027,"},{"Start":"00:26.970 ","End":"00:31.815","Text":"the initial condition is given at x=1."},{"Start":"00:31.815 ","End":"00:37.515","Text":"This would be a strong hint that x=1 is the point we\u0027re talking about."},{"Start":"00:37.515 ","End":"00:42.005","Text":"Before we start with power series just to make sure that x=1 is a regular point."},{"Start":"00:42.005 ","End":"00:43.939","Text":"If you look at the differential equation,"},{"Start":"00:43.939 ","End":"00:45.525","Text":"this coefficient is 1."},{"Start":"00:45.525 ","End":"00:49.175","Text":"We have a p and a q. Well, the p is 0 here because there\u0027s no y\u0027."},{"Start":"00:49.175 ","End":"00:52.310","Text":"This is the q and it\u0027s defined that x=1."},{"Start":"00:52.310 ","End":"00:55.070","Text":"No problem, all is okay."},{"Start":"00:55.070 ","End":"00:58.550","Text":"Now, how do we do around x=1?"},{"Start":"00:58.550 ","End":"01:03.200","Text":"Well, the system is to make a substitution and let me convert our problem to"},{"Start":"01:03.200 ","End":"01:08.195","Text":"around x= naught or maybe naught x but t. Well, you\u0027ll see what I mean."},{"Start":"01:08.195 ","End":"01:11.780","Text":"Here\u0027s the trick. We let t equals x=1."},{"Start":"01:11.780 ","End":"01:13.205","Text":"The 1 is the one from here."},{"Start":"01:13.205 ","End":"01:15.140","Text":"If we had to develop around x=3,"},{"Start":"01:15.140 ","End":"01:18.215","Text":"we put t=x minus 3 and so on."},{"Start":"01:18.215 ","End":"01:20.090","Text":"Now we convert everything."},{"Start":"01:20.090 ","End":"01:24.350","Text":"Well, we change y(x) to y(t) and at the end,"},{"Start":"01:24.350 ","End":"01:29.225","Text":"we\u0027re going to plug back t=x minus 1 at the very end."},{"Start":"01:29.225 ","End":"01:35.185","Text":"We have y\u0027\u0027(t) plus x minus 1 is t,"},{"Start":"01:35.185 ","End":"01:40.770","Text":"y(t) and here x is t plus 1."},{"Start":"01:40.770 ","End":"01:43.495","Text":"We write e^t plus 1."},{"Start":"01:43.495 ","End":"01:47.255","Text":"Also, the initial conditions when x is 1,"},{"Start":"01:47.255 ","End":"01:49.690","Text":"t is 0, 1 minus 1."},{"Start":"01:49.690 ","End":"01:51.335","Text":"Both here and here,"},{"Start":"01:51.335 ","End":"01:54.250","Text":"we get the condition around 0."},{"Start":"01:54.250 ","End":"01:57.875","Text":"Now we\u0027re all set up to solve around t= naught,"},{"Start":"01:57.875 ","End":"01:59.480","Text":"which we know how to do."},{"Start":"01:59.480 ","End":"02:03.005","Text":"But luckily, or maybe this was on purpose."},{"Start":"02:03.005 ","End":"02:06.545","Text":"This happens to be the same equation as an exercise,"},{"Start":"02:06.545 ","End":"02:11.150","Text":"I believe it\u0027s exercise 6 or one of the couple of exercises ago."},{"Start":"02:11.150 ","End":"02:13.085","Text":"We got the solution there."},{"Start":"02:13.085 ","End":"02:15.380","Text":"Go take a look at, hopefully, you\u0027ll find it."},{"Start":"02:15.380 ","End":"02:18.430","Text":"We\u0027ve got this as the solution with, of course,"},{"Start":"02:18.430 ","End":"02:22.160","Text":"we need the recursive condition on a_n and here it is"},{"Start":"02:22.160 ","End":"02:27.260","Text":"the condition that gives an in terms of previous values of a_n."},{"Start":"02:27.260 ","End":"02:31.880","Text":"Now, all we have to do is go back from the Lambda(t) to the Lambda(x)."},{"Start":"02:31.880 ","End":"02:35.785","Text":"As I said, we substitute back the t=x minus 1."},{"Start":"02:35.785 ","End":"02:38.510","Text":"Here I just basically copy this,"},{"Start":"02:38.510 ","End":"02:43.040","Text":"except that where I see t each place I put x minus 1."},{"Start":"02:43.040 ","End":"02:47.950","Text":"This is not Maclaurin series is a Taylor series around x=1."},{"Start":"02:47.950 ","End":"02:50.250","Text":"Here\u0027s the general term."},{"Start":"02:50.250 ","End":"02:52.380","Text":"But the coefficients don\u0027t change,"},{"Start":"02:52.380 ","End":"02:55.640","Text":"the an doesn\u0027t change so it\u0027s still the same recursive formula."},{"Start":"02:55.640 ","End":"02:57.200","Text":"This together with this,"},{"Start":"02:57.200 ","End":"03:00.229","Text":"define the solution as a power series,"},{"Start":"03:00.229 ","End":"03:05.380","Text":"specifically a Taylor series around x=1. We\u0027re done."}],"Thumbnail":null,"ID":7924},{"Watched":false,"Name":"Exercise 9","Duration":"2m 56s","ChapterTopicVideoID":7824,"CourseChapterTopicPlaylistID":4243,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.065","Text":"Here, we have this second-order differential equation to solve."},{"Start":"00:04.065 ","End":"00:08.858","Text":"It\u0027s homogeneous, it has non-constant coefficients,"},{"Start":"00:08.858 ","End":"00:10.590","Text":"and we have initial conditions."},{"Start":"00:10.590 ","End":"00:12.840","Text":"Now, because of the non-constant coefficients,"},{"Start":"00:12.840 ","End":"00:15.630","Text":"we\u0027re probably talking about power series."},{"Start":"00:15.630 ","End":"00:21.140","Text":"But notice that there\u0027s a minus 1 here and here in the initial conditions so"},{"Start":"00:21.140 ","End":"00:26.480","Text":"we\u0027re probably looking for a power series around x equals minus 1."},{"Start":"00:26.480 ","End":"00:32.870","Text":"We prefer to have power series around 0 so we\u0027re going to use our standard trick."},{"Start":"00:32.870 ","End":"00:38.165","Text":"Just before I start, I have to check that x equals minus 1 is a regular point of this."},{"Start":"00:38.165 ","End":"00:39.680","Text":"Here, the coefficient is 1."},{"Start":"00:39.680 ","End":"00:40.895","Text":"We have to check."},{"Start":"00:40.895 ","End":"00:43.325","Text":"This is our p and this is our q."},{"Start":"00:43.325 ","End":"00:45.290","Text":"They defined that x equals minus 1."},{"Start":"00:45.290 ","End":"00:46.490","Text":"Sure, no problem."},{"Start":"00:46.490 ","End":"00:47.840","Text":"We can proceed."},{"Start":"00:47.840 ","End":"00:49.280","Text":"Just wrote what I said over here."},{"Start":"00:49.280 ","End":"00:54.140","Text":"Yeah. The trick is to let t equals x plus 1."},{"Start":"00:54.140 ","End":"00:56.840","Text":"Because this is minus 1, it\u0027s x minus 1,"},{"Start":"00:56.840 ","End":"00:58.670","Text":"so it\u0027s x plus 1."},{"Start":"00:58.670 ","End":"01:04.025","Text":"What we get is we substitute everything and we let"},{"Start":"01:04.025 ","End":"01:09.440","Text":"y(t) and we replace x by t minus 1."},{"Start":"01:09.440 ","End":"01:11.255","Text":"I guess that\u0027s the reverse substitution,"},{"Start":"01:11.255 ","End":"01:13.325","Text":"x equals t minus 1."},{"Start":"01:13.325 ","End":"01:16.250","Text":"Here also, if we put x equals t minus 1,"},{"Start":"01:16.250 ","End":"01:18.785","Text":"we get 2t minus 2 minus 1, which is this."},{"Start":"01:18.785 ","End":"01:20.449","Text":"Now, the initial conditions,"},{"Start":"01:20.449 ","End":"01:23.670","Text":"when x is minus 1,"},{"Start":"01:23.670 ","End":"01:26.340","Text":"then t is,"},{"Start":"01:26.340 ","End":"01:28.800","Text":"minus 1 plus 1 is 0 so we have 0 here and here."},{"Start":"01:28.800 ","End":"01:30.890","Text":"This is our differential equation,"},{"Start":"01:30.890 ","End":"01:35.090","Text":"and this time, it\u0027s around t equals 0."},{"Start":"01:35.090 ","End":"01:36.840","Text":"Yeah, here it is again, print it."},{"Start":"01:36.840 ","End":"01:38.610","Text":"Now, this looks familiar."},{"Start":"01:38.610 ","End":"01:42.680","Text":"If you go and check a couple of exercises back,"},{"Start":"01:42.680 ","End":"01:44.540","Text":"we did this already."},{"Start":"01:44.540 ","End":"01:46.730","Text":"I think it was Exercise number 7."},{"Start":"01:46.730 ","End":"01:50.480","Text":"Anyway, the solution we got was this power series,"},{"Start":"01:50.480 ","End":"01:53.765","Text":"but this is not complete without the recursion formula."},{"Start":"01:53.765 ","End":"01:58.115","Text":"That was this together with the solution."},{"Start":"01:58.115 ","End":"01:59.870","Text":"Now in that exercise,"},{"Start":"01:59.870 ","End":"02:03.410","Text":"we didn\u0027t have an initial condition and here we do."},{"Start":"02:03.410 ","End":"02:05.360","Text":"We\u0027re given the value of y and y\u0027."},{"Start":"02:05.360 ","End":"02:07.535","Text":"When t is 0,"},{"Start":"02:07.535 ","End":"02:09.530","Text":"yeah, x is minus 1."},{"Start":"02:09.530 ","End":"02:15.780","Text":"But t is 0 and we have y(0) is a_0 and y\u0027(0) is a_1,"},{"Start":"02:15.780 ","End":"02:20.360","Text":"so we can actually be more specific here and we can"},{"Start":"02:20.360 ","End":"02:25.950","Text":"substitute a_0 and a_1 wherever they appear here,"},{"Start":"02:25.950 ","End":"02:29.705","Text":"and so we have the beginning of our series,"},{"Start":"02:29.705 ","End":"02:35.300","Text":"but we still have a_n which we have to define recursively."},{"Start":"02:35.300 ","End":"02:39.423","Text":"All that remains now is to replace t by x plus 1."},{"Start":"02:39.423 ","End":"02:40.820","Text":"If we do that,"},{"Start":"02:40.820 ","End":"02:44.195","Text":"we will get this power series."},{"Start":"02:44.195 ","End":"02:49.880","Text":"But of course, we have to still take it together with the recursion formula,"},{"Start":"02:49.880 ","End":"02:51.140","Text":"which is the same."},{"Start":"02:51.140 ","End":"02:54.245","Text":"This together with this defines our solution,"},{"Start":"02:54.245 ","End":"02:56.940","Text":"and we are done."}],"Thumbnail":null,"ID":7925}],"ID":4243},{"Name":"Homogeneous Equation around Regular-Singular Point","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Homogeneous Equation around Regular-Singular Point","Duration":"11m 38s","ChapterTopicVideoID":7825,"CourseChapterTopicPlaylistID":4244,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.555","Text":"This section is a bit similar to the previous section,"},{"Start":"00:03.555 ","End":"00:09.120","Text":"Second Order Linear Differential Equations solving with power series."},{"Start":"00:09.120 ","End":"00:11.100","Text":"The difference is that there we had"},{"Start":"00:11.100 ","End":"00:14.835","Text":"a regular point here we have something called a regular singular point."},{"Start":"00:14.835 ","End":"00:17.670","Text":"We also restrict it to just the homogeneous case,"},{"Start":"00:17.670 ","End":"00:20.310","Text":"whereas there we had also non-homogeneous."},{"Start":"00:20.310 ","End":"00:23.100","Text":"Let me show you what kind of equation I\u0027m talking about."},{"Start":"00:23.100 ","End":"00:28.965","Text":"This is what it looks like in the standard form where the coefficient of y\u0027\u0027 is 1."},{"Start":"00:28.965 ","End":"00:33.190","Text":"Later we\u0027ll discuss what happens if it\u0027s not 1 or what to do in general."},{"Start":"00:33.190 ","End":"00:37.800","Text":"The point x equals naught will be a regular singular point."},{"Start":"00:37.800 ","End":"00:39.510","Text":"We\u0027ll define that below."},{"Start":"00:39.510 ","End":"00:43.025","Text":"We\u0027re also going to assume that p and q are polynomials,"},{"Start":"00:43.025 ","End":"00:45.920","Text":"or if not polynomials then rational functions,"},{"Start":"00:45.920 ","End":"00:48.200","Text":"which are quotients of polynomials."},{"Start":"00:48.200 ","End":"00:51.860","Text":"I\u0027ll be giving an algorithm for solving this kind"},{"Start":"00:51.860 ","End":"00:56.200","Text":"of equation as a power series developed around x equals naught."},{"Start":"00:56.200 ","End":"01:00.815","Text":"It\u0027s going to be quite involved as opposed to the previous section."},{"Start":"01:00.815 ","End":"01:05.570","Text":"As before it can be generalized to not just around x equals naught,"},{"Start":"01:05.570 ","End":"01:10.070","Text":"but to any regular singular point, any x naught."},{"Start":"01:10.070 ","End":"01:11.480","Text":"I\u0027ve mentioned this term,"},{"Start":"01:11.480 ","End":"01:14.030","Text":"it\u0027s about time to define it."},{"Start":"01:14.030 ","End":"01:16.760","Text":"We\u0027ll start with the case where x equals naught."},{"Start":"01:16.760 ","End":"01:19.755","Text":"This is a regular singular point of the equation."},{"Start":"01:19.755 ","End":"01:25.970","Text":"If p(x) and or q(x) are undefined at x equals naught."},{"Start":"01:25.970 ","End":"01:32.810","Text":"But that both of these x times p(x) and x^2 times q(x) are defined at x equals naught."},{"Start":"01:32.810 ","End":"01:35.990","Text":"Of course, if p(x) and q(x) are defined at x equals naught,"},{"Start":"01:35.990 ","End":"01:38.675","Text":"then it\u0027s just a regular point without singular."},{"Start":"01:38.675 ","End":"01:40.790","Text":"Then we have the previous section."},{"Start":"01:40.790 ","End":"01:43.385","Text":"As mentioned, we can generalize this definition"},{"Start":"01:43.385 ","End":"01:46.670","Text":"to x equals x naught being a regular point,"},{"Start":"01:46.670 ","End":"01:52.460","Text":"basically, you just replace the x by x minus x naught."},{"Start":"01:52.460 ","End":"01:56.105","Text":"This definition is not precise."},{"Start":"01:56.105 ","End":"01:58.160","Text":"It\u0027s good enough for working with,"},{"Start":"01:58.160 ","End":"01:59.680","Text":"but if you look it up,"},{"Start":"01:59.680 ","End":"02:02.850","Text":"it\u0027s not defined this way."},{"Start":"02:02.850 ","End":"02:07.190","Text":"I will give something more exact at the end,"},{"Start":"02:07.190 ","End":"02:09.410","Text":"but we\u0027ll live with this for now."},{"Start":"02:09.410 ","End":"02:15.575","Text":"Next, I\u0027ll give you an example of a regular singular equation. Here it is."},{"Start":"02:15.575 ","End":"02:20.540","Text":"I\u0027ll just bring it into standard form by dividing throughout by 3x^2."},{"Start":"02:20.540 ","End":"02:23.300","Text":"If I do that, this is the equation I get."},{"Start":"02:23.300 ","End":"02:27.395","Text":"This bit is p(x) and this bit is q(x)."},{"Start":"02:27.395 ","End":"02:31.370","Text":"This is not a regular point because at x equals 0,"},{"Start":"02:31.370 ","End":"02:33.005","Text":"this is not defined,"},{"Start":"02:33.005 ","End":"02:36.560","Text":"but we are still in the case of a regular singular,"},{"Start":"02:36.560 ","End":"02:37.805","Text":"as I shall show,"},{"Start":"02:37.805 ","End":"02:45.260","Text":"x p(x) comes out to be just 2/3 and x^2 q(x) comes out to be this."},{"Start":"02:45.260 ","End":"02:46.865","Text":"When x equals naught,"},{"Start":"02:46.865 ","End":"02:48.875","Text":"there\u0027s no problems with these."},{"Start":"02:48.875 ","End":"02:50.240","Text":"I\u0027m sorry, by definition,"},{"Start":"02:50.240 ","End":"02:53.420","Text":"we are in the case of a regular singular equation."},{"Start":"02:53.420 ","End":"02:58.790","Text":"The general idea is to do a variation of the power series."},{"Start":"02:58.790 ","End":"03:01.925","Text":"The differential equation above will have a solution"},{"Start":"03:01.925 ","End":"03:05.585","Text":"also in terms of y1 and y2 a linear combination."},{"Start":"03:05.585 ","End":"03:13.640","Text":"But y1 is going to be of the form a power series times x to some power, call it Lambda."},{"Start":"03:13.640 ","End":"03:17.150","Text":"Now it\u0027s time to present the algorithm for the solution."},{"Start":"03:17.150 ","End":"03:18.545","Text":"How do we solve it?"},{"Start":"03:18.545 ","End":"03:23.815","Text":"This is going to be very abstract and you probably won\u0027t follow a lot of it."},{"Start":"03:23.815 ","End":"03:27.320","Text":"But when you look at the examples and there\u0027ll be quite a few,"},{"Start":"03:27.320 ","End":"03:32.015","Text":"everything will become clearer and you might want to go through the theory again."},{"Start":"03:32.015 ","End":"03:34.880","Text":"Before I start with the algorithm properly,"},{"Start":"03:34.880 ","End":"03:38.240","Text":"I\u0027ll just say that we want to bring it into a standard form,"},{"Start":"03:38.240 ","End":"03:40.385","Text":"not the usual standard form,"},{"Start":"03:40.385 ","End":"03:42.890","Text":"where the coefficient of y\u0027 is 1."},{"Start":"03:42.890 ","End":"03:46.610","Text":"It works out best if you make the coefficient of y\u0027"},{"Start":"03:46.610 ","End":"03:50.675","Text":"to be x squared or some constant times x squared."},{"Start":"03:50.675 ","End":"03:55.115","Text":"For example, if my equation was given to me in this form,"},{"Start":"03:55.115 ","End":"03:59.090","Text":"then I want it to be x squared in front of the y\u0027."},{"Start":"03:59.090 ","End":"04:03.380","Text":"I\u0027ll multiply everything by x and get this."},{"Start":"04:03.380 ","End":"04:06.020","Text":"Doesn\u0027t matter if it\u0027s x squared or 3x^2,"},{"Start":"04:06.020 ","End":"04:07.640","Text":"but it should be something x squared."},{"Start":"04:07.640 ","End":"04:08.795","Text":"If you don\u0027t do this,"},{"Start":"04:08.795 ","End":"04:11.555","Text":"you\u0027re likely to get stuck in the solution."},{"Start":"04:11.555 ","End":"04:13.565","Text":"You have been warned."},{"Start":"04:13.565 ","End":"04:16.280","Text":"In the first step we make a substitution,"},{"Start":"04:16.280 ","End":"04:18.665","Text":"not exactly a power series,"},{"Start":"04:18.665 ","End":"04:22.550","Text":"but x to the power of Lambda times a power series."},{"Start":"04:22.550 ","End":"04:24.650","Text":"The Sigma notation which I wrote earlier,"},{"Start":"04:24.650 ","End":"04:26.180","Text":"this would look like this,"},{"Start":"04:26.180 ","End":"04:28.465","Text":"but I\u0027m not going to use the sigma notation."},{"Start":"04:28.465 ","End":"04:30.875","Text":"Now for the substitution, we\u0027ll need y\u0027,"},{"Start":"04:30.875 ","End":"04:35.840","Text":"which is this, and will also need y\u0027\u0027."},{"Start":"04:35.840 ","End":"04:38.060","Text":"I\u0027ve included quite a few terms, so it looks messy."},{"Start":"04:38.060 ","End":"04:39.380","Text":"Notice that there\u0027s a dot,"},{"Start":"04:39.380 ","End":"04:40.805","Text":"dot, dot in all of them."},{"Start":"04:40.805 ","End":"04:45.035","Text":"Oh, and actually this should be at the end also because these are infinite series."},{"Start":"04:45.035 ","End":"04:46.775","Text":"I want to do that also."},{"Start":"04:46.775 ","End":"04:50.380","Text":"Now what we get if we do the substitution."},{"Start":"04:50.380 ","End":"04:52.140","Text":"This is part of step 2 already."},{"Start":"04:52.140 ","End":"04:56.420","Text":"We equate the coefficients of x to the lambda on"},{"Start":"04:56.420 ","End":"05:00.940","Text":"the left side to 0 because on the right-hand side we have 0."},{"Start":"05:00.940 ","End":"05:04.925","Text":"What you get is a quadratic equation in Lambda,"},{"Start":"05:04.925 ","End":"05:07.775","Text":"which is called the indicial equation."},{"Start":"05:07.775 ","End":"05:10.145","Text":"It actually comes from the root index."},{"Start":"05:10.145 ","End":"05:15.215","Text":"It will be a quadratic with 2 solutions, Lambda1 and Lambda2."},{"Start":"05:15.215 ","End":"05:16.655","Text":"They might be the same,"},{"Start":"05:16.655 ","End":"05:20.960","Text":"but we\u0027re going to assume that Lambda1 is bigger or equal to Lambda2."},{"Start":"05:20.960 ","End":"05:23.855","Text":"If not, then just change the order of them so that"},{"Start":"05:23.855 ","End":"05:27.410","Text":"the first formula always be at least as big as the second one."},{"Start":"05:27.410 ","End":"05:30.500","Text":"I repeat all this will become clearer in the examples."},{"Start":"05:30.500 ","End":"05:32.750","Text":"Now the next step is similar to this step."},{"Start":"05:32.750 ","End":"05:35.149","Text":"Here, we just took the very first coefficient,"},{"Start":"05:35.149 ","End":"05:36.680","Text":"which is x to the Lambda."},{"Start":"05:36.680 ","End":"05:40.250","Text":"Here we compare the nth coefficients in"},{"Start":"05:40.250 ","End":"05:43.880","Text":"the position where x is to the power of lambda plus n. Well,"},{"Start":"05:43.880 ","End":"05:46.190","Text":"the coefficient on the left-hand side has to"},{"Start":"05:46.190 ","End":"05:49.310","Text":"equal 0 because we have 0 on the right-hand side."},{"Start":"05:49.310 ","End":"05:53.030","Text":"What this will give us is a recursive formula"},{"Start":"05:53.030 ","End":"05:57.320","Text":"and we can extract a_n from it and a_n well,"},{"Start":"05:57.320 ","End":"06:00.125","Text":"of course, it depends on n because n varies,"},{"Start":"06:00.125 ","End":"06:02.090","Text":"but it also depends on Lambda."},{"Start":"06:02.090 ","End":"06:04.040","Text":"Again, you\u0027ll see this in the examples."},{"Start":"06:04.040 ","End":"06:05.780","Text":"The next step, step 4,"},{"Start":"06:05.780 ","End":"06:09.590","Text":"gives us just the first solution."},{"Start":"06:09.590 ","End":"06:15.725","Text":"We get y1 using this formula where Lambda1 is the larger of the 2 roots,"},{"Start":"06:15.725 ","End":"06:20.360","Text":"we just plug-in Lambda1 into the expression for a_n,"},{"Start":"06:20.360 ","End":"06:22.460","Text":"which we said depends on Lambda."},{"Start":"06:22.460 ","End":"06:25.625","Text":"Then we also put an x to the power of Lambda1 in front."},{"Start":"06:25.625 ","End":"06:28.025","Text":"This just gives us y1."},{"Start":"06:28.025 ","End":"06:29.480","Text":"That\u0027s the easy part."},{"Start":"06:29.480 ","End":"06:31.490","Text":"y2 is the hard part."},{"Start":"06:31.490 ","End":"06:36.425","Text":"For y2, we have to sub-divide into cases, actually 3 cases."},{"Start":"06:36.425 ","End":"06:40.325","Text":"Case 1 is actually the easiest case."},{"Start":"06:40.325 ","End":"06:45.905","Text":"It\u0027s where Lambda1 minus Lambda2 is not a whole number."},{"Start":"06:45.905 ","End":"06:47.600","Text":"This might be, I don\u0027t know,"},{"Start":"06:47.600 ","End":"06:50.315","Text":"3.5 and this might be 2."},{"Start":"06:50.315 ","End":"06:52.370","Text":"The difference wouldn\u0027t be a whole number."},{"Start":"06:52.370 ","End":"06:56.610","Text":"If I had this was 10 and this was 8, it would be a whole number."},{"Start":"06:56.610 ","End":"06:58.340","Text":"We wouldn\u0027t be in case 1."},{"Start":"06:58.340 ","End":"07:02.750","Text":"In this case, y2 is very similar to the expression we had for y1."},{"Start":"07:02.750 ","End":"07:07.670","Text":"It\u0027s just that instead of Lambda1 here and here, we have Lambda2."},{"Start":"07:07.670 ","End":"07:10.370","Text":"That\u0027s the easy case."},{"Start":"07:10.370 ","End":"07:14.319","Text":"The second case is where the 2 roots of the quadratic are equal."},{"Start":"07:14.319 ","End":"07:16.625","Text":"Lambda1 is equal to Lambda2."},{"Start":"07:16.625 ","End":"07:17.870","Text":"Well, in this case,"},{"Start":"07:17.870 ","End":"07:22.625","Text":"we leave the recursive formula for a_n remember it depended on Lambda,"},{"Start":"07:22.625 ","End":"07:23.885","Text":"also on n of course,"},{"Start":"07:23.885 ","End":"07:27.005","Text":"but we leave it as a function in terms of Lambda."},{"Start":"07:27.005 ","End":"07:30.710","Text":"We don\u0027t substitute Lambda2 or Lambda1 as we did earlier."},{"Start":"07:30.710 ","End":"07:36.300","Text":"We just leave it as a function of Lambda and we substitute all these a_n\u0027s in"},{"Start":"07:36.300 ","End":"07:43.455","Text":"the expansion of y. y depends on x and on Lambda."},{"Start":"07:43.455 ","End":"07:46.730","Text":"As I keep saying, it will be clearer in the examples."},{"Start":"07:46.730 ","End":"07:52.760","Text":"Then for case 2, the formula is that y2 is the derivative of this function, well,"},{"Start":"07:52.760 ","End":"07:56.045","Text":"the partial derivative with respect to Lambda,"},{"Start":"07:56.045 ","End":"08:01.235","Text":"and then we substitute Lambda equals Lambda2 or Lambda1."},{"Start":"08:01.235 ","End":"08:05.795","Text":"I mean, they\u0027re going to be the same and that gives us y2 for case 2."},{"Start":"08:05.795 ","End":"08:08.245","Text":"Next, we have case 3."},{"Start":"08:08.245 ","End":"08:13.630","Text":"Case 3 is where Lambda1 minus Lambda2 is a positive whole number."},{"Start":"08:13.630 ","End":"08:15.769","Text":"Now in this case, we first try,"},{"Start":"08:15.769 ","End":"08:16.910","Text":"it doesn\u0027t always work,"},{"Start":"08:16.910 ","End":"08:21.185","Text":"but we try like in case 1, sometimes it works."},{"Start":"08:21.185 ","End":"08:26.150","Text":"Often there\u0027s a problem of substituting Lambda2 into a_n,"},{"Start":"08:26.150 ","End":"08:29.435","Text":"it isn\u0027t always defined and there may be problems with this."},{"Start":"08:29.435 ","End":"08:32.540","Text":"In any event we can try, if it doesn\u0027t work,"},{"Start":"08:32.540 ","End":"08:38.690","Text":"we look at the expansion of y1(x) from step 4 as this."},{"Start":"08:38.690 ","End":"08:40.205","Text":"Well, it\u0027s also written here,"},{"Start":"08:40.205 ","End":"08:42.635","Text":"means we don\u0027t substitute Lambda,"},{"Start":"08:42.635 ","End":"08:44.525","Text":"we just leave Lambda."},{"Start":"08:44.525 ","End":"08:48.545","Text":"The coefficients a_n depend on lambda."},{"Start":"08:48.545 ","End":"08:50.645","Text":"Then we have a formula."},{"Start":"08:50.645 ","End":"08:55.130","Text":"We take this function y(x) and Lambda,"},{"Start":"08:55.130 ","End":"09:00.275","Text":"multiply it by Lambda minus Lambda2."},{"Start":"09:00.275 ","End":"09:03.140","Text":"This is known and this is just the variable."},{"Start":"09:03.140 ","End":"09:06.710","Text":"Then we differentiate this with respect to Lambda."},{"Start":"09:06.710 ","End":"09:08.345","Text":"It\u0027s a partial derivative,"},{"Start":"09:08.345 ","End":"09:12.980","Text":"and then substitute Lambda equals Lambda2 as always,"},{"Start":"09:12.980 ","End":"09:14.905","Text":"will be clear in the examples."},{"Start":"09:14.905 ","End":"09:17.570","Text":"That\u0027s definitions of the cases."},{"Start":"09:17.570 ","End":"09:22.925","Text":"Now I want to show you which exercises belong to which case,"},{"Start":"09:22.925 ","End":"09:26.780","Text":"unless someone has changed the numbering since I recorded this,"},{"Start":"09:26.780 ","End":"09:32.510","Text":"we have 8 exercises where exercises 1-4 belong to case 1,"},{"Start":"09:32.510 ","End":"09:34.795","Text":"which as you recall, is the easy case."},{"Start":"09:34.795 ","End":"09:41.855","Text":"Then we have a couple of exercises for case 2 and a couple of exercises for case 3."},{"Start":"09:41.855 ","End":"09:43.775","Text":"Now some remarks."},{"Start":"09:43.775 ","End":"09:48.575","Text":"I\u0027d like to remind you this here is just how we define the regular singular point."},{"Start":"09:48.575 ","End":"09:52.430","Text":"At that time I mentioned that it\u0027s not quite precise."},{"Start":"09:52.430 ","End":"09:54.560","Text":"I want to make it more precise."},{"Start":"09:54.560 ","End":"09:58.715","Text":"This is it, but the difference is that earlier I said that"},{"Start":"09:58.715 ","End":"10:04.130","Text":"these 2 functions are defined at x equals naught."},{"Start":"10:04.130 ","End":"10:08.870","Text":"But the more precise definition is that these 2 can"},{"Start":"10:08.870 ","End":"10:13.850","Text":"be expanded as a Maclaurin series around x equals naught."},{"Start":"10:13.850 ","End":"10:17.030","Text":"But in practice, whenever they\u0027re defined,"},{"Start":"10:17.030 ","End":"10:21.920","Text":"they will be expandable as Maclaurin series and in the cases that you will"},{"Start":"10:21.920 ","End":"10:27.140","Text":"see how to write this to conform to the conventional definition."},{"Start":"10:27.140 ","End":"10:31.295","Text":"Next remark is that we might be given an initial condition."},{"Start":"10:31.295 ","End":"10:34.355","Text":"Now as before, if we\u0027re given y of naught,"},{"Start":"10:34.355 ","End":"10:38.965","Text":"then y of naught happens to equal a_naught."},{"Start":"10:38.965 ","End":"10:44.930","Text":"If not, then we treat a_naught as a parameter like a general unknown as if we know it,"},{"Start":"10:44.930 ","End":"10:48.890","Text":"but we don\u0027t, and we express the other coefficients in terms of a_naught."},{"Start":"10:48.890 ","End":"10:52.850","Text":"But note that in contrast to the previous section,"},{"Start":"10:52.850 ","End":"10:57.080","Text":"there, we had the y\u0027 of a_naught was a_1."},{"Start":"10:57.080 ","End":"11:03.395","Text":"But here, the power series is replaced by this where we jack up the exponent by lambda."},{"Start":"11:03.395 ","End":"11:05.375","Text":"We still have this."},{"Start":"11:05.375 ","End":"11:07.100","Text":"That\u0027s the same as here,"},{"Start":"11:07.100 ","End":"11:08.780","Text":"that y naught is a naught,"},{"Start":"11:08.780 ","End":"11:13.880","Text":"but y\u0027 of naught is not equal to a_1."},{"Start":"11:13.880 ","End":"11:17.810","Text":"It\u0027s not even clear what happens when we substitute 0 here."},{"Start":"11:17.810 ","End":"11:24.200","Text":"Because for example, lambda minus 1 could be negative and then this isn\u0027t even defined."},{"Start":"11:24.200 ","End":"11:26.180","Text":"If lambda\u0027s bigger than 1,"},{"Start":"11:26.180 ","End":"11:27.830","Text":"then all these are zeros,"},{"Start":"11:27.830 ","End":"11:31.490","Text":"so we don\u0027t use the a_1 system here."},{"Start":"11:31.490 ","End":"11:32.975","Text":"It works differently."},{"Start":"11:32.975 ","End":"11:34.955","Text":"Don\u0027t worry about this too much."},{"Start":"11:34.955 ","End":"11:37.365","Text":"That\u0027s all I\u0027m going to say for the theory."},{"Start":"11:37.365 ","End":"11:39.880","Text":"Onto the examples."}],"Thumbnail":null,"ID":7885},{"Watched":false,"Name":"Exercise 1","Duration":"8m 30s","ChapterTopicVideoID":7834,"CourseChapterTopicPlaylistID":4244,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.690","Text":"In this exercise, we have to solve this differential equation."},{"Start":"00:03.690 ","End":"00:08.069","Text":"We\u0027re going to solve it using power series."},{"Start":"00:08.069 ","End":"00:12.810","Text":"We\u0027re going to expand y and we are going to show that x equals"},{"Start":"00:12.810 ","End":"00:15.885","Text":"naught is irregular singular point"},{"Start":"00:15.885 ","End":"00:19.470","Text":"and the power series will be expanded around x equals naught."},{"Start":"00:19.470 ","End":"00:21.240","Text":"I brought it into standard form."},{"Start":"00:21.240 ","End":"00:25.680","Text":"I divide it by the (3x^2) and now we can show it regular singular."},{"Start":"00:25.680 ","End":"00:29.759","Text":"First of all, note that p(x) is not defined at x equals"},{"Start":"00:29.759 ","End":"00:35.175","Text":"naught and it\u0027s required for singular,"},{"Start":"00:35.175 ","End":"00:38.910","Text":"that at least one of these is not defined and one of them isn\u0027t."},{"Start":"00:38.910 ","End":"00:42.255","Text":"But we have to have that."},{"Start":"00:42.255 ","End":"00:48.665","Text":"Well, I\u0027ll write it that x.p(x) and x^2.q(x) has to be defined at x equals naught."},{"Start":"00:48.665 ","End":"00:50.570","Text":"But if you look at what they are,"},{"Start":"00:50.570 ","End":"00:57.330","Text":"they\u0027re certainly defined and well behaved around x equals naught, constant polynomial."},{"Start":"00:57.330 ","End":"01:00.290","Text":"We are in the case of a regular singular point,"},{"Start":"01:00.290 ","End":"01:05.885","Text":"and then we can look for a solution of the form x to the power of Lambda times"},{"Start":"01:05.885 ","End":"01:11.570","Text":"the sum from 0 to infinity over power series like this."},{"Start":"01:11.570 ","End":"01:13.505","Text":"If we write it longhand,"},{"Start":"01:13.505 ","End":"01:14.840","Text":"it looks something like this."},{"Start":"01:14.840 ","End":"01:17.635","Text":"I\u0027ve probably written too many terms, but that\u0027s okay."},{"Start":"01:17.635 ","End":"01:22.400","Text":"Of course we\u0027ll also need y\u0027 and y\u0027\u0027 because we want to substitute,"},{"Start":"01:22.400 ","End":"01:24.260","Text":"well, in the original equation."},{"Start":"01:24.260 ","End":"01:26.780","Text":"Let me get some room for that."},{"Start":"01:26.780 ","End":"01:29.790","Text":"Here we are, y\u0027, y\u0027\u0027,"},{"Start":"01:29.790 ","End":"01:32.930","Text":"looks the mess is not so bad."},{"Start":"01:32.930 ","End":"01:38.030","Text":"What we have to do now is substitute in the original differential equation."},{"Start":"01:38.030 ","End":"01:39.830","Text":"Here it is again,"},{"Start":"01:39.830 ","End":"01:42.440","Text":"but we don\u0027t need everything,"},{"Start":"01:42.440 ","End":"01:44.540","Text":"I\u0027ve written way too many terms."},{"Start":"01:44.540 ","End":"01:51.810","Text":"If you remember, we just need the terms with x to the Lambda and the terms with x to"},{"Start":"01:51.810 ","End":"01:56.720","Text":"the Lambda plus n. Now note that the"},{"Start":"01:56.720 ","End":"02:02.030","Text":"y\u0027\u0027 will be multiplied by x^2 is also a constant 3,"},{"Start":"02:02.030 ","End":"02:04.010","Text":"but as far as exponents go,"},{"Start":"02:04.010 ","End":"02:06.650","Text":"these will all be raised by 2."},{"Start":"02:06.650 ","End":"02:09.605","Text":"Similarly for y\u0027, there\u0027s an x here."},{"Start":"02:09.605 ","End":"02:12.485","Text":"All these exponents are going to get raised by 1."},{"Start":"02:12.485 ","End":"02:15.815","Text":"For y they\u0027re going to get raised by 2."},{"Start":"02:15.815 ","End":"02:17.795","Text":"I can be quite selective."},{"Start":"02:17.795 ","End":"02:21.260","Text":"Let\u0027s take a look at the x to the Lambda first."},{"Start":"02:21.260 ","End":"02:23.785","Text":"Now here it\u0027s going come"},{"Start":"02:23.785 ","End":"02:28.380","Text":"from x to the Lambda minus 2 because it\u0027s going to be raised by 2."},{"Start":"02:28.380 ","End":"02:30.285","Text":"We\u0027ll need this piece,"},{"Start":"02:30.285 ","End":"02:32.910","Text":"and here it\u0027s going to be raised by 1,"},{"Start":"02:32.910 ","End":"02:38.775","Text":"we need this bit to get x to the Lambda and here,"},{"Start":"02:38.775 ","End":"02:41.930","Text":"well, we can\u0027t get x to the Lambda because everything\u0027s raised by 2,"},{"Start":"02:41.930 ","End":"02:44.225","Text":"it\u0027s going to start from Lambda plus 2."},{"Start":"02:44.225 ","End":"02:50.675","Text":"Nothing here. Now let\u0027s take the other one and see where we could get Lambda plus n from."},{"Start":"02:50.675 ","End":"02:53.660","Text":"Well, in here, because we are rising by 2,"},{"Start":"02:53.660 ","End":"02:57.530","Text":"we subtract 2, so we look at Lambda plus n minus 2."},{"Start":"02:57.530 ","End":"02:59.740","Text":"This is what we need,"},{"Start":"02:59.740 ","End":"03:03.120","Text":"here we need Lambda plus n minus 1."},{"Start":"03:03.120 ","End":"03:06.720","Text":"This is the one I\u0027m talking about."},{"Start":"03:06.720 ","End":"03:11.185","Text":"Here also, Lambda plus n minus 2,"},{"Start":"03:11.185 ","End":"03:14.795","Text":"because we\u0027re also multiplying by x^2."},{"Start":"03:14.795 ","End":"03:18.395","Text":"Let\u0027s collect all this stuff together."},{"Start":"03:18.395 ","End":"03:19.950","Text":"I\u0027ll do them 1 at a time."},{"Start":"03:19.950 ","End":"03:22.310","Text":"Let\u0027s do the y\u0027\u0027 first."},{"Start":"03:22.310 ","End":"03:24.650","Text":"From here I\u0027m going to take,"},{"Start":"03:24.650 ","End":"03:26.825","Text":"this one is here,"},{"Start":"03:26.825 ","End":"03:31.905","Text":"and this coefficient is here."},{"Start":"03:31.905 ","End":"03:35.840","Text":"Next from y\u0027 I get this and this,"},{"Start":"03:35.840 ","End":"03:42.240","Text":"which is this, and this. I\u0027ve lost the y."},{"Start":"03:42.240 ","End":"03:44.055","Text":"Let\u0027s just take a peek."},{"Start":"03:44.055 ","End":"03:49.580","Text":"I get just the a n minus 2 with the Lambda plus n,"},{"Start":"03:49.580 ","End":"03:51.835","Text":"but I don\u0027t get anything else."},{"Start":"03:51.835 ","End":"03:56.310","Text":"Just this a n minus 2,"},{"Start":"03:56.310 ","End":"03:58.160","Text":"the completeness, I also wrote this in,"},{"Start":"03:58.160 ","End":"04:01.130","Text":"but it\u0027s grayed out because I\u0027m not going to be collecting it."},{"Start":"04:01.130 ","End":"04:04.010","Text":"Now let\u0027s see, let\u0027s do a summary,"},{"Start":"04:04.010 ","End":"04:10.275","Text":"separately of the x to the Lambda and the x to the Lambda plus n,"},{"Start":"04:10.275 ","End":"04:12.085","Text":"for x to the Lambda,"},{"Start":"04:12.085 ","End":"04:13.700","Text":"I take from here and here."},{"Start":"04:13.700 ","End":"04:16.175","Text":"But there\u0027s also coefficients,"},{"Start":"04:16.175 ","End":"04:20.015","Text":"this 3 and this 2 which here and here,"},{"Start":"04:20.015 ","End":"04:22.370","Text":"and the set that equal to 0."},{"Start":"04:22.370 ","End":"04:26.135","Text":"Now this quadratic equation is the indicial equation."},{"Start":"04:26.135 ","End":"04:28.445","Text":"After you\u0027ve tidy it up and solve it,"},{"Start":"04:28.445 ","End":"04:33.605","Text":"the answers you get are Lambda equals naught or 1 1/3,"},{"Start":"04:33.605 ","End":"04:36.860","Text":"I think I\u0027ll highlight them then they\u0027re very important."},{"Start":"04:36.860 ","End":"04:41.720","Text":"When going to compare the x to the Lambda plus n. I need this,"},{"Start":"04:41.720 ","End":"04:45.890","Text":"this and this, but also there\u0027s the 3 and there\u0027s the 2."},{"Start":"04:45.890 ","End":"04:48.589","Text":"This is the equation I get."},{"Start":"04:48.589 ","End":"04:51.045","Text":"When I extract a n,"},{"Start":"04:51.045 ","End":"04:52.280","Text":"collect the an,"},{"Start":"04:52.280 ","End":"04:54.365","Text":"bring this to the other side and divide,"},{"Start":"04:54.365 ","End":"04:56.735","Text":"in short done the algebra for you."},{"Start":"04:56.735 ","End":"05:01.460","Text":"We get that an is an minus 2 over this."},{"Start":"05:01.460 ","End":"05:08.300","Text":"Well, we have to restrict n because this subscript n minus 2 has to be at least 0."},{"Start":"05:08.300 ","End":"05:11.815","Text":"This only works from n bigger or equal to 2."},{"Start":"05:11.815 ","End":"05:17.620","Text":"I don want to remind you that the Lambdas we got where 1 1/3 and 0."},{"Start":"05:17.620 ","End":"05:19.465","Text":"This is the bigger one,"},{"Start":"05:19.465 ","End":"05:23.860","Text":"and this is the case where the difference is not a whole number."},{"Start":"05:23.860 ","End":"05:27.380","Text":"In that case, we use the 1 1/3."},{"Start":"05:27.380 ","End":"05:30.320","Text":"If I plug Lambda equals a 1/3 into here,"},{"Start":"05:30.320 ","End":"05:34.310","Text":"then this boils down to this."},{"Start":"05:34.310 ","End":"05:39.615","Text":"I\u0027ll leave it to you to check the algebra and we can use this from 2 onwards."},{"Start":"05:39.615 ","End":"05:42.795","Text":"Let\u0027s try plugging in n=2."},{"Start":"05:42.795 ","End":"05:46.905","Text":"We get that a2 is a naught over,"},{"Start":"05:46.905 ","End":"05:49.800","Text":"you plug in 2 here, you get 14."},{"Start":"05:49.800 ","End":"05:56.610","Text":"Now I can\u0027t plug in n=3 because a3 would depend on a1 and we don\u0027t have"},{"Start":"05:56.610 ","End":"06:03.450","Text":"a1 but I could plug in n=4 and then I\u0027d get a4 in terms of a2 like this."},{"Start":"06:03.450 ","End":"06:07.025","Text":"But a2 depends on a naught and this is what we get."},{"Start":"06:07.025 ","End":"06:10.850","Text":"We can only get the even indexes this way."},{"Start":"06:10.850 ","End":"06:12.750","Text":"There\u0027s no restriction on a1."},{"Start":"06:12.750 ","End":"06:16.460","Text":"We could just take if we want a1=0,"},{"Start":"06:16.460 ","End":"06:19.280","Text":"then if we use this formula,"},{"Start":"06:19.280 ","End":"06:23.565","Text":"you\u0027ll see that a3 would also be 0."},{"Start":"06:23.565 ","End":"06:28.230","Text":"If a3 is 0, then a5 is 0, and so on."},{"Start":"06:28.230 ","End":"06:32.470","Text":"We just ignore the odd numbers and just continue with the evens."},{"Start":"06:32.470 ","End":"06:34.910","Text":"Recall that this gives us y1,"},{"Start":"06:34.910 ","End":"06:36.800","Text":"which is x to the Lambda,"},{"Start":"06:36.800 ","End":"06:40.460","Text":"times the sum of the power series with a n x^n."},{"Start":"06:40.460 ","End":"06:45.850","Text":"I\u0027ll just write that again, x to the Lambda, sum anx^n."},{"Start":"06:45.850 ","End":"06:47.880","Text":"With the a n\u0027s we have,"},{"Start":"06:47.880 ","End":"06:49.440","Text":"a naught, a2,"},{"Start":"06:49.440 ","End":"06:53.385","Text":"a4, they are all in terms of a naught."},{"Start":"06:53.385 ","End":"07:00.620","Text":"I can take the a naught outside the brackets and then we move on to the other Lambda."},{"Start":"07:00.620 ","End":"07:05.270","Text":"Remember in the case where the 2 Lambdas have a difference which is not a whole number,"},{"Start":"07:05.270 ","End":"07:08.675","Text":"we just do the same thing essentially with the other Lambda."},{"Start":"07:08.675 ","End":"07:11.270","Text":"The recursion formula, I\u0027m not going to go back."},{"Start":"07:11.270 ","End":"07:14.150","Text":"If you go back and substitute Lambda equals naught."},{"Start":"07:14.150 ","End":"07:16.775","Text":"This is the recursion formula we get."},{"Start":"07:16.775 ","End":"07:21.870","Text":"Again, each term depends on the one that\u0027s 2 before,"},{"Start":"07:21.870 ","End":"07:23.865","Text":"a n minus 2."},{"Start":"07:23.865 ","End":"07:26.005","Text":"If we plug in n=2,"},{"Start":"07:26.005 ","End":"07:29.540","Text":"we get a2 is minus a naught over 10."},{"Start":"07:29.540 ","End":"07:32.659","Text":"The same story with the odd numbered coefficients,"},{"Start":"07:32.659 ","End":"07:34.795","Text":"we can assume that those are 0."},{"Start":"07:34.795 ","End":"07:37.410","Text":"Then if we plug in n equals 4,"},{"Start":"07:37.410 ","End":"07:39.230","Text":"we get a4 in terms of a2,"},{"Start":"07:39.230 ","End":"07:41.330","Text":"which is then in terms of a naught."},{"Start":"07:41.330 ","End":"07:45.440","Text":"That gives us y2 x to the naught,"},{"Start":"07:45.440 ","End":"07:48.135","Text":"which is really just equal to 1."},{"Start":"07:48.135 ","End":"07:50.040","Text":"We can ignore that."},{"Start":"07:50.040 ","End":"07:52.760","Text":"Then we have the first few terms and again,"},{"Start":"07:52.760 ","End":"07:55.385","Text":"we can take a naught out the brackets."},{"Start":"07:55.385 ","End":"07:56.975","Text":"We get this."},{"Start":"07:56.975 ","End":"08:00.830","Text":"Strictly speaking, it\u0027s not the same a naught as it was with y1,"},{"Start":"08:00.830 ","End":"08:03.500","Text":"but it doesn\u0027t really matter as you\u0027ll see in a moment,"},{"Start":"08:03.500 ","End":"08:08.345","Text":"because in the following step we take a linear combination of y1 and y2."},{"Start":"08:08.345 ","End":"08:14.465","Text":"We really could ignore the a naught because a naught can be combined with the constant."},{"Start":"08:14.465 ","End":"08:19.190","Text":"We\u0027ll call this constant then k1 and this constant k2."},{"Start":"08:19.190 ","End":"08:23.330","Text":"This is the general shape of the solution with"},{"Start":"08:23.330 ","End":"08:27.980","Text":"the recursion formulas above to complete it."},{"Start":"08:27.980 ","End":"08:30.180","Text":"We\u0027re finally done."}],"Thumbnail":null,"ID":7886},{"Watched":false,"Name":"Exercise 2","Duration":"11m 13s","ChapterTopicVideoID":7835,"CourseChapterTopicPlaylistID":4244,"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":"00:07.350","Text":"second-order linear homogeneous non-constant coefficients."},{"Start":"00:07.350 ","End":"00:11.249","Text":"What we\u0027re going to do is solve it as a power series"},{"Start":"00:11.249 ","End":"00:15.855","Text":"around x=0 and first we\u0027ll show that this is a regular singular point."},{"Start":"00:15.855 ","End":"00:21.020","Text":"To show that, we put it in standard form where y’’ has a 1 coefficient."},{"Start":"00:21.020 ","End":"00:25.850","Text":"We divide by 2x^2 and we get p(x) and q(x)."},{"Start":"00:25.850 ","End":"00:29.795","Text":"Notice that these are not defined at x=0."},{"Start":"00:29.795 ","End":"00:32.240","Text":"That would\u0027ve been enough for one of them not to be defined."},{"Start":"00:32.240 ","End":"00:36.785","Text":"It\u0027s not a regular point but it is a regular singular point."},{"Start":"00:36.785 ","End":"00:40.610","Text":"We have to multiply this by x and this by x^2."},{"Start":"00:40.610 ","End":"00:44.655","Text":"So x p(x) comes out this x^2, q(x) comes out this."},{"Start":"00:44.655 ","End":"00:47.540","Text":"Both of these are not problematic,"},{"Start":"00:47.540 ","End":"00:49.180","Text":"they are well-behaved,"},{"Start":"00:49.180 ","End":"00:52.955","Text":"so we do indeed have a regular singular point."},{"Start":"00:52.955 ","End":"00:59.120","Text":"Now the theory goes that the solution to this will be a series of the form this."},{"Start":"00:59.120 ","End":"01:01.730","Text":"If you prefer a power series,"},{"Start":"01:01.730 ","End":"01:03.305","Text":"it explains it better."},{"Start":"01:03.305 ","End":"01:05.705","Text":"We have x^λ."},{"Start":"01:05.705 ","End":"01:11.210","Text":"Then regular power series of a_n x^n."},{"Start":"01:11.210 ","End":"01:13.885","Text":"We just raise all the powers by Lambda."},{"Start":"01:13.885 ","End":"01:17.390","Text":"Now in order to substitute in the differential equation,"},{"Start":"01:17.390 ","End":"01:21.865","Text":"we need not only y but we need also y’ and y’’."},{"Start":"01:21.865 ","End":"01:23.935","Text":"Let\u0027s compute those."},{"Start":"01:23.935 ","End":"01:28.730","Text":"Here they are, y’ and y’’, just technical."},{"Start":"01:28.730 ","End":"01:32.975","Text":"Here again is a differential equation into which we are going to substitute these"},{"Start":"01:32.975 ","End":"01:38.060","Text":"but this contains way too many terms that could be swallowed up in the dot, dot, dot."},{"Start":"01:38.060 ","End":"01:39.500","Text":"We\u0027re going to be very selective."},{"Start":"01:39.500 ","End":"01:44.030","Text":"If you remember, we only need the coefficients for x^λ and"},{"Start":"01:44.030 ","End":"01:49.750","Text":"the general x^λ plus n. Let\u0027s pick those but first,"},{"Start":"01:49.750 ","End":"01:52.700","Text":"I just want to split this middle term into 2."},{"Start":"01:52.700 ","End":"02:00.015","Text":"This is 7x^2y’ plus 7xy’."},{"Start":"02:00.015 ","End":"02:04.950","Text":"Notice that y’’ is multiplied by x^2 and also a"},{"Start":"02:04.950 ","End":"02:10.115","Text":"constant but what this means is that all these powers are going to be raised by 2."},{"Start":"02:10.115 ","End":"02:12.500","Text":"Similarly for y’,"},{"Start":"02:12.500 ","End":"02:15.720","Text":"part of it is multiplied by x^2."},{"Start":"02:15.720 ","End":"02:18.770","Text":"We\u0027re going to have the powers here raised by 2."},{"Start":"02:18.770 ","End":"02:23.255","Text":"Then once again, we\u0027re going to have the powers raised by 1 because the x,"},{"Start":"02:23.255 ","End":"02:26.150","Text":"but the power is in y, which we must multiply by a constant,"},{"Start":"02:26.150 ","End":"02:28.145","Text":"will stay the same. Let\u0027s see."},{"Start":"02:28.145 ","End":"02:31.640","Text":"Let\u0027s collect the x^λ terms first."},{"Start":"02:31.640 ","End":"02:34.890","Text":"I\u0027ll just make a highlight that in yellow."},{"Start":"02:34.890 ","End":"02:39.465","Text":"From y’’, since we\u0027re multiplying by x^2,"},{"Start":"02:39.465 ","End":"02:42.060","Text":"I need to look for λ-2."},{"Start":"02:42.060 ","End":"02:44.865","Text":"This coefficient is the one I want."},{"Start":"02:44.865 ","End":"02:48.690","Text":"Here, I\u0027ll need 2 of them because I have x^2 and the x,"},{"Start":"02:48.690 ","End":"02:52.770","Text":"I\u0027ll need λ-2 and λ-1."},{"Start":"02:52.770 ","End":"02:56.115","Text":"For λ-2, well,"},{"Start":"02:56.115 ","End":"03:00.225","Text":"I don\u0027t get a λ-2 here because it starts from λ-1."},{"Start":"03:00.225 ","End":"03:03.060","Text":"I just take this one."},{"Start":"03:03.060 ","End":"03:08.865","Text":"Here just y without any powers of x. I just get the x^λ just from here."},{"Start":"03:08.865 ","End":"03:16.140","Text":"Now, let\u0027s look at the x^λ + n. In this y’’,"},{"Start":"03:16.140 ","End":"03:18.945","Text":"I need to look for 2 less."},{"Start":"03:18.945 ","End":"03:22.650","Text":"I Need to look for λ+n-2. There it is."},{"Start":"03:22.650 ","End":"03:25.545","Text":"This is the coefficient that I want."},{"Start":"03:25.545 ","End":"03:28.620","Text":"Here I want 2 coefficients."},{"Start":"03:28.620 ","End":"03:32.460","Text":"I want -2 and -1 from this."},{"Start":"03:32.460 ","End":"03:37.319","Text":"λ+n-1. Yes, I need this."},{"Start":"03:37.319 ","End":"03:43.230","Text":"I also need λ+n-2, which is this."},{"Start":"03:43.230 ","End":"03:47.060","Text":"Each of these will be multiplied by a 7,"},{"Start":"03:47.060 ","End":"03:49.385","Text":"just like these will be multiplied by 2 and so on."},{"Start":"03:49.385 ","End":"03:51.020","Text":"We\u0027ll deal with this with the 2,"},{"Start":"03:51.020 ","End":"03:52.445","Text":"7, and -3 later."},{"Start":"03:52.445 ","End":"03:57.360","Text":"Here, I need just λ+n itself,"},{"Start":"03:57.360 ","End":"03:59.100","Text":"where it is the edge here."},{"Start":"03:59.100 ","End":"04:01.710","Text":"That would be this co-efficient."},{"Start":"04:01.710 ","End":"04:04.290","Text":"Now it\u0027s time to substitute."},{"Start":"04:04.290 ","End":"04:06.360","Text":"Here, we\u0027ll have 4 terms."},{"Start":"04:06.360 ","End":"04:08.585","Text":"We\u0027ll take each of the 4 separately."},{"Start":"04:08.585 ","End":"04:12.240","Text":"Now I claim that the y’’ term gives me this."},{"Start":"04:12.240 ","End":"04:16.580","Text":"You know what, I\u0027m going to highlight the coefficients also in a different color."},{"Start":"04:16.580 ","End":"04:19.115","Text":"I\u0027ll need those emphasized."},{"Start":"04:19.115 ","End":"04:26.935","Text":"Yes, from here I\u0027ll need the 2 and that\u0027s going to be 2x^2."},{"Start":"04:26.935 ","End":"04:31.235","Text":"This here comes over here, like so."},{"Start":"04:31.235 ","End":"04:35.865","Text":"This one here is this here."},{"Start":"04:35.865 ","End":"04:38.640","Text":"That\u0027s the y’’. You know what,"},{"Start":"04:38.640 ","End":"04:41.230","Text":"I\u0027ll take the 7xy’,"},{"Start":"04:41.230 ","End":"04:44.450","Text":"and that is this term here."},{"Start":"04:44.450 ","End":"04:47.525","Text":"We multiply it by x, we get x^λ."},{"Start":"04:47.525 ","End":"04:49.475","Text":"That\u0027s the right one here."},{"Start":"04:49.475 ","End":"04:51.890","Text":"I will also highlight this 7."},{"Start":"04:51.890 ","End":"04:54.500","Text":"Then we need this one but not this one."},{"Start":"04:54.500 ","End":"04:57.005","Text":"This one will be for the x^2y’."},{"Start":"04:57.005 ","End":"05:03.900","Text":"This one here gives us this. We did this one."},{"Start":"05:03.900 ","End":"05:06.780","Text":"We didn\u0027t do 7x^2y’."},{"Start":"05:06.780 ","End":"05:08.210","Text":"Let\u0027s highlight this."},{"Start":"05:08.210 ","End":"05:13.475","Text":"What we need here is just the λ+n-2."},{"Start":"05:13.475 ","End":"05:15.695","Text":"It is going to be multiplied by x^2."},{"Start":"05:15.695 ","End":"05:20.810","Text":"Here we need just this. We don\u0027t need this."},{"Start":"05:20.810 ","End":"05:24.230","Text":"I just wrote it in because it\u0027s the first in the series."},{"Start":"05:24.230 ","End":"05:26.875","Text":"The last one is -3y."},{"Start":"05:26.875 ","End":"05:31.880","Text":"Here, I\u0027ll highlight the -3 and the y term has disappeared,"},{"Start":"05:31.880 ","End":"05:34.520","Text":"so I\u0027ll scroll back. We can see it."},{"Start":"05:34.520 ","End":"05:36.280","Text":"What I need,"},{"Start":"05:36.280 ","End":"05:39.030","Text":"this and this. This is for the x^λ."},{"Start":"05:39.030 ","End":"05:41.200","Text":"This is for the x^λ+n."},{"Start":"05:41.200 ","End":"05:44.435","Text":"Let\u0027s scroll back down to the appropriate line."},{"Start":"05:44.435 ","End":"05:47.285","Text":"Here we are, the -3."},{"Start":"05:47.285 ","End":"05:50.875","Text":"We need this one and this one."},{"Start":"05:50.875 ","End":"05:57.035","Text":"Next we\u0027re going to compare coefficients and we\u0027ll start with x^λ."},{"Start":"05:57.035 ","End":"06:03.890","Text":"Now, notice that there\u0027s a_0 here and an a_0 here, and a_0 here."},{"Start":"06:03.890 ","End":"06:08.910","Text":"I add all these up but I didn\u0027t bother writing the a_0."},{"Start":"06:08.910 ","End":"06:12.405","Text":"It really should be this times a_0=0."},{"Start":"06:12.405 ","End":"06:14.160","Text":"We get this equation."},{"Start":"06:14.160 ","End":"06:18.445","Text":"In λ, this is the indicial equation, a quadratic equation."},{"Start":"06:18.445 ","End":"06:20.190","Text":"I\u0027ll tell you what solutions,"},{"Start":"06:20.190 ","End":"06:23.310","Text":"λ is either 1/2 or -3."},{"Start":"06:23.310 ","End":"06:24.739","Text":"These are the 2 lambdas."},{"Start":"06:24.739 ","End":"06:30.775","Text":"Next we\u0027ll get an equation for comparing coefficients of x^λ+n."},{"Start":"06:30.775 ","End":"06:32.430","Text":"That\u0027s these for this, this,"},{"Start":"06:32.430 ","End":"06:34.550","Text":"this and this but not to forget,"},{"Start":"06:34.550 ","End":"06:36.170","Text":"those are the coefficients."},{"Start":"06:36.170 ","End":"06:37.670","Text":"We have 2 times this,"},{"Start":"06:37.670 ","End":"06:39.590","Text":"7 times this, 7 times this,"},{"Start":"06:39.590 ","End":"06:43.280","Text":"-3 times this, equals 0 and this is what we get."},{"Start":"06:43.280 ","End":"06:51.500","Text":"What we can do though is get a recursion equation if we extract a_n in terms of a_n-1."},{"Start":"06:51.500 ","End":"06:54.560","Text":"Let\u0027s first of all collect like terms. These are terms for a_n."},{"Start":"06:54.560 ","End":"06:59.170","Text":"This is a_n-1. Then I\u0027ll bring this to the other side and divide it by this."},{"Start":"06:59.170 ","End":"07:04.185","Text":"This is our recursive equation for the coefficients. Here it is again."},{"Start":"07:04.185 ","End":"07:13.100","Text":"I\u0027ll just remind you that the 2 lambdas we got were 1/2 or 0.5 and also -3."},{"Start":"07:13.100 ","End":"07:15.985","Text":"It could be λ_1 and λ_2."},{"Start":"07:15.985 ","End":"07:18.620","Text":"I don\u0027t know if I said it earlier but"},{"Start":"07:18.620 ","End":"07:21.020","Text":"the difference between these 2 is not a whole number."},{"Start":"07:21.020 ","End":"07:22.610","Text":"That\u0027s one of the cases."},{"Start":"07:22.610 ","End":"07:23.930","Text":"That\u0027s the easier case."},{"Start":"07:23.930 ","End":"07:28.670","Text":"We can get y_1 by using this lambda and y_2 using this Lambda."},{"Start":"07:28.670 ","End":"07:30.425","Text":"If we take this lambda,"},{"Start":"07:30.425 ","End":"07:33.380","Text":"1/2 and plug it in here,"},{"Start":"07:33.380 ","End":"07:40.805","Text":"then what we get after simplification is that a_n is this expression in terms of a_n-1."},{"Start":"07:40.805 ","End":"07:43.370","Text":"Of course this index can\u0027t be negative,"},{"Start":"07:43.370 ","End":"07:45.415","Text":"so n has to be at least 1."},{"Start":"07:45.415 ","End":"07:47.610","Text":"That\u0027s one recursion formula."},{"Start":"07:47.610 ","End":"07:50.980","Text":"Later we\u0027ll have another one for λ-3."},{"Start":"07:50.980 ","End":"07:54.275","Text":"Anyway, for y_1 we are sticking to λ=1/2."},{"Start":"07:54.275 ","End":"07:57.070","Text":"If we plug in n=1,"},{"Start":"07:57.070 ","End":"08:01.140","Text":"we\u0027ll get a_1 in terms of a_0 and do the computation."},{"Start":"08:01.140 ","End":"08:05.114","Text":"Then we could plug in n=2 and get a_2 in terms of a_1"},{"Start":"08:05.114 ","End":"08:09.900","Text":"but we already have a_1 in terms of a_0 so we get a_2 in terms of a_0."},{"Start":"08:09.900 ","End":"08:14.835","Text":"We could continue to get a_3 in terms of a_2 and hence n terms of a_0 and so on."},{"Start":"08:14.835 ","End":"08:17.210","Text":"Get everything is a multiple of a_0."},{"Start":"08:17.210 ","End":"08:22.005","Text":"By the theory y_1 is x^λ."},{"Start":"08:22.005 ","End":"08:25.835","Text":"This time it\u0027s 1/2 times the power series we get."},{"Start":"08:25.835 ","End":"08:29.835","Text":"Really this should be taken together with this as a whole,"},{"Start":"08:29.835 ","End":"08:32.580","Text":"because this is just the first few coefficients"},{"Start":"08:32.580 ","End":"08:35.915","Text":"but this tells us how to keep getting extra co-efficient."},{"Start":"08:35.915 ","End":"08:37.910","Text":"That\u0027s the case for y_1."},{"Start":"08:37.910 ","End":"08:42.695","Text":"We could take the a_0 which appears everywhere outside the brackets."},{"Start":"08:42.695 ","End":"08:46.640","Text":"Next we\u0027re going to move on to the other one."},{"Start":"08:46.640 ","End":"08:48.589","Text":"We take the other lambda,"},{"Start":"08:48.589 ","End":"08:50.260","Text":"which was -3,"},{"Start":"08:50.260 ","End":"08:56.570","Text":"and this is the recursion formula we get for this lambda and once again n≥1."},{"Start":"08:56.570 ","End":"09:00.365","Text":"If we plug in n=1, we get a_1 in terms of a_0."},{"Start":"09:00.365 ","End":"09:03.420","Text":"Put n=2 and get a_2 in terms of a_1,"},{"Start":"09:03.420 ","End":"09:05.535","Text":"and n in terms of a_0 from here."},{"Start":"09:05.535 ","End":"09:07.905","Text":"A_3 in terms of a_2."},{"Start":"09:07.905 ","End":"09:11.330","Text":"Then I use this and get a_3 in terms of a_0."},{"Start":"09:11.330 ","End":"09:18.080","Text":"A_4 because I noticed that there\u0027s an n-4 here and when n is 4,"},{"Start":"09:18.080 ","End":"09:19.790","Text":"this comes out to be 0."},{"Start":"09:19.790 ","End":"09:22.565","Text":"Now, once a_4 is 0,"},{"Start":"09:22.565 ","End":"09:26.495","Text":"then a_5 will also be 0 because a_5 is something becomes a_4"},{"Start":"09:26.495 ","End":"09:30.695","Text":"and a_6 is something times a_5 so we get a dominal effect,"},{"Start":"09:30.695 ","End":"09:32.545","Text":"that when a_4 is 0,"},{"Start":"09:32.545 ","End":"09:36.060","Text":"then all the subsequent a_ns are also 0."},{"Start":"09:36.060 ","End":"09:39.015","Text":"A_n is 0 from 4 onwards."},{"Start":"09:39.015 ","End":"09:42.810","Text":"Now we can write what y_2 looks like."},{"Start":"09:42.810 ","End":"09:45.000","Text":"Y_2 is x^-3."},{"Start":"09:45.000 ","End":"09:48.785","Text":"This is the lambda times this. It ends here."},{"Start":"09:48.785 ","End":"09:51.345","Text":"I can delete to this bit."},{"Start":"09:51.345 ","End":"09:54.305","Text":"Then just take the a_0 in front."},{"Start":"09:54.305 ","End":"09:58.085","Text":"We have y_2 as a_0 times this function."},{"Start":"09:58.085 ","End":"10:00.455","Text":"Again, we don\u0027t need this."},{"Start":"10:00.455 ","End":"10:04.505","Text":"Now it\u0027s time to combine the y_1 and y_2."},{"Start":"10:04.505 ","End":"10:09.455","Text":"Remember that the general solution is the linear combination of y_1 and y_2."},{"Start":"10:09.455 ","End":"10:13.505","Text":"This is what we get when we plug in y_1 and y_2 to this."},{"Start":"10:13.505 ","End":"10:14.975","Text":"Yes, once again,"},{"Start":"10:14.975 ","End":"10:18.140","Text":"I don\u0027t need the dot here, the ellipses."},{"Start":"10:18.140 ","End":"10:22.265","Text":"Strictly speaking, it\u0027s a different a_0 here and here, but in any event,"},{"Start":"10:22.265 ","End":"10:25.295","Text":"it gets swallowed up because these 2 together are constant,"},{"Start":"10:25.295 ","End":"10:28.585","Text":"and these 2 together are general constant."},{"Start":"10:28.585 ","End":"10:30.360","Text":"Just name this k_1,"},{"Start":"10:30.360 ","End":"10:31.905","Text":"name this k_2,"},{"Start":"10:31.905 ","End":"10:35.010","Text":"and this gives us the solution."},{"Start":"10:35.010 ","End":"10:36.880","Text":"This part ends here."},{"Start":"10:36.880 ","End":"10:42.520","Text":"This part keeps going and we get the coefficients from the recursion formula above."},{"Start":"10:42.520 ","End":"10:45.015","Text":"That\u0027s it. Though we\u0027re finished,"},{"Start":"10:45.015 ","End":"10:47.074","Text":"I just want to add a word about the domain."},{"Start":"10:47.074 ","End":"10:55.480","Text":"This part\u0027s optional. Look, x^1.5 is the square root of x and x^-3 is 1/x^3."},{"Start":"10:55.480 ","End":"11:00.449","Text":"Now, this is defined for x ≥ 0."},{"Start":"11:00.449 ","End":"11:04.095","Text":"This is defined for x≠0."},{"Start":"11:04.095 ","End":"11:08.250","Text":"Together we get that x is bigger than 0."},{"Start":"11:08.250 ","End":"11:14.710","Text":"It\u0027s actually not defined as 0 but defined as x is bigger than 0, close to 0."}],"Thumbnail":null,"ID":7887},{"Watched":false,"Name":"Exercise 3","Duration":"9m 32s","ChapterTopicVideoID":7836,"CourseChapterTopicPlaylistID":4244,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"Here we have to solve this differential equation or"},{"Start":"00:03.270 ","End":"00:07.560","Text":"the 2 linear homogeneous non-constant coefficients."},{"Start":"00:07.560 ","End":"00:13.455","Text":"The plan is to solve it as a power series around x=0."},{"Start":"00:13.455 ","End":"00:17.340","Text":"After we show that x=0 is a regular singular point,"},{"Start":"00:17.340 ","End":"00:22.455","Text":"we divide by the coefficient of y\" to get it into standard form."},{"Start":"00:22.455 ","End":"00:24.045","Text":"This is what we call p(x)."},{"Start":"00:24.045 ","End":"00:29.490","Text":"This is q(x) and notice that this is not a regular point because at least one,"},{"Start":"00:29.490 ","End":"00:33.720","Text":"actually both of these are not defined at 0."},{"Start":"00:33.720 ","End":"00:38.235","Text":"However, if we take x times this and x^2 times this,"},{"Start":"00:38.235 ","End":"00:41.480","Text":"this expression and this expression then they"},{"Start":"00:41.480 ","End":"00:44.990","Text":"are defined at x equals naught and they\u0027re well-behaved."},{"Start":"00:44.990 ","End":"00:47.345","Text":"That makes it a regular singular point."},{"Start":"00:47.345 ","End":"00:51.320","Text":"The theory says we look for a solution of the form as follows,"},{"Start":"00:51.320 ","End":"00:56.434","Text":"which makes more sense actually if we write it in Sigma notation,"},{"Start":"00:56.434 ","End":"01:03.895","Text":"it\u0027s x to the Lambda times the usual infinite series a_nx^n."},{"Start":"01:03.895 ","End":"01:05.510","Text":"Everything starts from Lambda,"},{"Start":"01:05.510 ","End":"01:06.830","Text":"Lambda plus 1 and so on."},{"Start":"01:06.830 ","End":"01:09.020","Text":"Of course, we\u0027re going to need to substitute in here,"},{"Start":"01:09.020 ","End":"01:12.550","Text":"so we\u0027ll need y prime and y double-prime."},{"Start":"01:12.550 ","End":"01:15.890","Text":"Those turn out to be as follows,"},{"Start":"01:15.890 ","End":"01:18.020","Text":"it\u0027s the same as in the previous exercises."},{"Start":"01:18.020 ","End":"01:20.060","Text":"We\u0027ve lost the original load."},{"Start":"01:20.060 ","End":"01:22.280","Text":"Here it is again."},{"Start":"01:22.280 ","End":"01:24.725","Text":"We\u0027re going to substitute in here."},{"Start":"01:24.725 ","End":"01:27.890","Text":"But cautiously there\u0027s way too many terms here."},{"Start":"01:27.890 ","End":"01:29.990","Text":"Let\u0027s just take them one at a time."},{"Start":"01:29.990 ","End":"01:34.430","Text":"In fact, I\u0027m even going to break it up into 4 terms because this last one will"},{"Start":"01:34.430 ","End":"01:40.025","Text":"be xy minus 5y so we have 4 terms."},{"Start":"01:40.025 ","End":"01:43.880","Text":"Now we\u0027re only interested in 2 kinds of exponents."},{"Start":"01:43.880 ","End":"01:49.040","Text":"We want the x to the Lambda exponents and the x to"},{"Start":"01:49.040 ","End":"01:56.000","Text":"the Lambda plus n. Let\u0027s look in each of these and see how we might get these."},{"Start":"01:56.000 ","End":"01:58.280","Text":"Now let\u0027s go for the x to the Lambda first."},{"Start":"01:58.280 ","End":"02:00.693","Text":"If I look at the first term,"},{"Start":"02:00.693 ","End":"02:02.300","Text":"there\u0027s an x^2 y\","},{"Start":"02:02.300 ","End":"02:04.880","Text":"which is going to raise all the exponents by 2."},{"Start":"02:04.880 ","End":"02:06.334","Text":"The 2 which is a constant,"},{"Start":"02:06.334 ","End":"02:08.195","Text":"is not going to change the powers."},{"Start":"02:08.195 ","End":"02:15.600","Text":"Let me just highlight the ones that are not equal to 1 here I have a minus 5, minus 1."},{"Start":"02:15.600 ","End":"02:20.030","Text":"This is minus 1 and this is plus 1 x to the Lambda."},{"Start":"02:20.030 ","End":"02:26.100","Text":"In this I can only get from the x to the Lambda minus 2 because it\u0027s multiplied by x^2."},{"Start":"02:26.100 ","End":"02:28.890","Text":"This is of interest to me."},{"Start":"02:28.890 ","End":"02:33.000","Text":"For xy\u0027, the power is raised by 1."},{"Start":"02:33.000 ","End":"02:37.115","Text":"This is the one I need here."},{"Start":"02:37.115 ","End":"02:41.810","Text":"Well, it\u0027s both multiplied by x and by a constant."},{"Start":"02:41.810 ","End":"02:45.425","Text":"In order to get x to the Lambda."},{"Start":"02:45.425 ","End":"02:47.450","Text":"Well, I only can get it from here."},{"Start":"02:47.450 ","End":"02:50.060","Text":"If there was an x to the Lambda minus 1,"},{"Start":"02:50.060 ","End":"02:51.950","Text":"I would take that also, but there isn\u0027t."},{"Start":"02:51.950 ","End":"02:55.340","Text":"It\u0027s just that and now that settles this."},{"Start":"02:55.340 ","End":"03:00.590","Text":"Now let\u0027s collect places where we could get x to the Lambda plus n. Here,"},{"Start":"03:00.590 ","End":"03:04.355","Text":"like I said, everything gets raised by 2 because of the x^2."},{"Start":"03:04.355 ","End":"03:05.735","Text":"We need to subtract 2."},{"Start":"03:05.735 ","End":"03:07.190","Text":"This is the exponent."},{"Start":"03:07.190 ","End":"03:11.715","Text":"This is the one that\u0027s going to contribute in the xy\u0027."},{"Start":"03:11.715 ","End":"03:17.090","Text":"I just need to subtract 1 because x is going to raise it 1."},{"Start":"03:17.090 ","End":"03:21.740","Text":"This is the one that I need and here,"},{"Start":"03:21.740 ","End":"03:23.570","Text":"the Lambda plus n,"},{"Start":"03:23.570 ","End":"03:29.240","Text":"I can get either from Lambda plus n or when I multiply by the x,"},{"Start":"03:29.240 ","End":"03:31.810","Text":"I can also get from here."},{"Start":"03:31.810 ","End":"03:34.200","Text":"These are the ones of interest."},{"Start":"03:34.200 ","End":"03:38.165","Text":"Let\u0027s start collecting each of these 4 terms,"},{"Start":"03:38.165 ","End":"03:41.645","Text":"beginning with the 2x^2 y\"."},{"Start":"03:41.645 ","End":"03:45.890","Text":"I highlighted that I need this one,"},{"Start":"03:45.890 ","End":"03:48.965","Text":"and I need this one."},{"Start":"03:48.965 ","End":"03:51.605","Text":"Next, I\u0027ll go to the xy\u0027."},{"Start":"03:51.605 ","End":"03:56.895","Text":"I take this one and that comes out here."},{"Start":"03:56.895 ","End":"04:00.035","Text":"This one is here,"},{"Start":"04:00.035 ","End":"04:02.975","Text":"and there\u0027s a minus 1 constant."},{"Start":"04:02.975 ","End":"04:08.585","Text":"Next, we\u0027re going to take this term with the plus 1xy,"},{"Start":"04:08.585 ","End":"04:11.320","Text":"it\u0027s scrolled off screen."},{"Start":"04:11.320 ","End":"04:17.075","Text":"I\u0027ll just go and show you what we wanted is just this one."},{"Start":"04:17.075 ","End":"04:22.040","Text":"Because the only place where I can multiply by x is here."},{"Start":"04:22.040 ","End":"04:26.335","Text":"That gives me the a, n minus 1."},{"Start":"04:26.335 ","End":"04:28.565","Text":"We won\u0027t take this,"},{"Start":"04:28.565 ","End":"04:31.580","Text":"It\u0027s just there because that\u0027s the first term in the series,"},{"Start":"04:31.580 ","End":"04:33.980","Text":"but it won\u0027t come into the computations."},{"Start":"04:33.980 ","End":"04:36.245","Text":"The last term of the 4,"},{"Start":"04:36.245 ","End":"04:39.200","Text":"which is this one with the minus 5."},{"Start":"04:39.200 ","End":"04:43.400","Text":"Again, I\u0027ll just show you why I took the a naughts and the a_n."},{"Start":"04:43.400 ","End":"04:46.695","Text":"That was here and that was here."},{"Start":"04:46.695 ","End":"04:48.765","Text":"This is what we have."},{"Start":"04:48.765 ","End":"04:52.040","Text":"Now let\u0027s start doing some computations."},{"Start":"04:52.040 ","End":"04:55.010","Text":"Forgot to highlight this and this."},{"Start":"04:55.010 ","End":"04:57.635","Text":"If I compare the terms for x to the Lambda,"},{"Start":"04:57.635 ","End":"05:00.140","Text":"that will be all these yellow ones,"},{"Start":"05:00.140 ","End":"05:03.875","Text":"but we have to multiply them with the coefficients in light blue."},{"Start":"05:03.875 ","End":"05:05.075","Text":"Well, not exactly."},{"Start":"05:05.075 ","End":"05:09.800","Text":"I should also mention that a naught which is here and here,"},{"Start":"05:09.800 ","End":"05:12.830","Text":"and here, I just divide by it."},{"Start":"05:12.830 ","End":"05:14.930","Text":"We don\u0027t have to actually take it,"},{"Start":"05:14.930 ","End":"05:18.430","Text":"though strictly speaking, you should have multiplied by a naught and then divide by it."},{"Start":"05:18.430 ","End":"05:20.210","Text":"We get twice Lambda,"},{"Start":"05:20.210 ","End":"05:21.465","Text":"Lambda minus 1,"},{"Start":"05:21.465 ","End":"05:27.360","Text":"minus 1 Lambda and minus 5 from here."},{"Start":"05:27.360 ","End":"05:28.910","Text":"If we simplify it,"},{"Start":"05:28.910 ","End":"05:31.790","Text":"this is the quadratic equation we get which is called,"},{"Start":"05:31.790 ","End":"05:35.225","Text":"doesn\u0027t matter the name, but it\u0027s called the indicial equation."},{"Start":"05:35.225 ","End":"05:38.720","Text":"The solutions, I\u0027ll leave you to check this and this."},{"Start":"05:38.720 ","End":"05:42.260","Text":"Usually, we let Lambda 1 be the bigger one."},{"Start":"05:42.260 ","End":"05:44.165","Text":"This is Lambda 1, Lambda 2."},{"Start":"05:44.165 ","End":"05:47.735","Text":"Notice that the difference between them is not a whole number,"},{"Start":"05:47.735 ","End":"05:49.775","Text":"and that\u0027s the easy case."},{"Start":"05:49.775 ","End":"05:52.910","Text":"Now we\u0027ve finished with the comparing x to the Lambda."},{"Start":"05:52.910 ","End":"05:58.425","Text":"Let\u0027s go to the x to the Lambda plus n. Here we collect this color together."},{"Start":"05:58.425 ","End":"06:02.150","Text":"We\u0027ll just write each one times its coefficient here."},{"Start":"06:02.150 ","End":"06:06.730","Text":"If you follow it, we get the following 4 terms equals 0."},{"Start":"06:06.730 ","End":"06:08.940","Text":"Collect like terms the a_n separately,"},{"Start":"06:08.940 ","End":"06:10.920","Text":"the a_n minus 1 separately."},{"Start":"06:10.920 ","End":"06:16.475","Text":"Now we can write the recursion equation that a_n is equal to this."},{"Start":"06:16.475 ","End":"06:20.870","Text":"Let me just put a_n at the side and this is what we get."},{"Start":"06:20.870 ","End":"06:22.850","Text":"Let\u0027s see the restriction on n. Well,"},{"Start":"06:22.850 ","End":"06:26.270","Text":"it has to be bigger or equal to 1 because of this index."},{"Start":"06:26.270 ","End":"06:34.490","Text":"I\u0027d like to remind you that the 2 values of Lambda we got where minus 1 and 2.5,"},{"Start":"06:34.490 ","End":"06:37.145","Text":"here it doesn\u0027t matter which one we take first."},{"Start":"06:37.145 ","End":"06:38.905","Text":"Let\u0027s take the minus 1."},{"Start":"06:38.905 ","End":"06:41.130","Text":"If you plug minus 1 in here,"},{"Start":"06:41.130 ","End":"06:46.040","Text":"then we get that a_n is equal to this."},{"Start":"06:46.040 ","End":"06:47.270","Text":"This is important."},{"Start":"06:47.270 ","End":"06:49.120","Text":"I think I\u0027ll put it in a box."},{"Start":"06:49.120 ","End":"06:52.370","Text":"We don\u0027t have an initial condition so we can\u0027t find a naught,"},{"Start":"06:52.370 ","End":"06:54.740","Text":"but everything else we can find in terms of it,"},{"Start":"06:54.740 ","End":"06:55.910","Text":"I put n equals 1,"},{"Start":"06:55.910 ","End":"06:57.890","Text":"I get a_1 equals a naught."},{"Start":"06:57.890 ","End":"06:59.345","Text":"It comes out over 5,"},{"Start":"06:59.345 ","End":"07:00.920","Text":"is 1/5 a naught,"},{"Start":"07:00.920 ","End":"07:02.480","Text":"if I plug in n equals 2,"},{"Start":"07:02.480 ","End":"07:04.265","Text":"I get a_2 is a_1 over 6."},{"Start":"07:04.265 ","End":"07:05.690","Text":"But if I use this also,"},{"Start":"07:05.690 ","End":"07:07.870","Text":"it\u0027s a naught times 1 over 30."},{"Start":"07:07.870 ","End":"07:09.715","Text":"Similarly, if n is 3,"},{"Start":"07:09.715 ","End":"07:12.860","Text":"I get here the denominator 3,"},{"Start":"07:12.860 ","End":"07:16.045","Text":"and so we get a naught over 90."},{"Start":"07:16.045 ","End":"07:18.995","Text":"We have y_1, which is x to the Lambda,"},{"Start":"07:18.995 ","End":"07:22.535","Text":"max to the minus 1 here times the power series."},{"Start":"07:22.535 ","End":"07:25.880","Text":"Really this should be taken together with this because"},{"Start":"07:25.880 ","End":"07:29.254","Text":"this is the formula for finding out successive terms."},{"Start":"07:29.254 ","End":"07:33.980","Text":"Also, we could take a naught outside and this is what we get, and that\u0027s y1."},{"Start":"07:33.980 ","End":"07:36.160","Text":"Now let\u0027s go work on y2."},{"Start":"07:36.160 ","End":"07:37.460","Text":"The concept is the same,"},{"Start":"07:37.460 ","End":"07:40.220","Text":"it\u0027s just that into our recursion formula above with"},{"Start":"07:40.220 ","End":"07:43.475","Text":"the Lambda if you plug in Lambda as 2 and a 1/2,"},{"Start":"07:43.475 ","End":"07:45.725","Text":"we get this recursion,"},{"Start":"07:45.725 ","End":"07:47.510","Text":"just go and check the algebra."},{"Start":"07:47.510 ","End":"07:52.400","Text":"I claim it\u0027s this and I also think this is important so I want a box it."},{"Start":"07:52.400 ","End":"07:54.650","Text":"As before n bigger or equal to 1."},{"Start":"07:54.650 ","End":"07:58.205","Text":"If I put n equals 1 I get a_1 in terms of a naught."},{"Start":"07:58.205 ","End":"07:59.480","Text":"If i put n equals 2,"},{"Start":"07:59.480 ","End":"08:01.340","Text":"I get a_2 in terms of a_1,"},{"Start":"08:01.340 ","End":"08:03.589","Text":"but then I plug in a_1 in terms of a naught."},{"Start":"08:03.589 ","End":"08:06.040","Text":"So I\u0027ve got a_2 in terms of a_naught."},{"Start":"08:06.040 ","End":"08:08.730","Text":"Continuing this way with n equals 3,"},{"Start":"08:08.730 ","End":"08:12.265","Text":"we get a_3 in terms of a_2 and then in terms of a naught."},{"Start":"08:12.265 ","End":"08:15.980","Text":"So y2 is x to the other Lambda,"},{"Start":"08:15.980 ","End":"08:21.335","Text":"and then a naught minus a 1x plus a2x^2."},{"Start":"08:21.335 ","End":"08:24.260","Text":"But using these everything in terms of a naught."},{"Start":"08:24.260 ","End":"08:26.980","Text":"Then I put a naught out the brackets."},{"Start":"08:26.980 ","End":"08:28.535","Text":"This is what we get."},{"Start":"08:28.535 ","End":"08:31.895","Text":"Although really this should be taken together with this."},{"Start":"08:31.895 ","End":"08:34.520","Text":"This helps us to find successive terms."},{"Start":"08:34.520 ","End":"08:36.110","Text":"If we have each term we have,"},{"Start":"08:36.110 ","End":"08:40.265","Text":"we can get the next term in terms of the previous one using the recursion formula."},{"Start":"08:40.265 ","End":"08:43.865","Text":"Now we want to combine y_1 and y_2 together."},{"Start":"08:43.865 ","End":"08:47.915","Text":"Remember that we take a linear combination of y_1 and"},{"Start":"08:47.915 ","End":"08:51.780","Text":"y_2 with 2 constants,and this is what we get."},{"Start":"08:51.780 ","End":"08:55.225","Text":"Although strictly speaking, it\u0027s a different a naught here and here,"},{"Start":"08:55.225 ","End":"08:57.635","Text":"but I\u0027m ignoring that because in any event,"},{"Start":"08:57.635 ","End":"08:59.840","Text":"this combines to give another constant, call,"},{"Start":"08:59.840 ","End":"09:02.365","Text":"this one k1 and this one k2."},{"Start":"09:02.365 ","End":"09:04.790","Text":"This is the solution,"},{"Start":"09:04.790 ","End":"09:12.885","Text":"and I\u0027ll just remark in terms of domain that x to the minus 1 is not defined at 0."},{"Start":"09:12.885 ","End":"09:15.035","Text":"We have to take x not equal to 0,"},{"Start":"09:15.035 ","End":"09:19.160","Text":"but also x to the 2 and a 1/2 is x squared times square root of x."},{"Start":"09:19.160 ","End":"09:21.920","Text":"So x has to be bigger or equal to 0."},{"Start":"09:21.920 ","End":"09:25.325","Text":"It\u0027s defined at x bigger than 0,"},{"Start":"09:25.325 ","End":"09:28.100","Text":"at least near 0, but bigger than 0."},{"Start":"09:28.100 ","End":"09:30.200","Text":"Anyway, that\u0027s optional,"},{"Start":"09:30.200 ","End":"09:33.030","Text":"just mentioned it. We\u0027re done."}],"Thumbnail":null,"ID":7888},{"Watched":false,"Name":"Exercise 4","Duration":"7m 43s","ChapterTopicVideoID":7837,"CourseChapterTopicPlaylistID":4244,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.330","Text":"Here we have this second-order differential equation."},{"Start":"00:03.330 ","End":"00:05.160","Text":"There\u0027s more than one way to solve it."},{"Start":"00:05.160 ","End":"00:09.480","Text":"I happen to note that it\u0027s an Euler equation from Chapter 2,"},{"Start":"00:09.480 ","End":"00:12.240","Text":"but that\u0027s not what we\u0027re dealing with now."},{"Start":"00:12.240 ","End":"00:15.870","Text":"Now, we\u0027re dealing with regular singular points in power series solutions."},{"Start":"00:15.870 ","End":"00:19.230","Text":"Maybe at the end I\u0027ll solve it also as an Euler equation."},{"Start":"00:19.230 ","End":"00:24.360","Text":"Anyway, let\u0027s show that x=0 is a regular singular point."},{"Start":"00:24.360 ","End":"00:28.320","Text":"We divide by the coefficient of y\u0027\u0027 to get it in standard form."},{"Start":"00:28.320 ","End":"00:31.170","Text":"This part is p(x), this is q(x)."},{"Start":"00:31.170 ","End":"00:34.260","Text":"At least one of these is not defined that x=0,"},{"Start":"00:34.260 ","End":"00:36.240","Text":"in fact, both of them are not defined there,"},{"Start":"00:36.240 ","End":"00:41.850","Text":"but we have to make sure that x times p and x^2 times q are defined."},{"Start":"00:41.850 ","End":"00:44.150","Text":"These come out to be constant functions,"},{"Start":"00:44.150 ","End":"00:47.875","Text":"in fact of surely they\u0027re defined at 0 and everywhere."},{"Start":"00:47.875 ","End":"00:52.820","Text":"We expect a power series solution beginning with x to some Lambda."},{"Start":"00:52.820 ","End":"00:54.650","Text":"I prefer to write it in Sigma form,"},{"Start":"00:54.650 ","End":"00:56.540","Text":"it looks clearer what\u0027s happening."},{"Start":"00:56.540 ","End":"01:02.345","Text":"x to the Lambda times a regular power series, anx^n."},{"Start":"01:02.345 ","End":"01:08.225","Text":"Of course, we\u0027ll need y\u0027 and y\u0027\u0027 in order to substitute into the differential equation."},{"Start":"01:08.225 ","End":"01:11.110","Text":"Here they are, y\u0027 and y\u0027\u0027."},{"Start":"01:11.110 ","End":"01:14.270","Text":"We need the original ODE, here it is."},{"Start":"01:14.270 ","End":"01:18.125","Text":"We\u0027re going to be substituting these into here,"},{"Start":"01:18.125 ","End":"01:23.720","Text":"but we\u0027re going to do this selectively in a smart way because we really only need"},{"Start":"01:23.720 ","End":"01:31.190","Text":"the coefficients of x to the Lambda and also x to the Lambda plus n. Let\u0027s take a look."},{"Start":"01:31.190 ","End":"01:35.180","Text":"Well, look at this term, the x^2 y\u0027\u0027, forget the constant,"},{"Start":"01:35.180 ","End":"01:39.830","Text":"means that I\u0027m going to have to look for 2 less than the exponents here."},{"Start":"01:39.830 ","End":"01:43.175","Text":"I need Lambda minus 2 and Lambda plus n minus 2."},{"Start":"01:43.175 ","End":"01:46.685","Text":"Lambda plus 2 is here, so this I\u0027m going to need."},{"Start":"01:46.685 ","End":"01:49.505","Text":"Lambda plus n minus 2, here it is."},{"Start":"01:49.505 ","End":"01:51.274","Text":"I\u0027m going to need this."},{"Start":"01:51.274 ","End":"01:54.949","Text":"The next term has an x multiplier,"},{"Start":"01:54.949 ","End":"01:58.760","Text":"so I need to look for exponents that are 1 less than what I need."},{"Start":"01:58.760 ","End":"02:01.820","Text":"I need a Lambda minus 1 and Lambda plus n minus 1."},{"Start":"02:01.820 ","End":"02:05.660","Text":"I\u0027ve got Lambda minus 1 in this coefficient,"},{"Start":"02:05.660 ","End":"02:08.660","Text":"and I have Lambda plus n minus 1 here."},{"Start":"02:08.660 ","End":"02:10.430","Text":"I\u0027m going to need this one,"},{"Start":"02:10.430 ","End":"02:12.870","Text":"on my left arm is just straightforward, just y itself."},{"Start":"02:12.870 ","End":"02:19.430","Text":"I\u0027ve got x to the Lambda with this and x to the Lambda plus n with this."},{"Start":"02:19.430 ","End":"02:23.150","Text":"Before I substitute, I would like to just highlight also the coefficients here."},{"Start":"02:23.150 ","End":"02:24.680","Text":"I have a minus 1,"},{"Start":"02:24.680 ","End":"02:26.450","Text":"here I have a plus 1."},{"Start":"02:26.450 ","End":"02:29.350","Text":"I have a 3, a minus 1,"},{"Start":"02:29.350 ","End":"02:30.720","Text":"and a plus 1."},{"Start":"02:30.720 ","End":"02:32.540","Text":"I\u0027m going to take these 3 terms one at a time,"},{"Start":"02:32.540 ","End":"02:35.795","Text":"starting with the x^2y\u0027\u0027 term."},{"Start":"02:35.795 ","End":"02:38.435","Text":"What we need is,"},{"Start":"02:38.435 ","End":"02:39.800","Text":"on the ones I\u0027ve highlighted,"},{"Start":"02:39.800 ","End":"02:42.175","Text":"I need this which is here,"},{"Start":"02:42.175 ","End":"02:45.860","Text":"and I need this which is here."},{"Start":"02:45.860 ","End":"02:48.275","Text":"Now, let\u0027s move on to the next one."},{"Start":"02:48.275 ","End":"02:51.815","Text":"Here I\u0027m reading off from the y\u0027."},{"Start":"02:51.815 ","End":"02:53.660","Text":"I\u0027ve already collected the ones,"},{"Start":"02:53.660 ","End":"02:58.270","Text":"that\u0027s this one and this here,"},{"Start":"02:58.270 ","End":"03:00.620","Text":"and then the last one, well,"},{"Start":"03:00.620 ","End":"03:02.885","Text":"we\u0027ve scrolled off, so let\u0027s go back and look."},{"Start":"03:02.885 ","End":"03:06.005","Text":"We had a naught x to the Lambda an,"},{"Start":"03:06.005 ","End":"03:11.990","Text":"x to the Lambda plus n. That is this and this."},{"Start":"03:11.990 ","End":"03:14.555","Text":"Now, let\u0027s do some collecting."},{"Start":"03:14.555 ","End":"03:17.990","Text":"We\u0027ll start with the x to the Lambda terms."},{"Start":"03:17.990 ","End":"03:20.690","Text":"Forget about the a naught,"},{"Start":"03:20.690 ","End":"03:22.370","Text":"they always cancels out."},{"Start":"03:22.370 ","End":"03:23.615","Text":"Forgetting the a naught,"},{"Start":"03:23.615 ","End":"03:25.565","Text":"I got Lambda, Lambda minus 1,"},{"Start":"03:25.565 ","End":"03:27.430","Text":"but times the 3,"},{"Start":"03:27.430 ","End":"03:29.160","Text":"the coefficient, emphasizing."},{"Start":"03:29.160 ","End":"03:32.520","Text":"There is a 3 here minus 1 here, just a 1 here."},{"Start":"03:32.520 ","End":"03:35.540","Text":"These 3 together give me this."},{"Start":"03:35.540 ","End":"03:37.625","Text":"That\u0027s got to equal 0,"},{"Start":"03:37.625 ","End":"03:40.550","Text":"and this gives us the indicial equation,"},{"Start":"03:40.550 ","End":"03:45.835","Text":"which is this, and the solutions to this are 1 and 1/3."},{"Start":"03:45.835 ","End":"03:47.220","Text":"I\u0027ll call this one,"},{"Start":"03:47.220 ","End":"03:49.875","Text":"say Lambda 1 and this is Lambda 2."},{"Start":"03:49.875 ","End":"03:57.680","Text":"If I take the coefficients of Lambda plus n, from here, here,"},{"Start":"03:57.680 ","End":"04:00.470","Text":"and together with these constants,"},{"Start":"04:00.470 ","End":"04:02.135","Text":"the 3 and the minus 1 and so on,"},{"Start":"04:02.135 ","End":"04:06.720","Text":"I get that this is also got to equals 0."},{"Start":"04:06.720 ","End":"04:10.545","Text":"Here everything involves an which I can take out of the brackets,"},{"Start":"04:10.545 ","End":"04:14.210","Text":"and I claim that this thing in the square brackets is not 0."},{"Start":"04:14.210 ","End":"04:16.040","Text":"If you\u0027re not convinced I did a little checking,"},{"Start":"04:16.040 ","End":"04:18.755","Text":"if you substitute Lambda equals 1 in this,"},{"Start":"04:18.755 ","End":"04:23.530","Text":"then you get 3n^2 plus 2n,"},{"Start":"04:23.530 ","End":"04:25.370","Text":"and that\u0027s not going to be 0,"},{"Start":"04:25.370 ","End":"04:28.925","Text":"at least for n bigger or equal to 1 it\u0027s not going to be 0."},{"Start":"04:28.925 ","End":"04:31.565","Text":"If you plug in Lambda equals a 1/3,"},{"Start":"04:31.565 ","End":"04:32.690","Text":"I didn\u0027t make a mistake,"},{"Start":"04:32.690 ","End":"04:39.405","Text":"I got this to simplify to 3n^2 minus 2n or if you like,"},{"Start":"04:39.405 ","End":"04:41.745","Text":"n times 3n minus 2."},{"Start":"04:41.745 ","End":"04:43.850","Text":"Also when n is bigger or equal to 1,"},{"Start":"04:43.850 ","End":"04:46.370","Text":"each of these terms is non-zero,"},{"Start":"04:46.370 ","End":"04:50.425","Text":"so an is 0 for n bigger or equal to 1."},{"Start":"04:50.425 ","End":"04:54.075","Text":"We only have an a0 which might be non-zero."},{"Start":"04:54.075 ","End":"04:55.925","Text":"Where does that leave us?"},{"Start":"04:55.925 ","End":"05:00.770","Text":"I just want to remind you the Lambdas were equal to 1 or a 1/3."},{"Start":"05:00.770 ","End":"05:02.545","Text":"If Lambda equals 1,"},{"Start":"05:02.545 ","End":"05:04.710","Text":"then we\u0027ll get one of the solutions,"},{"Start":"05:04.710 ","End":"05:07.820","Text":"y1 is x^1 times the whole power series,"},{"Start":"05:07.820 ","End":"05:10.475","Text":"but the whole power series was just a naught."},{"Start":"05:10.475 ","End":"05:11.950","Text":"Everything else was 0."},{"Start":"05:11.950 ","End":"05:15.465","Text":"If I take Lambda equals a 1/3, then I\u0027ve got the other y,"},{"Start":"05:15.465 ","End":"05:18.320","Text":"y_2 is x^1/3 times the power series,"},{"Start":"05:18.320 ","End":"05:20.420","Text":"which is just a constant."},{"Start":"05:20.420 ","End":"05:22.265","Text":"Then we combine the two,"},{"Start":"05:22.265 ","End":"05:27.050","Text":"linear combination of the two solutions like so,"},{"Start":"05:27.050 ","End":"05:29.180","Text":"each one of them is just a constant."},{"Start":"05:29.180 ","End":"05:32.440","Text":"This is our solution, no power series,"},{"Start":"05:32.440 ","End":"05:34.350","Text":"it\u0027s an unusual one."},{"Start":"05:34.350 ","End":"05:36.140","Text":"We\u0027re basically done."},{"Start":"05:36.140 ","End":"05:37.850","Text":"But if you want to stay, I\u0027ll do it as"},{"Start":"05:37.850 ","End":"05:40.655","Text":"an Euler equation and see if we get the same thing."},{"Start":"05:40.655 ","End":"05:44.150","Text":"If you\u0027ve stayed, it means that you want me to solve it as"},{"Start":"05:44.150 ","End":"05:48.120","Text":"an Euler equation just for practice and for comparison."},{"Start":"05:48.120 ","End":"05:52.830","Text":"Let me just clear some space and the equation was 3x^2y\u0027\u0027"},{"Start":"05:52.830 ","End":"05:59.627","Text":"minus xy\u0027 plus y equals 0,"},{"Start":"05:59.627 ","End":"06:05.250","Text":"and it\u0027s of the form ax^2y\u0027\u0027 plus"},{"Start":"06:05.250 ","End":"06:11.760","Text":"bxy\u0027 plus c times y equals 0."},{"Start":"06:11.760 ","End":"06:17.225","Text":"The technique for solving such an equation is to write the characteristic equation,"},{"Start":"06:17.225 ","End":"06:23.630","Text":"ak(k) minus 1 plus bk plus c equals 0,"},{"Start":"06:23.630 ","End":"06:26.415","Text":"which in our case comes to be,"},{"Start":"06:26.415 ","End":"06:32.925","Text":"let\u0027s see, 3k(k) minus 1,"},{"Start":"06:32.925 ","End":"06:36.710","Text":"k plus 1 equals 0."},{"Start":"06:36.710 ","End":"06:39.295","Text":"This simplifies to, let\u0027s see,"},{"Start":"06:39.295 ","End":"06:45.340","Text":"3k^2 minus 3k minus k is minus 4k plus 1 equals 0."},{"Start":"06:45.340 ","End":"06:47.990","Text":"If we solve this equation,"},{"Start":"06:47.990 ","End":"06:53.275","Text":"we\u0027ll get k=1 or 1/3."},{"Start":"06:53.275 ","End":"06:56.179","Text":"No surprise, notice it\u0027s not a coincidence,"},{"Start":"06:56.179 ","End":"06:57.685","Text":"but is the same as this."},{"Start":"06:57.685 ","End":"07:00.230","Text":"Then the rule is that once we found the k,"},{"Start":"07:00.230 ","End":"07:03.235","Text":"so we find k_1 and k_2,"},{"Start":"07:03.235 ","End":"07:04.970","Text":"and if they\u0027re different,"},{"Start":"07:04.970 ","End":"07:06.140","Text":"which they are in our case,"},{"Start":"07:06.140 ","End":"07:16.075","Text":"the solution is y equals c_1^k_1 plus c_2x^k_2."},{"Start":"07:16.075 ","End":"07:21.470","Text":"But let\u0027s say we took x only positive just to make life easier,"},{"Start":"07:21.470 ","End":"07:25.100","Text":"and we replaced letter c with a letter k here and here,"},{"Start":"07:25.100 ","End":"07:28.010","Text":"then this is really just the same as this,"},{"Start":"07:28.010 ","End":"07:30.360","Text":"because k is 1 over a 1/3,"},{"Start":"07:30.360 ","End":"07:34.225","Text":"so we get that y is equal to c_1,"},{"Start":"07:34.225 ","End":"07:40.890","Text":"x^1 plus c_2x^1/3."},{"Start":"07:40.890 ","End":"07:44.290","Text":"Same thing, that\u0027s it."}],"Thumbnail":null,"ID":7889},{"Watched":false,"Name":"Exercise 5","Duration":"11m 57s","ChapterTopicVideoID":7838,"CourseChapterTopicPlaylistID":4244,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.590","Text":"n this exercise, we have this differential equation to solve and we\u0027re going to solve"},{"Start":"00:04.590 ","End":"00:09.555","Text":"it using the regular singular point and power series."},{"Start":"00:09.555 ","End":"00:12.450","Text":"X equals naught is a regular singular point"},{"Start":"00:12.450 ","End":"00:15.450","Text":"because if we write this equation in standard form,"},{"Start":"00:15.450 ","End":"00:18.210","Text":"dividing by the coefficient of y\u0027\u0027,"},{"Start":"00:18.210 ","End":"00:20.856","Text":"we get that this is p(x) and this is q(x),"},{"Start":"00:20.856 ","End":"00:23.300","Text":"and at least one of them is not defined at x equals naught."},{"Start":"00:23.300 ","End":"00:25.115","Text":"In fact, this one is not defined."},{"Start":"00:25.115 ","End":"00:30.260","Text":"But if I take x times this or x^2 times this as follows,"},{"Start":"00:30.260 ","End":"00:34.175","Text":"then both of these are defined at x=0,"},{"Start":"00:34.175 ","End":"00:37.430","Text":"so we can use the power series method and we write"},{"Start":"00:37.430 ","End":"00:41.240","Text":"y in the following form with powers beginning with x^Lambda."},{"Start":"00:41.240 ","End":"00:43.940","Text":"It\u0027s easier if we write it with the Sigma notation."},{"Start":"00:43.940 ","End":"00:50.060","Text":"It\u0027s x^Lambda times the sum of a power series an x^n."},{"Start":"00:50.060 ","End":"00:52.460","Text":"But of course we\u0027ll need the derivative and the second"},{"Start":"00:52.460 ","End":"00:56.210","Text":"derivative because we\u0027re going to substitute in the equation."},{"Start":"00:56.210 ","End":"00:59.701","Text":"Here they are, there\u0027s y\u0027 and there\u0027s y\u0027\u0027,"},{"Start":"00:59.701 ","End":"01:02.620","Text":"and here\u0027s the original equation."},{"Start":"01:02.620 ","End":"01:04.745","Text":"We\u0027re going to substitute in here,"},{"Start":"01:04.745 ","End":"01:08.120","Text":"but not the whole thing just selectively because all we"},{"Start":"01:08.120 ","End":"01:11.510","Text":"need is the coefficient of x to the lambda and"},{"Start":"01:11.510 ","End":"01:14.990","Text":"the coefficient of x to the lambda plus n."},{"Start":"01:14.990 ","End":"01:19.055","Text":"Let\u0027s see where these exponents could come from in each of these."},{"Start":"01:19.055 ","End":"01:21.210","Text":"In the case of x^2y\u0027\u0027,"},{"Start":"01:21.210 ","End":"01:26.060","Text":"x^2 is going to increase all these exponents by 2."},{"Start":"01:26.060 ","End":"01:27.230","Text":"To get x^Lambda,"},{"Start":"01:27.230 ","End":"01:30.335","Text":"we have to look for x^Lambda minus 2, which is here."},{"Start":"01:30.335 ","End":"01:34.616","Text":"We\u0027ll need this and for the Lambda plus n,"},{"Start":"01:34.616 ","End":"01:37.970","Text":"we\u0027ll need Lambda plus n minus 2 which is here,"},{"Start":"01:37.970 ","End":"01:40.490","Text":"and so we\u0027ll need this."},{"Start":"01:40.490 ","End":"01:42.380","Text":"The first of the three terms,"},{"Start":"01:42.380 ","End":"01:45.335","Text":"the x^2y\u0027\u0027 is this and I can highlight"},{"Start":"01:45.335 ","End":"01:48.845","Text":"this and this just so you see it\u0027s the same as here."},{"Start":"01:48.845 ","End":"01:52.765","Text":"Now, let\u0027s move on to the next term, which is xy\u0027."},{"Start":"01:52.765 ","End":"01:55.335","Text":"How did I get this?"},{"Start":"01:55.335 ","End":"01:57.860","Text":"I look in y\u0027 and this time,"},{"Start":"01:57.860 ","End":"02:01.865","Text":"since it\u0027s just x, I have to look for x^Lambda minus 1."},{"Start":"02:01.865 ","End":"02:03.355","Text":"I find it here,"},{"Start":"02:03.355 ","End":"02:04.805","Text":"so I need this,"},{"Start":"02:04.805 ","End":"02:07.730","Text":"and I look for Lambda plus n minus 1."},{"Start":"02:07.730 ","End":"02:11.135","Text":"I find that here and I need this because"},{"Start":"02:11.135 ","End":"02:15.445","Text":"these two will be raised by 1 after I multiply by x,"},{"Start":"02:15.445 ","End":"02:22.370","Text":"so this one is this and this one is this here."},{"Start":"02:22.370 ","End":"02:26.635","Text":"Now we\u0027ve got one more, is the x^2y."},{"Start":"02:26.635 ","End":"02:30.080","Text":"Let\u0027s go back and look because it\u0027s scrolled off screen."},{"Start":"02:30.080 ","End":"02:32.990","Text":"Here\u0027s y and what we need,"},{"Start":"02:32.990 ","End":"02:38.315","Text":"for x^Lambda, I need to find x^Lambda minus 2 here, but there is none."},{"Start":"02:38.315 ","End":"02:41.055","Text":"So we don\u0027t get anything for this,"},{"Start":"02:41.055 ","End":"02:42.488","Text":"and the Lambda plus n,"},{"Start":"02:42.488 ","End":"02:43.970","Text":"that I can find."},{"Start":"02:43.970 ","End":"02:46.370","Text":"I\u0027m looking for Lambda plus n minus 2."},{"Start":"02:46.370 ","End":"02:47.825","Text":"I find it here,"},{"Start":"02:47.825 ","End":"02:51.175","Text":"and so this is the thing I need."},{"Start":"02:51.175 ","End":"02:54.845","Text":"In the case of the last one,"},{"Start":"02:54.845 ","End":"02:57.500","Text":"this here is written in but I don\u0027t use it,"},{"Start":"02:57.500 ","End":"02:59.885","Text":"I just wrote it in because it\u0027s the first one in the series."},{"Start":"02:59.885 ","End":"03:03.680","Text":"But I do use this one and I\u0027ll highlight it."},{"Start":"03:03.680 ","End":"03:10.190","Text":"Now let\u0027s start gathering together all the coefficients for both x^Lambda,"},{"Start":"03:10.190 ","End":"03:12.560","Text":"and x^Lambda plus n. Now,"},{"Start":"03:12.560 ","End":"03:16.700","Text":"the x^Lambda is this color and what I get is"},{"Start":"03:16.700 ","End":"03:21.095","Text":"Lambda times Lambda minus 1 plus Lambda equals naught."},{"Start":"03:21.095 ","End":"03:22.670","Text":"The a naught, we ignore,"},{"Start":"03:22.670 ","End":"03:25.625","Text":"it cancels, so we just need the Lambda."},{"Start":"03:25.625 ","End":"03:28.100","Text":"Now, if you expand this, what is this?"},{"Start":"03:28.100 ","End":"03:33.245","Text":"This is Lambda squared minus Lambda plus Lambda equals 0,"},{"Start":"03:33.245 ","End":"03:35.735","Text":"so Lambda squared equals 0."},{"Start":"03:35.735 ","End":"03:37.595","Text":"0 is a double root,"},{"Start":"03:37.595 ","End":"03:40.615","Text":"Lambda equals 0 and Lambda equals 0."},{"Start":"03:40.615 ","End":"03:42.650","Text":"In this exercise we have a double root,"},{"Start":"03:42.650 ","End":"03:45.725","Text":"so we have to treat it accordingly."},{"Start":"03:45.725 ","End":"03:47.225","Text":"But the next step is the same,"},{"Start":"03:47.225 ","End":"03:51.860","Text":"the one we look for the x^Lambda plus n coefficients."},{"Start":"03:51.860 ","End":"03:53.645","Text":"I\u0027m writing all this here,"},{"Start":"03:53.645 ","End":"03:56.685","Text":"this copied here, this copied here."},{"Start":"03:56.685 ","End":"03:58.500","Text":"Then if we collect terms,"},{"Start":"03:58.500 ","End":"04:01.770","Text":"we collect the an separately and the an minus 2 and"},{"Start":"04:01.770 ","End":"04:05.353","Text":"bring the an minus 2 to the other side and divide,"},{"Start":"04:05.353 ","End":"04:08.710","Text":"we extract an and we get this,"},{"Start":"04:08.710 ","End":"04:11.850","Text":"and this only works for n bigger or"},{"Start":"04:11.850 ","End":"04:15.230","Text":"equal to 2 because we can\u0027t have a negative index here."},{"Start":"04:15.230 ","End":"04:16.520","Text":"I don\u0027t think this is important,"},{"Start":"04:16.520 ","End":"04:18.710","Text":"so I\u0027ll put it in a box."},{"Start":"04:18.710 ","End":"04:22.870","Text":"Now I need some more space and here we are."},{"Start":"04:22.870 ","End":"04:24.765","Text":"Now, in this step we\u0027re going to find y_1,"},{"Start":"04:24.765 ","End":"04:27.830","Text":"and this is the same as when we had different roots."},{"Start":"04:27.830 ","End":"04:29.075","Text":"You just take one of them."},{"Start":"04:29.075 ","End":"04:34.925","Text":"They\u0027re both 0, so we take 0 and plug it into the recursion equation."},{"Start":"04:34.925 ","End":"04:38.030","Text":"If Lambda is 0, it\u0027s just n^2 in the denominator."},{"Start":"04:38.030 ","End":"04:41.510","Text":"The first thing I can plug in is 2 and if I do that, here,"},{"Start":"04:41.510 ","End":"04:45.054","Text":"we get a_2 in terms of a naught."},{"Start":"04:45.054 ","End":"04:49.585","Text":"Similarly, I can get a_4 in terms of a_2 and then in terms of a naught."},{"Start":"04:49.585 ","End":"04:55.230","Text":"For the odd ones, I can\u0027t find them because if I want to find a_3,"},{"Start":"04:55.230 ","End":"04:56.850","Text":"it gives me in terms of a_1."},{"Start":"04:56.850 ","End":"04:58.350","Text":"But I don\u0027t know what a_1 is."},{"Start":"04:58.350 ","End":"04:59.880","Text":"A_1 could be anything."},{"Start":"04:59.880 ","End":"05:05.640","Text":"We can actually let a_1 equal 0 and then a_3 is something times a_1."},{"Start":"05:05.640 ","End":"05:07.470","Text":"a_3 is 0,"},{"Start":"05:07.470 ","End":"05:09.170","Text":"a_5 is 0,"},{"Start":"05:09.170 ","End":"05:11.090","Text":"and so on, so just ignore the odd ones."},{"Start":"05:11.090 ","End":"05:14.360","Text":"We\u0027ll just continue with the even ones. Here\u0027s another one."},{"Start":"05:14.360 ","End":"05:17.030","Text":"I am deliberately writing quite a few elements,"},{"Start":"05:17.030 ","End":"05:19.474","Text":"and I do this because I can find a pattern."},{"Start":"05:19.474 ","End":"05:21.500","Text":"I don\u0027t just have a recursion formula,"},{"Start":"05:21.500 ","End":"05:25.055","Text":"I can find a closed form for the even terms."},{"Start":"05:25.055 ","End":"05:28.230","Text":"An even term, so we just write it as a_2n."},{"Start":"05:28.230 ","End":"05:34.090","Text":"Like here, n would be 3 and we have a_2 times 3 or a_6."},{"Start":"05:34.790 ","End":"05:39.450","Text":"I forgot to write the a naught here."},{"Start":"05:39.450 ","End":"05:41.750","Text":"First off, we get this thing."},{"Start":"05:41.750 ","End":"05:44.075","Text":"We see that the signs alternate,"},{"Start":"05:44.075 ","End":"05:46.955","Text":"a_2 is minus then this is plus, this is minus."},{"Start":"05:46.955 ","End":"05:49.505","Text":"We have minus 1^n."},{"Start":"05:49.505 ","End":"05:51.320","Text":"Then on the denominator,"},{"Start":"05:51.320 ","End":"05:53.850","Text":"we can see it from here."},{"Start":"05:53.850 ","End":"05:57.000","Text":"It\u0027s like 2 times 4 times 6 all squared."},{"Start":"05:57.000 ","End":"06:01.985","Text":"In general, we have 2 times 4 times 6 up to the index here."},{"Start":"06:01.985 ","End":"06:05.030","Text":"Then we process it by writing this as 2 times 1,"},{"Start":"06:05.030 ","End":"06:06.470","Text":"2 times 2, 2 times 3,"},{"Start":"06:06.470 ","End":"06:11.370","Text":"2 times n, 2, 2, 2, 2."},{"Start":"06:11.370 ","End":"06:14.160","Text":"I have n factors 2, that\u0027s 2^n."},{"Start":"06:14.160 ","End":"06:16.380","Text":"Then I have 1 times 2 times 3 up to n,"},{"Start":"06:16.380 ","End":"06:18.030","Text":"which is n factorial."},{"Start":"06:18.030 ","End":"06:21.174","Text":"We have a closed form for a_2n,"},{"Start":"06:21.174 ","End":"06:23.670","Text":"and of course I have to put the a naught here."},{"Start":"06:23.670 ","End":"06:27.950","Text":"We can get y1(x) as x^Lambda,"},{"Start":"06:27.950 ","End":"06:30.485","Text":"which happens to be naught 0."},{"Start":"06:30.485 ","End":"06:33.035","Text":"Then we just take all these terms here."},{"Start":"06:33.035 ","End":"06:40.250","Text":"What we can do now is take out the a naught from the brackets and x^naught is 1,"},{"Start":"06:40.250 ","End":"06:41.975","Text":"so what we get is this,"},{"Start":"06:41.975 ","End":"06:44.390","Text":"and it looks neater in Sigma form."},{"Start":"06:44.390 ","End":"06:50.540","Text":"I just take this general term and put it here and since we go on to infinity,"},{"Start":"06:50.540 ","End":"06:54.050","Text":"we just say n goes from naught to infinity."},{"Start":"06:54.050 ","End":"06:58.595","Text":"However, I don\u0027t really need an a naught or I can let a naught equal one."},{"Start":"06:58.595 ","End":"07:00.470","Text":"I just need a particular y_1."},{"Start":"07:00.470 ","End":"07:02.210","Text":"In any event, I\u0027m going to multiply it by a"},{"Start":"07:02.210 ","End":"07:05.885","Text":"constant at the end with a linear combination with y_2."},{"Start":"07:05.885 ","End":"07:10.700","Text":"I\u0027m just crossing out the a naught and I have y_1 as written here,"},{"Start":"07:10.700 ","End":"07:12.230","Text":"and I\u0027ll just put it in a box."},{"Start":"07:12.230 ","End":"07:14.270","Text":"That\u0027s an important result."},{"Start":"07:14.270 ","End":"07:16.480","Text":"Next we need to find y_2."},{"Start":"07:16.480 ","End":"07:19.790","Text":"But it\u0027s not going to be like before where we just use"},{"Start":"07:19.790 ","End":"07:23.525","Text":"the other Lambda because there is no other Lambda, there\u0027s only one Lambda."},{"Start":"07:23.525 ","End":"07:26.690","Text":"What do we do to find the other y_2?"},{"Start":"07:26.690 ","End":"07:29.630","Text":"Because like I said, we can\u0027t substitute Lambda again,"},{"Start":"07:29.630 ","End":"07:31.235","Text":"the two Lambdas were the same."},{"Start":"07:31.235 ","End":"07:33.185","Text":"Here\u0027s how it goes."},{"Start":"07:33.185 ","End":"07:37.528","Text":"We just leave Lambda as is and don\u0027t substitute anything,"},{"Start":"07:37.528 ","End":"07:42.915","Text":"and then we compute a few coefficients using the recursion rule but keeping Lambda."},{"Start":"07:42.915 ","End":"07:45.370","Text":"For example, if I let n equals 2,"},{"Start":"07:45.370 ","End":"07:49.540","Text":"I get a_2 equals a naught over Lambda plus 2^2 squared,"},{"Start":"07:49.540 ","End":"07:55.035","Text":"and then n equals 4 and we get a_4 in terms of a_2 and then in terms of a naught,"},{"Start":"07:55.035 ","End":"07:57.255","Text":"and we identify a pattern."},{"Start":"07:57.255 ","End":"08:02.745","Text":"We see the alternating sign gives us the minus 1^n."},{"Start":"08:02.745 ","End":"08:05.400","Text":"Here we have like Lambda plus 2^2,"},{"Start":"08:05.400 ","End":"08:06.865","Text":"Lambda plus 4^2,"},{"Start":"08:06.865 ","End":"08:11.510","Text":"and we keep on adding until we get to Lambda plus the last one squared."},{"Start":"08:11.510 ","End":"08:15.185","Text":"This is the general term for a_2n. The odd ones are 0."},{"Start":"08:15.185 ","End":"08:16.880","Text":"A_1, there\u0027s no restriction,"},{"Start":"08:16.880 ","End":"08:22.580","Text":"I could take a 0 and then a_3 in terms of a_1 is something times 0, and so on."},{"Start":"08:22.580 ","End":"08:24.575","Text":"Just forget about the odd numbers."},{"Start":"08:24.575 ","End":"08:27.950","Text":"Now if we substitute all these a_2,"},{"Start":"08:27.950 ","End":"08:32.270","Text":"a_4 and so on into the expression for y,"},{"Start":"08:32.270 ","End":"08:35.570","Text":"the series, we get y in terms of x and Lambda."},{"Start":"08:35.570 ","End":"08:38.200","Text":"We don\u0027t substitute Lambda at this stage,"},{"Start":"08:38.200 ","End":"08:43.970","Text":"so we get a naught and then we have minus a naught over Lambda plus 2^2."},{"Start":"08:43.970 ","End":"08:50.000","Text":"Then from here we have plus this from here and so on and so on."},{"Start":"08:50.000 ","End":"08:53.630","Text":"The general term is what we have here,"},{"Start":"08:53.630 ","End":"08:56.975","Text":"and it continues plus and so on."},{"Start":"08:56.975 ","End":"09:01.261","Text":"I can take a naught outside the brackets,"},{"Start":"09:01.261 ","End":"09:02.810","Text":"but in fact we don\u0027t need a naught."},{"Start":"09:02.810 ","End":"09:04.730","Text":"It could be any non-zero constant."},{"Start":"09:04.730 ","End":"09:06.530","Text":"We\u0027re just looking for a particular y,"},{"Start":"09:06.530 ","End":"09:09.013","Text":"so let it be 1 and be done with it."},{"Start":"09:09.013 ","End":"09:10.985","Text":"Next is the key step."},{"Start":"09:10.985 ","End":"09:16.680","Text":"This is how we find the other y in the case of two equal Lambdas."},{"Start":"09:16.680 ","End":"09:19.395","Text":"We let y_2 equal this, and I\u0027ll explain."},{"Start":"09:19.395 ","End":"09:21.860","Text":"We have y as a function of Lambda and x,"},{"Start":"09:21.860 ","End":"09:26.060","Text":"so it has partial derivatives with respect to x and with respect to Lambda."},{"Start":"09:26.060 ","End":"09:32.525","Text":"We differentiate this with respect to Lambda and then substitute Lambda equals 0."},{"Start":"09:32.525 ","End":"09:35.880","Text":"That\u0027s what gives us our other y."},{"Start":"09:35.880 ","End":"09:38.690","Text":"We need to do the derivative with respect to Lambda."},{"Start":"09:38.690 ","End":"09:40.220","Text":"This we treat as a product."},{"Start":"09:40.220 ","End":"09:42.290","Text":"The x^Lambda is like f,"},{"Start":"09:42.290 ","End":"09:47.765","Text":"and all the rest of it is like g and of course I\u0027m using the formula that fg\u0027."},{"Start":"09:47.765 ","End":"09:49.550","Text":"Each time we differentiate one,"},{"Start":"09:49.550 ","End":"09:50.840","Text":"leave the other untouched,"},{"Start":"09:50.840 ","End":"09:54.735","Text":"product, then the other way round, fg\u0027,"},{"Start":"09:54.735 ","End":"09:56.775","Text":"and If we do that here,"},{"Start":"09:56.775 ","End":"10:02.225","Text":"the first part is the derivative of x with respect to Lambda,"},{"Start":"10:02.225 ","End":"10:05.090","Text":"which is x^Lambda natural log of x."},{"Start":"10:05.090 ","End":"10:08.540","Text":"Then we take the series as is."},{"Start":"10:08.540 ","End":"10:11.765","Text":"I just wrote this part and then dot dot dot."},{"Start":"10:11.765 ","End":"10:15.620","Text":"Then the other way round x^Lambda as is."},{"Start":"10:15.620 ","End":"10:17.345","Text":"Then the derivative of this,"},{"Start":"10:17.345 ","End":"10:23.405","Text":"which comes out to be quite messy once we get beyond the third term,"},{"Start":"10:23.405 ","End":"10:25.775","Text":"1 is easier, it becomes 0."},{"Start":"10:25.775 ","End":"10:27.830","Text":"This is just like an exponent,"},{"Start":"10:27.830 ","End":"10:31.240","Text":"it\u0027s like Lambda to the minus 2 or a variation of it,"},{"Start":"10:31.240 ","End":"10:35.150","Text":"and we get minus 2 Lambda to the minus 3 and it isn\u0027t Lambda, it\u0027s Lambda plus 2."},{"Start":"10:35.150 ","End":"10:37.470","Text":"Anyway, we get this with a plus."},{"Start":"10:37.470 ","End":"10:39.800","Text":"This one you have to work a bit,"},{"Start":"10:39.800 ","End":"10:43.565","Text":"either do it as a quotient or as something to the minus 1."},{"Start":"10:43.565 ","End":"10:46.310","Text":"Anyway, after simplification we get this."},{"Start":"10:46.310 ","End":"10:48.550","Text":"I won\u0027t do any more terms."},{"Start":"10:48.550 ","End":"10:51.140","Text":"Now that we\u0027ve differentiated with respect to Lambda,"},{"Start":"10:51.140 ","End":"10:54.440","Text":"we still have to substitute Lambda equals 0 from here."},{"Start":"10:54.440 ","End":"10:57.950","Text":"If we substitute Lambda equals 0, we get this."},{"Start":"10:57.950 ","End":"11:01.130","Text":"X^0 is 1,"},{"Start":"11:01.130 ","End":"11:03.260","Text":"both here and here."},{"Start":"11:03.260 ","End":"11:05.600","Text":"What we\u0027re left with is natural log of x."},{"Start":"11:05.600 ","End":"11:11.225","Text":"Here if Lambda is 0, it\u0027s 1 minus 1 over 2^2 plus 1 over 4^2,"},{"Start":"11:11.225 ","End":"11:13.820","Text":"2^2 with x^4 even powers."},{"Start":"11:13.820 ","End":"11:17.905","Text":"This I just substituted Lambda equals naught or 2 over 2^3."},{"Start":"11:17.905 ","End":"11:20.168","Text":"It could be simplified, but why bother?"},{"Start":"11:20.168 ","End":"11:24.875","Text":"Minus 4 times 3 and then 4^3, 2^3."},{"Start":"11:24.875 ","End":"11:27.935","Text":"Just notice that this bit here,"},{"Start":"11:27.935 ","End":"11:31.765","Text":"if we go back and look, is just y_1(x)."},{"Start":"11:31.765 ","End":"11:34.490","Text":"It might be written with Sigma notation,"},{"Start":"11:34.490 ","End":"11:36.200","Text":"but essentially this is what we have,"},{"Start":"11:36.200 ","End":"11:38.150","Text":"so that can simplify it a bit."},{"Start":"11:38.150 ","End":"11:41.930","Text":"I just want to put it in an nice box like I did with y_1 and be consistent."},{"Start":"11:41.930 ","End":"11:43.670","Text":"We\u0027ve got y_1 and y_2."},{"Start":"11:43.670 ","End":"11:45.410","Text":"Then finally, I just mentioned"},{"Start":"11:45.410 ","End":"11:49.820","Text":"that the general solution to the linear combination of y_1 and y_2,"},{"Start":"11:49.820 ","End":"11:52.310","Text":"I\u0027m not going to actually substitute them in here."},{"Start":"11:52.310 ","End":"11:53.870","Text":"This is quite enough."},{"Start":"11:53.870 ","End":"11:57.630","Text":"So difficult exercise, but we\u0027re done."}],"Thumbnail":null,"ID":7890},{"Watched":false,"Name":"Exercise 6","Duration":"7m 39s","ChapterTopicVideoID":7839,"CourseChapterTopicPlaylistID":4244,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"In this exercise, we have this differential equation to solve."},{"Start":"00:03.450 ","End":"00:05.370","Text":"There\u0027s more than one way of solving it."},{"Start":"00:05.370 ","End":"00:08.805","Text":"For example, you could solve it as an Euler equation,"},{"Start":"00:08.805 ","End":"00:18.660","Text":"which is ax^2 y\u0027\u0027 plus bxy\u0027 plus cy=0."},{"Start":"00:18.660 ","End":"00:20.310","Text":"Do it like that. In fact,"},{"Start":"00:20.310 ","End":"00:25.650","Text":"I\u0027d recommend it as an exercise to also solve it afterwards as an Euler\u0027s equation,"},{"Start":"00:25.650 ","End":"00:27.225","Text":"and then check what we get."},{"Start":"00:27.225 ","End":"00:30.839","Text":"But, we\u0027re on the section on power series solutions,"},{"Start":"00:30.839 ","End":"00:38.210","Text":"and we\u0027re going to develop the solution y as a power series in x around x=0,"},{"Start":"00:38.210 ","End":"00:41.070","Text":"and we first have to show that this is a singular point."},{"Start":"00:41.070 ","End":"00:44.810","Text":"We bring this equation to the standard form by dividing by this,"},{"Start":"00:44.810 ","End":"00:49.295","Text":"and then we get the coefficient of y\u0027 p(x) and here q(x)."},{"Start":"00:49.295 ","End":"00:56.530","Text":"Note that x is naught is not a regular point because neither of these is defined as x=0."},{"Start":"00:56.530 ","End":"00:57.650","Text":"Even if one of them wasn\u0027t,"},{"Start":"00:57.650 ","End":"00:59.170","Text":"then it\u0027s already not regular."},{"Start":"00:59.170 ","End":"01:01.535","Text":"But, it is what we call regular singular."},{"Start":"01:01.535 ","End":"01:03.140","Text":"If you multiply this by x,"},{"Start":"01:03.140 ","End":"01:04.550","Text":"and this by x^2,"},{"Start":"01:04.550 ","End":"01:06.290","Text":"like here and here,"},{"Start":"01:06.290 ","End":"01:08.300","Text":"then what we get is minus 1 and 1."},{"Start":"01:08.300 ","End":"01:09.470","Text":"These are constant functions,"},{"Start":"01:09.470 ","End":"01:10.910","Text":"and of course they are defined,"},{"Start":"01:10.910 ","End":"01:13.655","Text":"and well-behaved at x=0."},{"Start":"01:13.655 ","End":"01:15.730","Text":"We\u0027re going to try a power series,"},{"Start":"01:15.730 ","End":"01:19.595","Text":"but a power series starting at x to the power of Lambda, and we don\u0027t know Lambda."},{"Start":"01:19.595 ","End":"01:25.310","Text":"This is easier seen when we write it in Sigma notation."},{"Start":"01:25.310 ","End":"01:29.875","Text":"We have x to the Lambda times a regular power series."},{"Start":"01:29.875 ","End":"01:32.760","Text":"That\u0027s this. Of course we\u0027ll need the first,"},{"Start":"01:32.760 ","End":"01:35.180","Text":"and second derivatives because after all,"},{"Start":"01:35.180 ","End":"01:37.960","Text":"we\u0027re going to substitute in the differential equation."},{"Start":"01:37.960 ","End":"01:41.404","Text":"Here they are, first derivative and second derivative."},{"Start":"01:41.404 ","End":"01:44.810","Text":"We also need the original equation. Here it is."},{"Start":"01:44.810 ","End":"01:47.495","Text":"We\u0027re going to be substituting these into here,"},{"Start":"01:47.495 ","End":"01:51.020","Text":"but not everything, a lot of these terms are unnecessary."},{"Start":"01:51.020 ","End":"01:52.705","Text":"The part of the dot."},{"Start":"01:52.705 ","End":"01:56.720","Text":"We want to collect x to the Lambda and we want to collect x to"},{"Start":"01:56.720 ","End":"02:00.910","Text":"the Lambda plus n. Let\u0027s check where we might get these."},{"Start":"02:00.910 ","End":"02:04.580","Text":"I\u0027ll take the terms one at a time, the x^2y\u0027\u0027."},{"Start":"02:04.580 ","End":"02:07.580","Text":"All the powers here are raised by 2."},{"Start":"02:07.580 ","End":"02:12.420","Text":"I have to look for these exponents minus 2. Lambda minus 2 I have here,"},{"Start":"02:12.420 ","End":"02:13.995","Text":"so I\u0027ll need this,"},{"Start":"02:13.995 ","End":"02:17.340","Text":"and x to the Lambda plus n minus 2,"},{"Start":"02:17.340 ","End":"02:18.615","Text":"I have here,"},{"Start":"02:18.615 ","End":"02:21.330","Text":"so I\u0027m going to need this one."},{"Start":"02:21.330 ","End":"02:24.180","Text":"I\u0027ll write the x^2y\u0027\u0027 as x^2,"},{"Start":"02:24.180 ","End":"02:26.833","Text":"and then I just collect the terms that are useful to me,"},{"Start":"02:26.833 ","End":"02:31.310","Text":"and everything else is swallowed up in the ellipsis, the dot."},{"Start":"02:31.310 ","End":"02:33.060","Text":"That\u0027s this one done."},{"Start":"02:33.060 ","End":"02:34.835","Text":"Let\u0027s get on to the next one."},{"Start":"02:34.835 ","End":"02:39.880","Text":"Here we\u0027re multiplying y\u0027 by x and it raises all the powers by 1."},{"Start":"02:39.880 ","End":"02:43.940","Text":"We can get x to the Lambda by looking for x to the lambda minus 1."},{"Start":"02:43.940 ","End":"02:45.395","Text":"This is what we need."},{"Start":"02:45.395 ","End":"02:50.025","Text":"We also want to look for x to the lambda plus n minus 1. That\u0027s here."},{"Start":"02:50.025 ","End":"02:51.990","Text":"This is what we need."},{"Start":"02:51.990 ","End":"02:54.630","Text":"That gives us this."},{"Start":"02:54.630 ","End":"02:57.315","Text":"I\u0027ll highlight this over here."},{"Start":"02:57.315 ","End":"02:59.860","Text":"You can see I took the right thing and this,"},{"Start":"02:59.860 ","End":"03:02.150","Text":"I\u0027ll highlight over here."},{"Start":"03:02.150 ","End":"03:07.055","Text":"Now we have the term with y to do."},{"Start":"03:07.055 ","End":"03:09.620","Text":"These exponents are just as is."},{"Start":"03:09.620 ","End":"03:12.135","Text":"I\u0027ve lost y. Here it is."},{"Start":"03:12.135 ","End":"03:15.257","Text":"Well, we need this for x to the lambda,"},{"Start":"03:15.257 ","End":"03:17.480","Text":"and we need this for x to the Lambda plus"},{"Start":"03:17.480 ","End":"03:22.240","Text":"n. We\u0027ll just write that over here and highlight that."},{"Start":"03:22.240 ","End":"03:23.970","Text":"It was this and this."},{"Start":"03:23.970 ","End":"03:25.490","Text":"Now that we\u0027ve got these,"},{"Start":"03:25.490 ","End":"03:28.685","Text":"we want to compare the coefficients to 0."},{"Start":"03:28.685 ","End":"03:34.670","Text":"We have x to the Lambda and x to the Lambda plus n. For this color,"},{"Start":"03:34.670 ","End":"03:40.560","Text":"we get this equation and we get this plus this plus this."},{"Start":"03:40.560 ","End":"03:42.270","Text":"It\u0027s Lambda Lambda minus 1,"},{"Start":"03:42.270 ","End":"03:44.310","Text":"minus Lambda plus 1."},{"Start":"03:44.310 ","End":"03:45.405","Text":"I should have said,"},{"Start":"03:45.405 ","End":"03:49.455","Text":"we don\u0027t need the a_0 because they will cancel."},{"Start":"03:49.455 ","End":"03:51.920","Text":"We\u0027re used to that, so we just ignore those,"},{"Start":"03:51.920 ","End":"03:55.685","Text":"could let a_0 equal 1. We get this equation."},{"Start":"03:55.685 ","End":"04:00.470","Text":"If you simplify it, it comes out to Lambda squared minus 2 Lambda plus 1,"},{"Start":"04:00.470 ","End":"04:03.245","Text":"which is Lambda minus 1 squared."},{"Start":"04:03.245 ","End":"04:05.000","Text":"We get two of the same root."},{"Start":"04:05.000 ","End":"04:07.550","Text":"We get Lambda equals 1 and Lambda equals 1."},{"Start":"04:07.550 ","End":"04:09.020","Text":"Leave that aside a moment."},{"Start":"04:09.020 ","End":"04:13.040","Text":"The other thing we want to do is to compare the coefficients of x to the lambda plus"},{"Start":"04:13.040 ","End":"04:17.765","Text":"n. What we get is this and this and this,"},{"Start":"04:17.765 ","End":"04:20.090","Text":"which is here, is 0."},{"Start":"04:20.090 ","End":"04:24.529","Text":"Now, they all contain a to the n. We get this."},{"Start":"04:24.529 ","End":"04:27.590","Text":"Now, if we substitute lambda equals 1,"},{"Start":"04:27.590 ","End":"04:29.884","Text":"which is the double root we found,"},{"Start":"04:29.884 ","End":"04:31.745","Text":"then this is what we get."},{"Start":"04:31.745 ","End":"04:33.170","Text":"Perhaps I should explain this."},{"Start":"04:33.170 ","End":"04:34.850","Text":"If we put lambda equals 1,"},{"Start":"04:34.850 ","End":"04:44.121","Text":"I get 1 plus n times n. 1 plus n minus 1 is just n. Here I get minus 1 minus n,"},{"Start":"04:44.121 ","End":"04:45.975","Text":"and here plus 1."},{"Start":"04:45.975 ","End":"04:48.675","Text":"If you look at it, n^2 plus n,"},{"Start":"04:48.675 ","End":"04:50.790","Text":"and then the minus n cancels with"},{"Start":"04:50.790 ","End":"04:53.510","Text":"the plus n and the 1 and the minus 1 cancel each other out."},{"Start":"04:53.510 ","End":"04:55.450","Text":"We just get n^2."},{"Start":"04:55.450 ","End":"04:57.840","Text":"If a_n times n^2 is 0,"},{"Start":"04:57.840 ","End":"05:02.295","Text":"then for n which is not 0 from 1 onwards,"},{"Start":"05:02.295 ","End":"05:05.045","Text":"we have that all the coefficients a_n are 0."},{"Start":"05:05.045 ","End":"05:07.370","Text":"Except as I said for the first,"},{"Start":"05:07.370 ","End":"05:08.650","Text":"which is a_0,"},{"Start":"05:08.650 ","End":"05:11.360","Text":"but a_1, a_2, a_3 are all 0."},{"Start":"05:11.360 ","End":"05:15.260","Text":"Now, y_1 is x to the Lambda."},{"Start":"05:15.260 ","End":"05:19.525","Text":"I got this from x to the Lambda times the power series,"},{"Start":"05:19.525 ","End":"05:23.510","Text":"a_0 plus a_1x plus and so on."},{"Start":"05:23.510 ","End":"05:26.030","Text":"But, all are 0 except for the first one,"},{"Start":"05:26.030 ","End":"05:27.350","Text":"so we end up with just a_0,"},{"Start":"05:27.350 ","End":"05:34.330","Text":"and x to the 1 is x. I don\u0027t really need a_0 because I just need any non-zero y_1."},{"Start":"05:34.330 ","End":"05:37.335","Text":"I could take y_1(x) as just x."},{"Start":"05:37.335 ","End":"05:39.920","Text":"Then eventually we\u0027re going to multiply it by a constant when we take"},{"Start":"05:39.920 ","End":"05:42.965","Text":"a linear combination of y_1 and y_2 at the end."},{"Start":"05:42.965 ","End":"05:45.620","Text":"That\u0027s y_1, how do we find y_2?"},{"Start":"05:45.620 ","End":"05:50.985","Text":"Just want to remind you that a_n is naught for all but the 0th term,"},{"Start":"05:50.985 ","End":"05:53.200","Text":"because lambda is still 1."},{"Start":"05:53.200 ","End":"05:56.660","Text":"But, what we do for the other root is instead of substituting"},{"Start":"05:56.660 ","End":"06:00.095","Text":"Lambda equals 1 here as we did here to get y_1,"},{"Start":"06:00.095 ","End":"06:02.315","Text":"we don\u0027t substitute Lambda,"},{"Start":"06:02.315 ","End":"06:04.925","Text":"we keep lambda as it is."},{"Start":"06:04.925 ","End":"06:07.354","Text":"Of course these are still 0."},{"Start":"06:07.354 ","End":"06:09.650","Text":"If we don\u0027t substitute lambda,"},{"Start":"06:09.650 ","End":"06:13.805","Text":"we get y as a function of lambda and x."},{"Start":"06:13.805 ","End":"06:16.309","Text":"As I said, these are all 0,"},{"Start":"06:16.309 ","End":"06:18.560","Text":"so we just get a_0 x to the Lambda,"},{"Start":"06:18.560 ","End":"06:20.605","Text":"but this time we keep the Lambda."},{"Start":"06:20.605 ","End":"06:26.870","Text":"The rule for this case is that we differentiate this function with respect to lambda,"},{"Start":"06:26.870 ","End":"06:29.045","Text":"and then substitute the value,"},{"Start":"06:29.045 ","End":"06:33.010","Text":"that common value of the double root in this case is 1."},{"Start":"06:33.010 ","End":"06:34.695","Text":"Let\u0027s clear some space."},{"Start":"06:34.695 ","End":"06:38.810","Text":"What I\u0027m doing next is I\u0027m going to differentiate this with respect to Lambda."},{"Start":"06:38.810 ","End":"06:41.900","Text":"The a_0 is a constant and the derivative of x to"},{"Start":"06:41.900 ","End":"06:45.725","Text":"the Lambda is x to the Lambda natural log of x."},{"Start":"06:45.725 ","End":"06:52.565","Text":"There\u0027s a rule that the derivative of a^x is a^x natural log of x only here,"},{"Start":"06:52.565 ","End":"06:55.970","Text":"x is lambda and a is x."},{"Start":"06:55.970 ","End":"06:59.950","Text":"We get this and if we plug lambda equals 1,"},{"Start":"06:59.950 ","End":"07:03.375","Text":"x to the lambda is just 1, so it\u0027s just x."},{"Start":"07:03.375 ","End":"07:07.200","Text":"We get x natural log of x and we don\u0027t need the a_0."},{"Start":"07:07.200 ","End":"07:08.410","Text":"We could let it be 1,"},{"Start":"07:08.410 ","End":"07:10.705","Text":"could be any non-zero value."},{"Start":"07:10.705 ","End":"07:12.210","Text":"We have y_2,"},{"Start":"07:12.210 ","End":"07:17.795","Text":"the second solution or a second solution such that these two are linearly independent."},{"Start":"07:17.795 ","End":"07:21.080","Text":"Finally, the general solution comes from a linear combination of"},{"Start":"07:21.080 ","End":"07:29.650","Text":"these two and specifically c_1 times x and c_2 times xlnx."},{"Start":"07:29.650 ","End":"07:31.160","Text":"I mentioned at the beginning,"},{"Start":"07:31.160 ","End":"07:33.230","Text":"you could have solved it as an Euler\u0027s equation."},{"Start":"07:33.230 ","End":"07:35.765","Text":"Anyway, I\u0027ll leave that as an exercise to solve it using"},{"Start":"07:35.765 ","End":"07:40.260","Text":"Euler\u0027s equation and see that you get the same thing. We\u0027re done."}],"Thumbnail":null,"ID":7891},{"Watched":false,"Name":"Exercise 7","Duration":"16m 6s","ChapterTopicVideoID":7840,"CourseChapterTopicPlaylistID":4244,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.220","Text":"Here we have this differential equation to solve"},{"Start":"00:02.220 ","End":"00:06.120","Text":"it\u0027s second-order linear, non-homogeneous,"},{"Start":"00:06.120 ","End":"00:09.630","Text":"non-constant coefficients, and we\u0027ll solve it by"},{"Start":"00:09.630 ","End":"00:14.430","Text":"developing y as the power series around x=0."},{"Start":"00:14.430 ","End":"00:19.200","Text":"But first, we need to show that this is a regular singular point."},{"Start":"00:19.200 ","End":"00:23.468","Text":"We rewrite the equation in this standard form divided by x^2,"},{"Start":"00:23.468 ","End":"00:25.170","Text":"and this we call p,"},{"Start":"00:25.170 ","End":"00:26.520","Text":"and this we call q."},{"Start":"00:26.520 ","End":"00:29.680","Text":"Notice that neither of these is defined at x equals naught,"},{"Start":"00:29.680 ","End":"00:33.315","Text":"and if even one of them is not defined and this is not a regular point."},{"Start":"00:33.315 ","End":"00:36.200","Text":"Our hope is that it\u0027s irregular singular point,"},{"Start":"00:36.200 ","End":"00:40.035","Text":"and we do that by checking xp of x,"},{"Start":"00:40.035 ","End":"00:41.550","Text":"and x^2 q of x,"},{"Start":"00:41.550 ","End":"00:44.030","Text":"like so, and the first one comes out x plus 2,"},{"Start":"00:44.030 ","End":"00:46.759","Text":"the second one comes up minus 2 both these functions"},{"Start":"00:46.759 ","End":"00:50.140","Text":"are defined at x=0 and very well-behaved."},{"Start":"00:50.140 ","End":"00:52.760","Text":"We\u0027re all right for regular singular point."},{"Start":"00:52.760 ","End":"00:56.975","Text":"Then we look for a solution that\u0027s a power series of this form."},{"Start":"00:56.975 ","End":"01:01.850","Text":"But, I think it\u0027s clearer if we just write it with Sigma notation,"},{"Start":"01:01.850 ","End":"01:03.080","Text":"it\u0027s x to some Lambda,"},{"Start":"01:03.080 ","End":"01:04.140","Text":"that\u0027s the Lambda here,"},{"Start":"01:04.140 ","End":"01:09.410","Text":"and then the sum of irregular power series a_nx^n."},{"Start":"01:09.410 ","End":"01:10.883","Text":"If we had irregular points,"},{"Start":"01:10.883 ","End":"01:12.845","Text":"you wouldn\u0027t bother with the x to the Lambda,"},{"Start":"01:12.845 ","End":"01:15.530","Text":"but it is a regular singular point."},{"Start":"01:15.530 ","End":"01:17.960","Text":"Of course, we\u0027re going to need the first,"},{"Start":"01:17.960 ","End":"01:23.515","Text":"and second order derivatives of y so we can substitute in the original OD,"},{"Start":"01:23.515 ","End":"01:25.880","Text":"and here they both are those y-prime,"},{"Start":"01:25.880 ","End":"01:27.290","Text":"there\u0027s y double-prime,"},{"Start":"01:27.290 ","End":"01:30.200","Text":"and the original OD is here."},{"Start":"01:30.200 ","End":"01:32.419","Text":"I just copied it. After we substitute,"},{"Start":"01:32.419 ","End":"01:37.265","Text":"we\u0027re going to compare coefficients on both sides of x to the Lambda,"},{"Start":"01:37.265 ","End":"01:40.100","Text":"but also of x to the Lambda plus"},{"Start":"01:40.100 ","End":"01:43.610","Text":"n on the left will get some expression on the right, we\u0027ll get 0."},{"Start":"01:43.610 ","End":"01:47.840","Text":"I don\u0027t need all these extra terms that should be swallowed up in the ellipses,"},{"Start":"01:47.840 ","End":"01:50.780","Text":"the dot, let\u0027s just select the ones we"},{"Start":"01:50.780 ","End":"01:53.885","Text":"really need and they\u0027re color-coded them to distinguish."},{"Start":"01:53.885 ","End":"01:55.640","Text":"Let\u0027s take the first term,"},{"Start":"01:55.640 ","End":"01:59.390","Text":"the x^2 y\u0027\u0027 and look for these here."},{"Start":"01:59.390 ","End":"02:01.940","Text":"Now the x^2, when I multiply,"},{"Start":"02:01.940 ","End":"02:04.460","Text":"it, will raise all the powers by 2."},{"Start":"02:04.460 ","End":"02:06.590","Text":"To get these, I have to subtract 2."},{"Start":"02:06.590 ","End":"02:08.945","Text":"In other words, I need this one"},{"Start":"02:08.945 ","End":"02:12.030","Text":"because it\u0027s Lambda minus 2 will give me x to the Lambda,"},{"Start":"02:12.030 ","End":"02:15.605","Text":"and also I need plus n minus 2, which is here."},{"Start":"02:15.605 ","End":"02:20.780","Text":"I will need this co-efficient and the appropriate colors."},{"Start":"02:20.780 ","End":"02:22.625","Text":"Now let\u0027s go to this one."},{"Start":"02:22.625 ","End":"02:30.470","Text":"Going to have to break up the middle term and write it as x^2 y\u0027 plus 2y\u0027"},{"Start":"02:30.470 ","End":"02:39.365","Text":"These are both different because x^2 y\u0027 will erase the exponents by 2x y\u0027."},{"Start":"02:39.365 ","End":"02:41.690","Text":"That will raise the exponent by 1."},{"Start":"02:41.690 ","End":"02:44.150","Text":"To get x to the Lambda,"},{"Start":"02:44.150 ","End":"02:48.530","Text":"we have to start either by Lambda minus 2,"},{"Start":"02:48.530 ","End":"02:52.050","Text":"or Lambda minus 1 because is x^2 and this x,"},{"Start":"02:52.050 ","End":"02:53.290","Text":"for the x^2,"},{"Start":"02:53.290 ","End":"02:57.095","Text":"I don\u0027t find anything because there is no Lambda minus 2 in this list."},{"Start":"02:57.095 ","End":"02:59.750","Text":"But for 2xy\u0027,"},{"Start":"02:59.750 ","End":"03:05.650","Text":"I can use the coefficient here because this is going to get raised by 1."},{"Start":"03:05.650 ","End":"03:08.675","Text":"For this, I\u0027ve got two contributing terms."},{"Start":"03:08.675 ","End":"03:12.200","Text":"I\u0027ve got x to the Lambda plus n minus 1."},{"Start":"03:12.200 ","End":"03:13.805","Text":"That\u0027s over here."},{"Start":"03:13.805 ","End":"03:17.320","Text":"We\u0027ll be able to use for x to the Lambda plus n,"},{"Start":"03:17.320 ","End":"03:19.620","Text":"we\u0027re multiplying once by x^2,"},{"Start":"03:19.620 ","End":"03:22.035","Text":"and once by x we get the constant."},{"Start":"03:22.035 ","End":"03:27.575","Text":"We can make use of Lambda plus n minus 1 and also minus 2,"},{"Start":"03:27.575 ","End":"03:29.090","Text":"and if we look here,"},{"Start":"03:29.090 ","End":"03:31.985","Text":"we see the minus 1 here."},{"Start":"03:31.985 ","End":"03:34.490","Text":"But we also can use the minus 2,"},{"Start":"03:34.490 ","End":"03:36.025","Text":"which is this,"},{"Start":"03:36.025 ","End":"03:39.725","Text":"and then for the y part,"},{"Start":"03:39.725 ","End":"03:41.495","Text":"it\u0027s just multiplied by a constant."},{"Start":"03:41.495 ","End":"03:43.805","Text":"Things are as they are."},{"Start":"03:43.805 ","End":"03:45.560","Text":"We have this one,"},{"Start":"03:45.560 ","End":"03:49.505","Text":"Lambda plus n from here and x to the Lambda."},{"Start":"03:49.505 ","End":"03:55.385","Text":"This is the coefficient and now we want to put these together for the first term,"},{"Start":"03:55.385 ","End":"03:56.840","Text":"I got this and this,"},{"Start":"03:56.840 ","End":"03:59.850","Text":"and that is here and here."},{"Start":"03:59.850 ","End":"04:03.620","Text":"Next, for the 2xy\u0027,"},{"Start":"04:03.620 ","End":"04:05.600","Text":"which is the second one of these two,"},{"Start":"04:05.600 ","End":"04:07.940","Text":"we have this because looking at"},{"Start":"04:07.940 ","End":"04:11.570","Text":"the minus 1 and we have this Gamma looking at the minus 1."},{"Start":"04:11.570 ","End":"04:12.860","Text":"Let\u0027s take these two,"},{"Start":"04:12.860 ","End":"04:16.915","Text":"that gives me this and this,"},{"Start":"04:16.915 ","End":"04:21.555","Text":"and next, we\u0027ll take the x^2 y\u0027."},{"Start":"04:21.555 ","End":"04:26.285","Text":"We need the minus 2s in the y\u0027."},{"Start":"04:26.285 ","End":"04:29.105","Text":"Now here we have a minus 2, but here we didn\u0027t,"},{"Start":"04:29.105 ","End":"04:33.350","Text":"because Lambda minus 2 is before the beginning of the list, it\u0027s not there."},{"Start":"04:33.350 ","End":"04:38.900","Text":"We have this, but we do not have this, this doesn\u0027t contribute,"},{"Start":"04:38.900 ","End":"04:40.130","Text":"but it\u0027s the beginning of the list,"},{"Start":"04:40.130 ","End":"04:41.390","Text":"so I put it in,"},{"Start":"04:41.390 ","End":"04:44.135","Text":"and now the last of those four terms,"},{"Start":"04:44.135 ","End":"04:46.190","Text":"y is scrolled off the screen."},{"Start":"04:46.190 ","End":"04:49.798","Text":"But, I remember that we had just a note,"},{"Start":"04:49.798 ","End":"04:51.420","Text":"and a_n here,"},{"Start":"04:51.420 ","End":"04:54.440","Text":"and those are just like to highlight the coefficients while here it\u0027s 1,"},{"Start":"04:54.440 ","End":"04:57.050","Text":"I won\u0027t bother highlighting it here I have a 2."},{"Start":"04:57.050 ","End":"04:59.345","Text":"Here I have a minus 2."},{"Start":"04:59.345 ","End":"05:02.090","Text":"When we do the arithmetic, that could help."},{"Start":"05:02.090 ","End":"05:03.770","Text":"Let\u0027s see now."},{"Start":"05:03.770 ","End":"05:05.270","Text":"We need both kinds."},{"Start":"05:05.270 ","End":"05:07.940","Text":"Let\u0027s go with the x to the Lambda first."},{"Start":"05:07.940 ","End":"05:11.090","Text":"For the x to the Lambda, we need this,"},{"Start":"05:11.090 ","End":"05:14.135","Text":"this and this, but multiplied appropriately,"},{"Start":"05:14.135 ","End":"05:17.345","Text":"this one by 2, and this one by minus 2,"},{"Start":"05:17.345 ","End":"05:18.775","Text":"and add them up,"},{"Start":"05:18.775 ","End":"05:20.840","Text":"and we get this equation."},{"Start":"05:20.840 ","End":"05:22.130","Text":"It\u0027s a quadratic."},{"Start":"05:22.130 ","End":"05:26.015","Text":"I\u0027m not going to go into the technical details of expanding it and finding its roots."},{"Start":"05:26.015 ","End":"05:27.320","Text":"Just tell you the answer."},{"Start":"05:27.320 ","End":"05:29.650","Text":"We\u0027ve got 1 and minus 2."},{"Start":"05:29.650 ","End":"05:33.795","Text":"The larger one is Lambda 1 on the smaller one is Lambda 2."},{"Start":"05:33.795 ","End":"05:36.110","Text":"Smaller I mean, in the absolute sense."},{"Start":"05:36.110 ","End":"05:38.477","Text":"Well, I mean, minus 2 is less than 1."},{"Start":"05:38.477 ","End":"05:40.700","Text":"I\u0027ll return to that in a moment."},{"Start":"05:40.700 ","End":"05:42.560","Text":"Meanwhile, let\u0027s just do the other one,"},{"Start":"05:42.560 ","End":"05:45.170","Text":"the x to the Lambda plus n and compare."},{"Start":"05:45.170 ","End":"05:47.300","Text":"Here we have four bits,"},{"Start":"05:47.300 ","End":"05:49.130","Text":"this times 1, this time 2,"},{"Start":"05:49.130 ","End":"05:50.240","Text":"this times 1,"},{"Start":"05:50.240 ","End":"05:54.050","Text":"this times minus 2 and I\u0027ll leave you to check that this is what we get."},{"Start":"05:54.050 ","End":"05:55.220","Text":"Now if we separate,"},{"Start":"05:55.220 ","End":"05:59.340","Text":"we see we have three terms with a_n and we have this one with a_n minus 1,"},{"Start":"05:59.340 ","End":"06:03.545","Text":"and then we\u0027re collecting, I\u0027m putting these on the left a_n minus 1 on the right,"},{"Start":"06:03.545 ","End":"06:08.630","Text":"and then we just want to extract a_n so we divide by"},{"Start":"06:08.630 ","End":"06:13.730","Text":"its coefficient and we get this recursion equation, which is important."},{"Start":"06:13.730 ","End":"06:14.930","Text":"Here it is again,"},{"Start":"06:14.930 ","End":"06:18.350","Text":"and I wanted to just remind you that one of the Lambdas was 1,"},{"Start":"06:18.350 ","End":"06:22.720","Text":"the larger one on the smaller one was minus 2."},{"Start":"06:22.720 ","End":"06:25.100","Text":"To find y1, we use the larger,"},{"Start":"06:25.100 ","End":"06:26.375","Text":"we use Lambda 1,"},{"Start":"06:26.375 ","End":"06:32.180","Text":"and if we let Lambda equals 1 in this recursive formula for the coefficients,"},{"Start":"06:32.180 ","End":"06:36.530","Text":"then we get this a_n in terms of a n minus 1."},{"Start":"06:36.530 ","End":"06:39.620","Text":"But, this only holds for n bigger than or equal to 1."},{"Start":"06:39.620 ","End":"06:43.954","Text":"I can substitute 0 because I\u0027ll get a minus 1 and then not defined."},{"Start":"06:43.954 ","End":"06:46.640","Text":"Now, but as simplifying well, the denominator here,"},{"Start":"06:46.640 ","End":"06:50.330","Text":"n^2 plus 3n plus 2 minus 2 comes out right square root of 3,"},{"Start":"06:50.330 ","End":"06:53.990","Text":"and now I can divide top and bottom by n, which is okay."},{"Start":"06:53.990 ","End":"06:56.314","Text":"It\u0027s bigger equal to 1, so it\u0027s not 0."},{"Start":"06:56.314 ","End":"06:59.690","Text":"Let\u0027s talk computing a few terms dot with a_1."},{"Start":"06:59.690 ","End":"07:01.790","Text":"We know that a_0, we can\u0027t find,"},{"Start":"07:01.790 ","End":"07:04.505","Text":"we treat it as if it was given,"},{"Start":"07:04.505 ","End":"07:06.050","Text":"but everything else in terms of it,"},{"Start":"07:06.050 ","End":"07:07.445","Text":"here we get a_1."},{"Start":"07:07.445 ","End":"07:11.090","Text":"If we put an equals 1 is minus 1/4 a naught."},{"Start":"07:11.090 ","End":"07:14.060","Text":"Next we have a_2 in terms of a_1,"},{"Start":"07:14.060 ","End":"07:17.945","Text":"which we then convert to express in terms of a_0."},{"Start":"07:17.945 ","End":"07:20.570","Text":"A_3 is given in terms of a_2,"},{"Start":"07:20.570 ","End":"07:22.145","Text":"which I substitute from here,"},{"Start":"07:22.145 ","End":"07:24.575","Text":"and the reason I\u0027m continuing,"},{"Start":"07:24.575 ","End":"07:28.430","Text":"usually don\u0027t do a lot is that there\u0027s a pattern emerging."},{"Start":"07:28.430 ","End":"07:31.985","Text":"I think you can see that there\u0027s definitely a pattern here."},{"Start":"07:31.985 ","End":"07:34.185","Text":"Let\u0027s try and write this rule."},{"Start":"07:34.185 ","End":"07:36.030","Text":"Look for example, at a_3."},{"Start":"07:36.030 ","End":"07:37.670","Text":"There\u0027s a 3 here and there\u0027s a minus,"},{"Start":"07:37.670 ","End":"07:40.085","Text":"minus, minus the 3 minuses."},{"Start":"07:40.085 ","End":"07:42.725","Text":"That would be minus 1^3, and in general,"},{"Start":"07:42.725 ","End":"07:45.575","Text":"can see it\u0027s minus 1^n, it alternates signs."},{"Start":"07:45.575 ","End":"07:48.350","Text":"That\u0027s this bit. Now the denominators,"},{"Start":"07:48.350 ","End":"07:51.020","Text":"we start from n plus 3, because look,"},{"Start":"07:51.020 ","End":"07:52.280","Text":"1 plus 3 is 4,"},{"Start":"07:52.280 ","End":"07:54.665","Text":"2 plus 3 is 5, 3 plus 3 is 6."},{"Start":"07:54.665 ","End":"07:57.305","Text":"That\u0027s the starting point in all of these,"},{"Start":"07:57.305 ","End":"08:01.350","Text":"and the ending point in all of these denominators is 4, 4."},{"Start":"08:01.350 ","End":"08:06.640","Text":"4. Multiply down from n plus 3 consecutively down to 4."},{"Start":"08:06.640 ","End":"08:08.425","Text":"That\u0027s the rule."},{"Start":"08:08.425 ","End":"08:10.630","Text":"We can maybe simplify this."},{"Start":"08:10.630 ","End":"08:12.880","Text":"Just get some more space here."},{"Start":"08:12.880 ","End":"08:16.420","Text":"I want to just say something that the reason I\u0027m continuing with this is that"},{"Start":"08:16.420 ","End":"08:20.065","Text":"generally or sometimes we can\u0027t find a general rule."},{"Start":"08:20.065 ","End":"08:22.960","Text":"But, when we can find the general pattern for a_n,"},{"Start":"08:22.960 ","End":"08:26.170","Text":"it\u0027s considered to be a much better solution than just writing"},{"Start":"08:26.170 ","End":"08:29.620","Text":"2 or 3 or 4 terms and the recursion formula."},{"Start":"08:29.620 ","End":"08:31.570","Text":"If we can find the closed formula,"},{"Start":"08:31.570 ","End":"08:33.490","Text":"that is definitely better."},{"Start":"08:33.490 ","End":"08:38.845","Text":"What I\u0027m going to do here is this looks like a factorial product of numbers."},{"Start":"08:38.845 ","End":"08:41.020","Text":"Only factorial goes down to 1."},{"Start":"08:41.020 ","End":"08:43.870","Text":"I fix this by putting after the 4 or 3, 2,"},{"Start":"08:43.870 ","End":"08:47.920","Text":"1 and compensating by putting in the numerator also."},{"Start":"08:47.920 ","End":"08:53.395","Text":"Now this is n plus 3 factorial because of n plus 3 products down to 1."},{"Start":"08:53.395 ","End":"08:56.560","Text":"In the numerator, 1 times 2 times 3 is 6,"},{"Start":"08:56.560 ","End":"08:58.447","Text":"and the a naught in here."},{"Start":"08:58.447 ","End":"09:01.390","Text":"Recall that y_1 is x to the Lambda."},{"Start":"09:01.390 ","End":"09:03.970","Text":"I mean, this bit is the x to the Lambda."},{"Start":"09:03.970 ","End":"09:05.560","Text":"Then times the series,"},{"Start":"09:05.560 ","End":"09:07.765","Text":"the sum of a_n x^n."},{"Start":"09:07.765 ","End":"09:11.020","Text":"But they all contain a naught which I brought out front."},{"Start":"09:11.020 ","End":"09:14.905","Text":"Then I just need to copy the coefficients we had before."},{"Start":"09:14.905 ","End":"09:16.210","Text":"If you multiply, we had a 4,"},{"Start":"09:16.210 ","End":"09:17.905","Text":"then we had 5 times 4,"},{"Start":"09:17.905 ","End":"09:23.470","Text":"and in general, we have this which is 6 over n plus 3 factorial."},{"Start":"09:23.470 ","End":"09:25.690","Text":"Really much clearer in Sigma form."},{"Start":"09:25.690 ","End":"09:30.115","Text":"This is the general term and it works also for n equals naught."},{"Start":"09:30.115 ","End":"09:35.710","Text":"When n equals naught, you see we have 6 over 3 factorial, which is 1."},{"Start":"09:35.710 ","End":"09:39.925","Text":"As I said, this 1 is the Lambda 1 we\u0027re using."},{"Start":"09:39.925 ","End":"09:42.775","Text":"I\u0027ll pull the 6 out front."},{"Start":"09:42.775 ","End":"09:45.220","Text":"I\u0027ll drop the 1 from x^1,"},{"Start":"09:45.220 ","End":"09:46.765","Text":"and this is what I get."},{"Start":"09:46.765 ","End":"09:48.100","Text":"Now, in the past,"},{"Start":"09:48.100 ","End":"09:51.490","Text":"I\u0027ve often just thrown a naught out or said we can let it equal 1,"},{"Start":"09:51.490 ","End":"09:52.990","Text":"it could be anything non-zero."},{"Start":"09:52.990 ","End":"09:56.650","Text":"Here\u0027s suits me to take a naught equal 1/6,"},{"Start":"09:56.650 ","End":"10:02.395","Text":"then I can get rid of this thing altogether because y_1 is only up to a constant anyway."},{"Start":"10:02.395 ","End":"10:09.820","Text":"I have a closed expression with the sigma infinite series for y_1(x) to equal this."},{"Start":"10:09.820 ","End":"10:11.500","Text":"But we don\u0027t stop here,"},{"Start":"10:11.500 ","End":"10:12.940","Text":"prepare yourself for a ride."},{"Start":"10:12.940 ","End":"10:15.610","Text":"I\u0027m going to manipulate this and simplify it and even"},{"Start":"10:15.610 ","End":"10:18.415","Text":"get rid of the Sigma and write it in a much simpler form."},{"Start":"10:18.415 ","End":"10:21.250","Text":"I\u0027m going to bring in a formula and you"},{"Start":"10:21.250 ","End":"10:24.220","Text":"probably wondering why am I bringing this in, you\u0027ll see."},{"Start":"10:24.220 ","End":"10:26.890","Text":"We know the formula for power series for e^x."},{"Start":"10:26.890 ","End":"10:29.590","Text":"It\u0027s this with all pluses where if we put a minus x,"},{"Start":"10:29.590 ","End":"10:32.200","Text":"then we get an alternating sign series."},{"Start":"10:32.200 ","End":"10:38.460","Text":"What made me think of this is that I look here and I see this looks like there\u0027s a power,"},{"Start":"10:38.460 ","End":"10:40.005","Text":"there is a factorial,"},{"Start":"10:40.005 ","End":"10:42.135","Text":"and there\u0027s an alternating sign."},{"Start":"10:42.135 ","End":"10:44.670","Text":"The minus 1 to the n means alternating sign."},{"Start":"10:44.670 ","End":"10:47.430","Text":"So I thought maybe I can somehow get this in terms of e"},{"Start":"10:47.430 ","End":"10:50.360","Text":"to the minus x with a bit of manipulation."},{"Start":"10:50.360 ","End":"10:51.790","Text":"I don\u0027t know why I wrote this again,"},{"Start":"10:51.790 ","End":"10:54.025","Text":"it seems to be the same as [inaudible]."},{"Start":"10:54.025 ","End":"10:56.800","Text":"Moving on, what I wanted to do is see"},{"Start":"10:56.800 ","End":"10:59.440","Text":"here they perfectly match there\u0027s a 4 with a 4 factorial,"},{"Start":"10:59.440 ","End":"11:01.225","Text":"5 with a 5 factorial,"},{"Start":"11:01.225 ","End":"11:02.740","Text":"3 with 3 factorial."},{"Start":"11:02.740 ","End":"11:08.410","Text":"We would like the n to be with n factorial or the same thing upstairs as downstairs."},{"Start":"11:08.410 ","End":"11:11.200","Text":"I\u0027m going to fix this by,"},{"Start":"11:11.200 ","End":"11:13.420","Text":"if I multiply here by x^3,"},{"Start":"11:13.420 ","End":"11:16.375","Text":"my idea is I\u0027m going to get n plus 3 factorial and that will"},{"Start":"11:16.375 ","End":"11:19.495","Text":"synchronize the exponent on the denominator."},{"Start":"11:19.495 ","End":"11:22.180","Text":"Of course, I can\u0027t just multiply it by x^3,"},{"Start":"11:22.180 ","End":"11:23.815","Text":"I have to divide by 2,"},{"Start":"11:23.815 ","End":"11:25.060","Text":"but it doesn\u0027t depend on n,"},{"Start":"11:25.060 ","End":"11:28.645","Text":"so I can bring it in front of the Sigma, we get this."},{"Start":"11:28.645 ","End":"11:32.530","Text":"In the next line, I just combine the n and the 3 to n plus 3."},{"Start":"11:32.530 ","End":"11:36.745","Text":"Also x over x^3 is 1 over x squared."},{"Start":"11:36.745 ","End":"11:39.295","Text":"If we now expand the sigma,"},{"Start":"11:39.295 ","End":"11:40.915","Text":"I mean, write some of the terms."},{"Start":"11:40.915 ","End":"11:42.355","Text":"When n is naught,"},{"Start":"11:42.355 ","End":"11:47.200","Text":"we get plus and then we get x^3 over 3 factorial."},{"Start":"11:47.200 ","End":"11:51.024","Text":"Their signs keep alternating and I get x^4 over 4 factorial,"},{"Start":"11:51.024 ","End":"11:53.350","Text":"x^5 over 5 factorial, and so on."},{"Start":"11:53.350 ","End":"11:56.740","Text":"X^n over n factorial with the alternating sign,"},{"Start":"11:56.740 ","End":"11:58.600","Text":"and there\u0027s a dot missing here."},{"Start":"11:58.600 ","End":"12:01.990","Text":"Look what I\u0027m going to do with what this in brackets, I have to go back a bit."},{"Start":"12:01.990 ","End":"12:08.605","Text":"This here is the same as what I have here that I\u0027m highlighting,"},{"Start":"12:08.605 ","End":"12:13.060","Text":"except that this is minus of what is this,"},{"Start":"12:13.060 ","End":"12:14.860","Text":"here I have plus minus and so on,"},{"Start":"12:14.860 ","End":"12:17.170","Text":"here I thought minus plus."},{"Start":"12:17.170 ","End":"12:20.740","Text":"This is slightly wrong."},{"Start":"12:20.740 ","End":"12:22.690","Text":"This should be a minus."},{"Start":"12:22.690 ","End":"12:27.970","Text":"The reason this is a minus is because if I make this n plus 3,"},{"Start":"12:27.970 ","End":"12:33.280","Text":"then I have to compensate by minus 1 to the power of 3 is minus,"},{"Start":"12:33.280 ","End":"12:36.340","Text":"it\u0027s actually a minus this."},{"Start":"12:36.340 ","End":"12:40.975","Text":"Anyway, what\u0027s written in this brackets is the negative of what\u0027s here."},{"Start":"12:40.975 ","End":"12:42.370","Text":"If I take this equation,"},{"Start":"12:42.370 ","End":"12:44.710","Text":"I bring what\u0027s in yellow to the other side,"},{"Start":"12:44.710 ","End":"12:47.260","Text":"it now becomes plus just like here."},{"Start":"12:47.260 ","End":"12:51.585","Text":"I take the e to the minus x and throw it over to the right,"},{"Start":"12:51.585 ","End":"12:55.800","Text":"then I\u0027ll get that what this thing equals is 1 minus"},{"Start":"12:55.800 ","End":"13:00.370","Text":"x minus e to the minus x. I have to scroll,"},{"Start":"13:00.370 ","End":"13:01.960","Text":"just disappeared off-screen,"},{"Start":"13:01.960 ","End":"13:03.070","Text":"but if you look back,"},{"Start":"13:03.070 ","End":"13:05.815","Text":"you\u0027ll see that this is what we get in the brackets here."},{"Start":"13:05.815 ","End":"13:10.180","Text":"We\u0027ve got the first 3 terms and then minus e to the minus x."},{"Start":"13:10.180 ","End":"13:16.960","Text":"We now have y_1 in closed form and without any sigma or just elementary functions."},{"Start":"13:16.960 ","End":"13:19.360","Text":"Now we have the task of finding y_2."},{"Start":"13:19.360 ","End":"13:20.680","Text":"I just wanted to remind you,"},{"Start":"13:20.680 ","End":"13:25.150","Text":"here\u0027s the recursion formula also that we found Lambda 1 equals"},{"Start":"13:25.150 ","End":"13:30.355","Text":"1 and we also had Lambda 2 is minus 2 and the difference is a whole number,"},{"Start":"13:30.355 ","End":"13:31.885","Text":"so we have to be careful."},{"Start":"13:31.885 ","End":"13:34.930","Text":"But if you recall in the tutorial, we said first of all,"},{"Start":"13:34.930 ","End":"13:38.320","Text":"let\u0027s just try doing y_2 the same way we did y_1,"},{"Start":"13:38.320 ","End":"13:40.630","Text":"just using this value of Lambda."},{"Start":"13:40.630 ","End":"13:44.020","Text":"We plugged this minus 2 into here,"},{"Start":"13:44.020 ","End":"13:49.450","Text":"and what we get is a_n equals this times a n minus 1,"},{"Start":"13:49.450 ","End":"13:52.870","Text":"but it only applies from n bigger or equal to 1 of course,"},{"Start":"13:52.870 ","End":"13:56.995","Text":"and multiply out the denominator and we get this."},{"Start":"13:56.995 ","End":"14:00.370","Text":"Now we can take from here n out the brackets and we get n minus"},{"Start":"14:00.370 ","End":"14:04.210","Text":"3 and it\u0027ll cancel with the 3 minus n. We have to put in a minus,"},{"Start":"14:04.210 ","End":"14:06.130","Text":"this is n minus 3, this is 3 minus n,"},{"Start":"14:06.130 ","End":"14:08.710","Text":"and we\u0027re left with 1 over n times n minus 1."},{"Start":"14:08.710 ","End":"14:10.315","Text":"That\u0027s pretty straightforward."},{"Start":"14:10.315 ","End":"14:12.340","Text":"If we substitute n equals 1,"},{"Start":"14:12.340 ","End":"14:14.410","Text":"we get a_1 in terms of a naught."},{"Start":"14:14.410 ","End":"14:17.050","Text":"We only can expect to get things in terms of a naught, of course."},{"Start":"14:17.050 ","End":"14:20.935","Text":"Next, a_2 minus 1/2 times the previous,"},{"Start":"14:20.935 ","End":"14:23.425","Text":"we get minus a 1/2 minus 1 over 1."},{"Start":"14:23.425 ","End":"14:26.260","Text":"Next, we\u0027ve got minus 1 over n is minus a 1/3,"},{"Start":"14:26.260 ","End":"14:28.750","Text":"so we\u0027ve minus a 1/3 minus a 1/2 minus 1 over 1."},{"Start":"14:28.750 ","End":"14:32.170","Text":"I think you can see that a pattern is emerging here also."},{"Start":"14:32.170 ","End":"14:34.285","Text":"I claim that this is the rule."},{"Start":"14:34.285 ","End":"14:36.475","Text":"Let\u0027s just look at the case of 3, for example,"},{"Start":"14:36.475 ","End":"14:39.205","Text":"I could write it as minus, minus, minus."},{"Start":"14:39.205 ","End":"14:40.780","Text":"The number of minuses is 3,"},{"Start":"14:40.780 ","End":"14:42.340","Text":"just like here there\u0027s 2 minuses,"},{"Start":"14:42.340 ","End":"14:45.010","Text":"it\u0027s minus 1 to the n. As for the rest of it,"},{"Start":"14:45.010 ","End":"14:47.380","Text":"3 times 2 times 1 is 3 factorial,"},{"Start":"14:47.380 ","End":"14:49.405","Text":"and that\u0027s how the pattern goes."},{"Start":"14:49.405 ","End":"14:54.640","Text":"We\u0027ve got the general formula for a_n in terms of a naught."},{"Start":"14:54.640 ","End":"14:56.725","Text":"What do we do with this?"},{"Start":"14:56.725 ","End":"14:58.255","Text":"Well, we plug it in."},{"Start":"14:58.255 ","End":"15:00.370","Text":"This is the x to the Lambda part,"},{"Start":"15:00.370 ","End":"15:02.470","Text":"and this is the power series part,"},{"Start":"15:02.470 ","End":"15:05.515","Text":"the a_n x^n summation."},{"Start":"15:05.515 ","End":"15:07.915","Text":"Now, this is not guaranteed to work this method."},{"Start":"15:07.915 ","End":"15:09.310","Text":"But in the tutorial,"},{"Start":"15:09.310 ","End":"15:13.855","Text":"we said that when we have 2 Lambdas that differ by a whole number, it\u0027s worth trying."},{"Start":"15:13.855 ","End":"15:15.715","Text":"It might or might not work."},{"Start":"15:15.715 ","End":"15:18.610","Text":"Usually, if nothing goes wrong in the recursion formula,"},{"Start":"15:18.610 ","End":"15:20.515","Text":"then it\u0027s going to work till the end."},{"Start":"15:20.515 ","End":"15:25.750","Text":"I believe the next exercise is one where things do go wrong and how to overcome that."},{"Start":"15:25.750 ","End":"15:31.660","Text":"Anyway, here, we take the a naught outside the brackets and write it in sigma form,"},{"Start":"15:31.660 ","End":"15:35.230","Text":"sigma basically, this is just the general term which we write here."},{"Start":"15:35.230 ","End":"15:39.505","Text":"This is exactly the series for e to the minus x that we had above."},{"Start":"15:39.505 ","End":"15:41.125","Text":"This is what we have."},{"Start":"15:41.125 ","End":"15:43.960","Text":"We can let a naught be anything non-zero we want."},{"Start":"15:43.960 ","End":"15:46.030","Text":"I could take a naught equals 1,"},{"Start":"15:46.030 ","End":"15:48.865","Text":"and now I can write down y_2,"},{"Start":"15:48.865 ","End":"15:50.590","Text":"y_2 x to the minus 2,"},{"Start":"15:50.590 ","End":"15:52.255","Text":"e to the minus x."},{"Start":"15:52.255 ","End":"15:56.815","Text":"All that remains to do is to take the linear combination of y_1 and y_2."},{"Start":"15:56.815 ","End":"15:58.000","Text":"Y_2, we have here,"},{"Start":"15:58.000 ","End":"15:59.935","Text":"go back and see what y_1 is."},{"Start":"15:59.935 ","End":"16:02.080","Text":"Y_1 was 1 over x^2 times this thing."},{"Start":"16:02.080 ","End":"16:03.985","Text":"Anyway, this is the final answer,"},{"Start":"16:03.985 ","End":"16:07.160","Text":"and now we\u0027re really done."}],"Thumbnail":null,"ID":7892},{"Watched":false,"Name":"Exercise 8","Duration":"13m 24s","ChapterTopicVideoID":7841,"CourseChapterTopicPlaylistID":4244,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"We have this differential equation to solve,"},{"Start":"00:02.550 ","End":"00:09.510","Text":"and we\u0027re going do it by showing that x=naught to the regular singular point,"},{"Start":"00:09.510 ","End":"00:13.785","Text":"and developing the solution y as a power series around x= naught."},{"Start":"00:13.785 ","End":"00:16.500","Text":"I need to justify that this is a regular singular points."},{"Start":"00:16.500 ","End":"00:18.525","Text":"We divide by this,"},{"Start":"00:18.525 ","End":"00:20.700","Text":"and get it in standard form."},{"Start":"00:20.700 ","End":"00:22.763","Text":"This we call p,"},{"Start":"00:22.763 ","End":"00:24.240","Text":"and this we call q."},{"Start":"00:24.240 ","End":"00:27.270","Text":"Now both of these are not defined as x= naught,"},{"Start":"00:27.270 ","End":"00:31.650","Text":"and any one of those not being defined would make it not a regular point."},{"Start":"00:31.650 ","End":"00:33.614","Text":"Then to be regular singular,"},{"Start":"00:33.614 ","End":"00:36.750","Text":"you\u0027d have to show that x times p,"},{"Start":"00:36.750 ","End":"00:40.890","Text":"and x^2 times q are both defined,"},{"Start":"00:40.890 ","End":"00:43.045","Text":"well-behaved at x=naught,"},{"Start":"00:43.045 ","End":"00:45.785","Text":"x times p comes out to be this,"},{"Start":"00:45.785 ","End":"00:47.900","Text":"an x^2 q comes out to be this."},{"Start":"00:47.900 ","End":"00:50.135","Text":"A nice linear, and a constant function."},{"Start":"00:50.135 ","End":"00:52.880","Text":"It can\u0027t get any better than that certainly defined"},{"Start":"00:52.880 ","End":"00:55.790","Text":"and everything around the x equals naught."},{"Start":"00:55.790 ","End":"00:59.375","Text":"We can go ahead and write y,"},{"Start":"00:59.375 ","End":"01:00.770","Text":"which is one of those power series,"},{"Start":"01:00.770 ","End":"01:03.635","Text":"but it starts with x to the Lambda. We don\u0027t know Lambda."},{"Start":"01:03.635 ","End":"01:06.650","Text":"It just looks a bit nicer if I write it in Sigma form,"},{"Start":"01:06.650 ","End":"01:13.620","Text":"it\u0027s x to the Lambda times a regular power series, a_nx^n we used to."},{"Start":"01:13.620 ","End":"01:20.370","Text":"Now we\u0027ll need y\u0027 and y\" in order to substitute in the differential equation."},{"Start":"01:20.370 ","End":"01:27.975","Text":"Here they are y\u0027, y\" will also like the original equation. Here this is."},{"Start":"01:27.975 ","End":"01:30.620","Text":"We\u0027re not going to substitute every term that\u0027s written out"},{"Start":"01:30.620 ","End":"01:33.500","Text":"here as is some of these are just unnecessary."},{"Start":"01:33.500 ","End":"01:35.735","Text":"They\u0027re part of the ellipsis, the dot, dot, dot."},{"Start":"01:35.735 ","End":"01:38.000","Text":"We\u0027re just going to be very judicious and choosing."},{"Start":"01:38.000 ","End":"01:43.888","Text":"We only need the coefficients of x to the Lambda in this result,"},{"Start":"01:43.888 ","End":"01:47.370","Text":"and also x to the Lambda plus n. I\u0027m"},{"Start":"01:47.370 ","End":"01:51.140","Text":"going to have to expand this middle term into 2 separate pieces."},{"Start":"01:51.140 ","End":"01:56.285","Text":"This is x squared y\u0027 minus 2xy\u0027,"},{"Start":"01:56.285 ","End":"02:00.490","Text":"so actually 4 pieces to the left-hand side, 4 terms."},{"Start":"02:00.490 ","End":"02:02.423","Text":"I color-coded these,"},{"Start":"02:02.423 ","End":"02:04.370","Text":"and I\u0027m going to highlight stuff here."},{"Start":"02:04.370 ","End":"02:08.735","Text":"Let\u0027s look at the x to the Lambda in the first term, x squared y\"."},{"Start":"02:08.735 ","End":"02:11.045","Text":"We\u0027re going to be multiplying this by x squared."},{"Start":"02:11.045 ","End":"02:13.520","Text":"All the powers are going to get raised by 2."},{"Start":"02:13.520 ","End":"02:15.130","Text":"To get x to the Lambda,"},{"Start":"02:15.130 ","End":"02:17.715","Text":"I\u0027ll be looking x to the Lambda minus 2."},{"Start":"02:17.715 ","End":"02:20.040","Text":"I\u0027ll need this coefficient."},{"Start":"02:20.040 ","End":"02:24.150","Text":"Also for this, I\u0027ll need Lambda plus n minus 2 and I find it here."},{"Start":"02:24.150 ","End":"02:26.640","Text":"I\u0027m going to need this one."},{"Start":"02:26.640 ","End":"02:29.555","Text":"Now let\u0027s go on to the next one."},{"Start":"02:29.555 ","End":"02:33.310","Text":"This, and this is here and here."},{"Start":"02:33.310 ","End":"02:36.990","Text":"Now the next one I want to do is the minus 2xy\u0027."},{"Start":"02:36.990 ","End":"02:38.685","Text":"Never mind the minus 2."},{"Start":"02:38.685 ","End":"02:43.490","Text":"Xy\u0027, we\u0027ll raise all these exponents by 1."},{"Start":"02:43.490 ","End":"02:48.195","Text":"I\u0027m looking for x to the Lambda minus 1 and Lambda plus n minus 1."},{"Start":"02:48.195 ","End":"02:50.850","Text":"I locate the x to the Lambda minus 1 here,"},{"Start":"02:50.850 ","End":"02:54.300","Text":"so I\u0027ll need this and Lambda plus n minus 1 here."},{"Start":"02:54.300 ","End":"02:56.475","Text":"I need this."},{"Start":"02:56.475 ","End":"02:58.640","Text":"Let me just mark them here."},{"Start":"02:58.640 ","End":"03:04.880","Text":"This one from here and this one here, like so."},{"Start":"03:04.880 ","End":"03:07.250","Text":"Next I\u0027ll look at x squared y\u0027,"},{"Start":"03:07.250 ","End":"03:09.605","Text":"which raises all these exponents by 2."},{"Start":"03:09.605 ","End":"03:13.760","Text":"I\u0027m looking for x to the Lambda minus 2 and Lambda plus n minus 2."},{"Start":"03:13.760 ","End":"03:19.235","Text":"Well, Lambda minus 2 is absent here because the first one is lambda minus 1,"},{"Start":"03:19.235 ","End":"03:21.725","Text":"so we don\u0027t get anything in this color."},{"Start":"03:21.725 ","End":"03:28.010","Text":"But for this, we\u0027ve got Lambda plus n minus 2 here."},{"Start":"03:28.010 ","End":"03:32.405","Text":"This one is going to be useful and I\u0027d like to mark them off."},{"Start":"03:32.405 ","End":"03:35.195","Text":"This term is here,"},{"Start":"03:35.195 ","End":"03:41.195","Text":"but this term, I\u0027m not going to use it because there is no Lambda minus 2 here."},{"Start":"03:41.195 ","End":"03:43.025","Text":"Now we have the fourth one,"},{"Start":"03:43.025 ","End":"03:45.095","Text":"which is the 2y."},{"Start":"03:45.095 ","End":"03:47.060","Text":"Y is scrolled offscreen,"},{"Start":"03:47.060 ","End":"03:48.775","Text":"I need to back to it."},{"Start":"03:48.775 ","End":"03:52.470","Text":"We\u0027ll need this for x to the Lambda and"},{"Start":"03:52.470 ","End":"03:57.615","Text":"this coefficient of x to the Lambda plus n. Now back down."},{"Start":"03:57.615 ","End":"04:00.840","Text":"It\u0027s this one, and this one."},{"Start":"04:00.840 ","End":"04:04.110","Text":"Also note that there are coefficients."},{"Start":"04:04.110 ","End":"04:07.715","Text":"Here, it\u0027s one. I\u0027m going to mark the ones that are important. This is this."},{"Start":"04:07.715 ","End":"04:09.898","Text":"Now we\u0027re going to do some arithmetic,"},{"Start":"04:09.898 ","End":"04:12.650","Text":"and we\u0027re going to equate the coefficients of"},{"Start":"04:12.650 ","End":"04:16.880","Text":"x Lambda on left and right and also this left and right but first this."},{"Start":"04:16.880 ","End":"04:18.485","Text":"For x to the Lambda,"},{"Start":"04:18.485 ","End":"04:20.330","Text":"we take terms from here."},{"Start":"04:20.330 ","End":"04:25.010","Text":"We need this and we need minus twice this and plus twice this."},{"Start":"04:25.010 ","End":"04:30.860","Text":"But I don\u0027t need the I naught it appears everywhere and we just divide by it."},{"Start":"04:30.860 ","End":"04:34.730","Text":"Now if you expand this equation and solve it and omitting the details,"},{"Start":"04:34.730 ","End":"04:38.440","Text":"we end up with 2 solutions, 2 and 1."},{"Start":"04:38.440 ","End":"04:41.770","Text":"We see we\u0027re in the case where the difference is a whole number."},{"Start":"04:41.770 ","End":"04:46.560","Text":"Also the larger one is Lambda 1 and the smaller one is Lambda 2."},{"Start":"04:46.560 ","End":"04:51.390","Text":"We\u0027re going to use this value of Lambda when looking for y1."},{"Start":"04:51.390 ","End":"04:52.940","Text":"I will return here in a moment."},{"Start":"04:52.940 ","End":"04:55.700","Text":"Let\u0027s now compare the coefficients of"},{"Start":"04:55.700 ","End":"04:59.375","Text":"x to the Lambda plus n from here, that\u0027s this color."},{"Start":"04:59.375 ","End":"05:01.280","Text":"Collecting these 4 here,"},{"Start":"05:01.280 ","End":"05:05.120","Text":"and with coefficients like minus 2 and 2 where necessary,"},{"Start":"05:05.120 ","End":"05:07.160","Text":"we get this equation."},{"Start":"05:07.160 ","End":"05:11.559","Text":"Next we collect like terms and throw all the a_n terms on the left,"},{"Start":"05:11.559 ","End":"05:14.125","Text":"and the a_n minus 1 on the right."},{"Start":"05:14.125 ","End":"05:16.980","Text":"Then we can extract a_n."},{"Start":"05:16.980 ","End":"05:23.890","Text":"This gives us a recursion for a_n in terms of a_n minus 1 for a given Lambda."},{"Start":"05:23.890 ","End":"05:28.010","Text":"I just made some space and I want to substitute the value of Lambda 1."},{"Start":"05:28.010 ","End":"05:30.988","Text":"I just want to remind you we had Lambda 1 was 2,"},{"Start":"05:30.988 ","End":"05:32.540","Text":"and Lambda 2 was 1."},{"Start":"05:32.540 ","End":"05:36.620","Text":"The first Lambda, which is 2 substitute that in here and that gives us"},{"Start":"05:36.620 ","End":"05:42.870","Text":"the specific recursive equation for a_n in terms of a_n minus 1 for this Lambda,"},{"Start":"05:42.870 ","End":"05:46.300","Text":"and as usual has a certain restriction on a_n."},{"Start":"05:46.300 ","End":"05:50.570","Text":"We want to simplify this. This comes out to be n squared plus n minus"},{"Start":"05:50.570 ","End":"05:55.100","Text":"2 plus 2 so it\u0027s n squared plus n. Then n squared plus n,"},{"Start":"05:55.100 ","End":"05:57.910","Text":"I can take n out and I\u0027ll have n plus 1."},{"Start":"05:57.910 ","End":"05:59.340","Text":"We have a simple formula."},{"Start":"05:59.340 ","End":"06:06.480","Text":"A_n is minus 1 over n times the previous a_n minus 1 and this holds from 1 onwards."},{"Start":"06:06.480 ","End":"06:09.210","Text":"As usual, we\u0027re going to plug in a bunch of values."},{"Start":"06:09.210 ","End":"06:12.540","Text":"We\u0027re going to plug in n equals 1 and get a_1 in terms of a naught."},{"Start":"06:12.540 ","End":"06:14.130","Text":"Then we\u0027re going to plug in 2,"},{"Start":"06:14.130 ","End":"06:15.960","Text":"and get a_2 in terms of a_1,"},{"Start":"06:15.960 ","End":"06:18.960","Text":"which is then in terms of a naught then a_3,"},{"Start":"06:18.960 ","End":"06:20.805","Text":"and so on and so on."},{"Start":"06:20.805 ","End":"06:24.010","Text":"If we look at it, we\u0027ve got the general form of it."},{"Start":"06:24.010 ","End":"06:25.315","Text":"We\u0027ll look at a_3."},{"Start":"06:25.315 ","End":"06:28.405","Text":"With 3 minuses to get this minus with a minus."},{"Start":"06:28.405 ","End":"06:31.345","Text":"That\u0027s minus 1^3 here."},{"Start":"06:31.345 ","End":"06:35.805","Text":"Then we have a 1/3 and 1/2 and the next one is going to be a 1/4."},{"Start":"06:35.805 ","End":"06:38.460","Text":"On the denominator we\u0027re going to have 3 times 2 times 1,"},{"Start":"06:38.460 ","End":"06:40.140","Text":"the next 4 times 3 times 2 times 1."},{"Start":"06:40.140 ","End":"06:42.165","Text":"In general, n factorial,"},{"Start":"06:42.165 ","End":"06:44.830","Text":"and the a naught they\u0027re everywhere."},{"Start":"06:45.380 ","End":"06:50.310","Text":"Y_1 is x to the Lambda times the power series,"},{"Start":"06:50.310 ","End":"06:52.290","Text":"but the Lambda was 2 in our cases,"},{"Start":"06:52.290 ","End":"06:56.090","Text":"y it\u0027s x squared, and the power series we computed the first few terms."},{"Start":"06:56.090 ","End":"07:01.290","Text":"In fact, we computed the general term even and we can take a naught in front."},{"Start":"07:01.290 ","End":"07:03.890","Text":"This series should look familiar,"},{"Start":"07:03.890 ","End":"07:05.270","Text":"especially if I instead of 2,"},{"Start":"07:05.270 ","End":"07:10.040","Text":"I wrote 2 factorial and instead of 6 I wrote 3 factorial and Sigma form,"},{"Start":"07:10.040 ","End":"07:11.810","Text":"it should be even more obvious."},{"Start":"07:11.810 ","End":"07:13.340","Text":"But in case you don\u0027t recognize it,"},{"Start":"07:13.340 ","End":"07:15.140","Text":"I\u0027ll tell you this series,"},{"Start":"07:15.140 ","End":"07:17.600","Text":"the Sigma as e to the minus x."},{"Start":"07:17.600 ","End":"07:20.615","Text":"That\u0027s the sum and the x-squared is here."},{"Start":"07:20.615 ","End":"07:22.160","Text":"Now, a naught,"},{"Start":"07:22.160 ","End":"07:24.170","Text":"we can take anything non-zero."},{"Start":"07:24.170 ","End":"07:26.780","Text":"Y_1 is only up to a constant anyway,"},{"Start":"07:26.780 ","End":"07:29.165","Text":"I\u0027ll take a naught equals 1."},{"Start":"07:29.165 ","End":"07:33.785","Text":"Then we can get Y_1 as x squared e to the minus x,"},{"Start":"07:33.785 ","End":"07:36.170","Text":"or any other non-zero constant times this."},{"Start":"07:36.170 ","End":"07:39.005","Text":"This is the simplest. Now we\u0027re going to go find y_2."},{"Start":"07:39.005 ","End":"07:41.029","Text":"Remember we\u0027re in a difficult situation"},{"Start":"07:41.029 ","End":"07:43.610","Text":"where the difference of the 2 roots is a whole number."},{"Start":"07:43.610 ","End":"07:49.070","Text":"Back to the recursion formula and the reminder again that we had the solutions"},{"Start":"07:49.070 ","End":"07:54.575","Text":"where 2 and 1 and in the tutorial we said,"},{"Start":"07:54.575 ","End":"07:57.530","Text":"first of all, try your luck and just see what happens if"},{"Start":"07:57.530 ","End":"08:00.695","Text":"we take Lambda 2 equals 1 and proceed as before."},{"Start":"08:00.695 ","End":"08:04.820","Text":"This time we plug in Lambda equals 1 and we"},{"Start":"08:04.820 ","End":"08:09.335","Text":"get that the recursion formula if we plug in Lambda comes out to be this."},{"Start":"08:09.335 ","End":"08:11.630","Text":"It holds from 1 onwards."},{"Start":"08:11.630 ","End":"08:14.285","Text":"If we plug in n equals 1,"},{"Start":"08:14.285 ","End":"08:17.870","Text":"what we get in the denominator is 1 plus 1 is 2,"},{"Start":"08:17.870 ","End":"08:19.085","Text":"1 minus 2 is minus 1,"},{"Start":"08:19.085 ","End":"08:21.305","Text":"minus 2 plus 2, 0."},{"Start":"08:21.305 ","End":"08:24.515","Text":"Plug in in n equals 1, we get 0 on the denominator."},{"Start":"08:24.515 ","End":"08:26.950","Text":"We\u0027re already stuck."},{"Start":"08:26.950 ","End":"08:29.630","Text":"This method is failed us,"},{"Start":"08:29.630 ","End":"08:35.045","Text":"so we\u0027re going to have to use Plan B. I indicate this method does not work."},{"Start":"08:35.045 ","End":"08:36.650","Text":"Actually, if you simplify this,"},{"Start":"08:36.650 ","End":"08:39.140","Text":"you get minus 1 over n minus 1."},{"Start":"08:39.140 ","End":"08:42.980","Text":"Then more clearly for n equals 1 can\u0027t be done."},{"Start":"08:42.980 ","End":"08:44.869","Text":"If you remember in the tutorial,"},{"Start":"08:44.869 ","End":"08:46.880","Text":"perhaps you don\u0027t remember, I\u0027ll refresh your memory."},{"Start":"08:46.880 ","End":"08:51.980","Text":"What we do in such a case is we let y_2 be as follows."},{"Start":"08:51.980 ","End":"08:54.650","Text":"We don\u0027t evaluate Lambda,"},{"Start":"08:54.650 ","End":"08:56.855","Text":"we leave Lambda as a variable."},{"Start":"08:56.855 ","End":"09:00.080","Text":"We get y in terms of Lambda and x."},{"Start":"09:00.080 ","End":"09:04.080","Text":"Then we multiply by Lambda minus Lambda 2,"},{"Start":"09:04.080 ","End":"09:07.270","Text":"which in this case is 1. This is 1."},{"Start":"09:07.270 ","End":"09:10.385","Text":"Then we take the partial derivative with respect to Lambda,"},{"Start":"09:10.385 ","End":"09:15.130","Text":"and then we plug in Lambda equals the Lambda 2,"},{"Start":"09:15.130 ","End":"09:17.465","Text":"which is 1 in our case."},{"Start":"09:17.465 ","End":"09:19.640","Text":"Slowly we have to build this thing up."},{"Start":"09:19.640 ","End":"09:23.075","Text":"Let\u0027s first of all compute y in terms of Lambda and x."},{"Start":"09:23.075 ","End":"09:27.370","Text":"Well, perhaps I\u0027ll do a side computation to show you how I got to this n equals 1."},{"Start":"09:27.370 ","End":"09:33.780","Text":"What I have is minus Lambda because 1 minus 1 is 0, it cancels."},{"Start":"09:33.780 ","End":"09:40.575","Text":"On the denominator, n is 1 so we have Lambda plus 1 here and n is 1."},{"Start":"09:40.575 ","End":"09:48.135","Text":"Here we have Lambda minus 2 and then there\u0027s a plus 2 and this equals minus Lambda over,"},{"Start":"09:48.135 ","End":"09:53.925","Text":"this is Lambda squared minus Lambda minus 2 plus 2."},{"Start":"09:53.925 ","End":"09:55.695","Text":"The 2\u0027s just cancel out."},{"Start":"09:55.695 ","End":"09:57.435","Text":"If we divide by Lambda,"},{"Start":"09:57.435 ","End":"10:00.810","Text":"we get minus 1 over Lambda minus 1."},{"Start":"10:00.810 ","End":"10:06.450","Text":"Next we plug in n equals 2 and it simplifies to a_2 is minus"},{"Start":"10:06.450 ","End":"10:12.675","Text":"1 over Lambda a_1 but we already have a_1 from here so that\u0027s what we get."},{"Start":"10:12.675 ","End":"10:15.155","Text":"Let\u0027s just continue."},{"Start":"10:15.155 ","End":"10:16.850","Text":"Here\u0027s the result of a_3,"},{"Start":"10:16.850 ","End":"10:19.024","Text":"I won\u0027t go into all the details."},{"Start":"10:19.024 ","End":"10:21.785","Text":"Now y in terms of Lambda annex,"},{"Start":"10:21.785 ","End":"10:24.265","Text":"we don\u0027t substitute the Lambda this time."},{"Start":"10:24.265 ","End":"10:27.040","Text":"It\u0027s x to the Lambda times the power series and"},{"Start":"10:27.040 ","End":"10:29.900","Text":"we\u0027ve got the first few terms of the power series."},{"Start":"10:29.900 ","End":"10:35.950","Text":"Notice again that we can\u0027t substitute Lambda equals 1 because Lambda minus 1 is 0."},{"Start":"10:35.950 ","End":"10:37.590","Text":"That approach didn\u0027t work."},{"Start":"10:37.590 ","End":"10:43.835","Text":"Now, just about see what we have to compute is Lambda minus 1 times this."},{"Start":"10:43.835 ","End":"10:47.344","Text":"First I\u0027ll bring a naught out in front that will make it easier."},{"Start":"10:47.344 ","End":"10:50.385","Text":"Next, I\u0027ll compute Lambda minus,"},{"Start":"10:50.385 ","End":"10:52.995","Text":"this is 1 times this expression."},{"Start":"10:52.995 ","End":"10:55.002","Text":"Multiply by Lambda minus 1."},{"Start":"10:55.002 ","End":"10:56.280","Text":"Here we get Lambda minus 1."},{"Start":"10:56.280 ","End":"10:57.420","Text":"Here it cancels."},{"Start":"10:57.420 ","End":"10:59.505","Text":"Here we get 1 over Lambda."},{"Start":"10:59.505 ","End":"11:02.820","Text":"Here 1 over Lambda times Lambda plus 1."},{"Start":"11:02.820 ","End":"11:05.160","Text":"The Lambda minus 1 disappears."},{"Start":"11:05.160 ","End":"11:10.640","Text":"What we\u0027re going to do next is to take this expression which is this bit here,"},{"Start":"11:10.640 ","End":"11:13.040","Text":"that\u0027s what this is and differentiate it,"},{"Start":"11:13.040 ","End":"11:14.150","Text":"plus the function of Lambda,"},{"Start":"11:14.150 ","End":"11:15.940","Text":"treat x like a constant."},{"Start":"11:15.940 ","End":"11:20.475","Text":"After we\u0027ve differentiated, we\u0027re going to substitute Lambda equals 1."},{"Start":"11:20.475 ","End":"11:22.640","Text":"I\u0027m going to use the product rule,"},{"Start":"11:22.640 ","End":"11:24.860","Text":"this will be this times this."},{"Start":"11:24.860 ","End":"11:26.870","Text":"I\u0027ll need the derivative of this times this,"},{"Start":"11:26.870 ","End":"11:30.360","Text":"and then this times the derivative of this. Here\u0027s the calculation."},{"Start":"11:30.360 ","End":"11:32.120","Text":"The a naught gets left outside,"},{"Start":"11:32.120 ","End":"11:33.650","Text":"so before the substitution,"},{"Start":"11:33.650 ","End":"11:35.150","Text":"but after the differentiation."},{"Start":"11:35.150 ","End":"11:39.395","Text":"The derivative of x to the Lambda is x to the Lambda natural log of x."},{"Start":"11:39.395 ","End":"11:41.380","Text":"Then this was what was written here."},{"Start":"11:41.380 ","End":"11:45.110","Text":"Then we\u0027ve got the x to the Lambda as is and the derivative of this,"},{"Start":"11:45.110 ","End":"11:47.815","Text":"which this thing comes out to be 1."},{"Start":"11:47.815 ","End":"11:50.610","Text":"Here this is the constant that\u0027s nothing."},{"Start":"11:50.610 ","End":"11:53.645","Text":"Here 1 over the Lambda gives minus 1 over Lambda squared."},{"Start":"11:53.645 ","End":"11:56.855","Text":"We\u0027re differentiating with respect to Lambda remember not x."},{"Start":"11:56.855 ","End":"11:58.160","Text":"This one, well,"},{"Start":"11:58.160 ","End":"11:59.945","Text":"if you do the computation of this,"},{"Start":"11:59.945 ","End":"12:02.275","Text":"we get this and so on."},{"Start":"12:02.275 ","End":"12:05.250","Text":"Now the substitution."},{"Start":"12:05.250 ","End":"12:10.800","Text":"What we get this Lambda minus 1 is 0."},{"Start":"12:10.800 ","End":"12:13.520","Text":"Let\u0027s see for the top we get x natural log of x."},{"Start":"12:13.520 ","End":"12:16.560","Text":"We get minus x, 1 over 1 here,"},{"Start":"12:16.560 ","End":"12:19.190","Text":"1 over 2 here, and so on."},{"Start":"12:19.190 ","End":"12:20.750","Text":"You\u0027ve got a first few terms."},{"Start":"12:20.750 ","End":"12:27.735","Text":"Then here also x^1 natural log of x and Lambda is 1."},{"Start":"12:27.735 ","End":"12:36.410","Text":"These coefficients are 1 minus 1 with the x^2 and then minus 3/4 with x^3 and so on."},{"Start":"12:36.410 ","End":"12:39.110","Text":"As usual, I often like to substitute a naught."},{"Start":"12:39.110 ","End":"12:40.655","Text":"Anything non-zero."},{"Start":"12:40.655 ","End":"12:43.340","Text":"I could choose 1 or I could choose minus 1."},{"Start":"12:43.340 ","End":"12:47.225","Text":"I\u0027ll just go with a naught being 1 and I can just put a line through it."},{"Start":"12:47.225 ","End":"12:51.395","Text":"Also notice and you might want to go back and look what y_1 was."},{"Start":"12:51.395 ","End":"12:58.275","Text":"But this x^2 together with this was actually what our y_1 was."},{"Start":"12:58.275 ","End":"13:04.580","Text":"It\u0027s simpler to write this first bit as y_1 times natural log of x."},{"Start":"13:04.580 ","End":"13:06.400","Text":"Then the second bit,"},{"Start":"13:06.400 ","End":"13:08.250","Text":"a naught here also."},{"Start":"13:08.250 ","End":"13:10.280","Text":"Just get x and then this series,"},{"Start":"13:10.280 ","End":"13:12.470","Text":"which is not clear what the pattern is."},{"Start":"13:12.470 ","End":"13:15.980","Text":"That\u0027s basically it. We found y_1 and y_2 here."},{"Start":"13:15.980 ","End":"13:17.840","Text":"Then we have to take a linear combination of them."},{"Start":"13:17.840 ","End":"13:20.600","Text":"I don\u0027t see any point in plugging y_1 and y_2 into this."},{"Start":"13:20.600 ","End":"13:24.960","Text":"We\u0027ll just leave it at that. We\u0027ve worked hard enough already. We are done."}],"Thumbnail":null,"ID":7893}],"ID":4244}]

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