Graphical Methods
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Numerical Methods
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[{"Name":"Graphical Methods","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Direction Fields","Duration":"18m 58s","ChapterTopicVideoID":25329,"CourseChapterTopicPlaylistID":27385,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/25329.jpeg","UploadDate":"2021-03-30T09:07:39.9470000","DurationForVideoObject":"PT18M58S","Description":null,"MetaTitle":"Direction Fields: Video + Workbook | Proprep","MetaDescription":"Graphical and Numerical Methods - Graphical Methods. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/ordinary-differential-equations/graphical-and-numerical-methods/graphical-methods/vid26146","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.175","Text":"Starting a new topic in ordinary differential equations,"},{"Start":"00:05.175 ","End":"00:10.020","Text":"graphical and numerical methods for solving them."},{"Start":"00:10.020 ","End":"00:20.895","Text":"It turns out most ordinary differential equations can\u0027t be solved analytically."},{"Start":"00:20.895 ","End":"00:25.620","Text":"We have to use graphical or numerical methods."},{"Start":"00:25.620 ","End":"00:32.325","Text":"But we\u0027re not going to consider the most general ordinary differential equations."},{"Start":"00:32.325 ","End":"00:42.250","Text":"Only ones of the form that y\u0027 is a function of x and y."},{"Start":"00:42.250 ","End":"00:46.780","Text":"A first order differential equation."},{"Start":"00:46.780 ","End":"00:51.674","Text":"f(x,y) is some continuous function."},{"Start":"00:51.674 ","End":"00:54.620","Text":"y\u0027 means derivative with respect to x,"},{"Start":"00:54.620 ","End":"00:57.785","Text":"so just in case you were wondering."},{"Start":"00:57.785 ","End":"01:02.220","Text":"That\u0027s the kind of equation we\u0027ll solve."},{"Start":"01:02.650 ","End":"01:07.310","Text":"You probably know that the general solution to"},{"Start":"01:07.310 ","End":"01:13.280","Text":"ordinary differential equations is typically a family of solutions."},{"Start":"01:13.280 ","End":"01:16.175","Text":"In other words, if we could solve it analytically it would have"},{"Start":"01:16.175 ","End":"01:21.770","Text":"a constant C in it unless we give it an initial condition."},{"Start":"01:21.770 ","End":"01:25.865","Text":"Then the ODE is called an IVP."},{"Start":"01:25.865 ","End":"01:30.154","Text":"We provide an initial condition and say that the solution"},{"Start":"01:30.154 ","End":"01:35.990","Text":"y(x) satisfies this condition that it passes through x_0,"},{"Start":"01:35.990 ","End":"01:40.535","Text":"y_0, or the solution at x_0 is equal to y_0."},{"Start":"01:40.535 ","End":"01:44.570","Text":"We give a point that the solution goes through in that"},{"Start":"01:44.570 ","End":"01:49.215","Text":"restricted to just 1 solution usually."},{"Start":"01:49.215 ","End":"01:51.660","Text":"I kind of repeated myself here."},{"Start":"01:51.660 ","End":"01:54.050","Text":"That\u0027s in the case of initial value problem."},{"Start":"01:54.050 ","End":"01:58.730","Text":"We want the solution that passes through the point x_0,"},{"Start":"01:58.730 ","End":"02:01.865","Text":"y_0, and only that solution."},{"Start":"02:01.865 ","End":"02:04.805","Text":"Here we\u0027ll learn 2 techniques,"},{"Start":"02:04.805 ","End":"02:09.680","Text":"one of them will be graphical and one of them will be numerical."},{"Start":"02:09.680 ","End":"02:12.860","Text":"There\u0027ll be a graphical method for solving"},{"Start":"02:12.860 ","End":"02:17.810","Text":"the general ODE using something called direction fields,"},{"Start":"02:17.810 ","End":"02:20.060","Text":"which we shall define."},{"Start":"02:20.060 ","End":"02:26.255","Text":"We\u0027ll also learn a numerical method for solving the initial value problem,"},{"Start":"02:26.255 ","End":"02:30.170","Text":"i.e the differential equation with initial condition."},{"Start":"02:30.170 ","End":"02:33.094","Text":"It\u0027s going to be called Euler\u0027s method,"},{"Start":"02:33.094 ","End":"02:36.520","Text":"named after the mathematician Euler."},{"Start":"02:36.520 ","End":"02:40.159","Text":"That was an introduction and next,"},{"Start":"02:40.159 ","End":"02:45.965","Text":"we\u0027ll go and look at the graphical method using direction fields."},{"Start":"02:45.965 ","End":"02:49.580","Text":"Now we come to the first technique,"},{"Start":"02:49.580 ","End":"02:57.020","Text":"a graphical technique for solving ODEs using"},{"Start":"02:57.020 ","End":"03:04.670","Text":"direction fields and we\u0027ll demonstrate the method using an example ODE."},{"Start":"03:04.670 ","End":"03:10.060","Text":"Remember in general, we wanted y\u0027=f(x,y)."},{"Start":"03:10.060 ","End":"03:17.710","Text":"In our case, we\u0027re taking f(x,y) to be a simple function, x minus y."},{"Start":"03:17.710 ","End":"03:22.214","Text":"Now, remember y\u0027 has a geometric meaning."},{"Start":"03:22.214 ","End":"03:26.720","Text":"y\u0027 at a given point,"},{"Start":"03:26.720 ","End":"03:32.486","Text":"x,y would represent the slope of the solution curve,"},{"Start":"03:32.486 ","End":"03:33.845","Text":"call it y(x),"},{"Start":"03:33.845 ","End":"03:35.860","Text":"it passes through it."},{"Start":"03:35.860 ","End":"03:39.920","Text":"Now we need to define a concept, a line element,"},{"Start":"03:39.920 ","End":"03:47.940","Text":"which is a short line segment beginning at the given point x,y with slope y\u0027."},{"Start":"03:47.940 ","End":"03:51.500","Text":"What it actually is is a little bit of the tangent line."},{"Start":"03:51.500 ","End":"03:58.280","Text":"As such, it\u0027s an approximation to the solution curve through the point."},{"Start":"03:58.280 ","End":"03:59.720","Text":"We take a little piece of"},{"Start":"03:59.720 ","End":"04:03.620","Text":"the tangent line at"},{"Start":"04:03.620 ","End":"04:05.359","Text":"the point to approximate"},{"Start":"04:05.359 ","End":"04:08.330","Text":"the solution and the shorter the line or the closer to the point,"},{"Start":"04:08.330 ","End":"04:10.555","Text":"the better the approximation."},{"Start":"04:10.555 ","End":"04:13.490","Text":"We\u0027ll soon draw a picture and it\u0027ll be clearer."},{"Start":"04:13.490 ","End":"04:16.084","Text":"Meanwhile, let me just introduce another definition."},{"Start":"04:16.084 ","End":"04:19.850","Text":"A direction field is a collection of a whole bunch of"},{"Start":"04:19.850 ","End":"04:24.860","Text":"these line elements sketched graphically."},{"Start":"04:24.860 ","End":"04:28.610","Text":"We\u0027ll see what I mean by a lot."},{"Start":"04:28.610 ","End":"04:30.320","Text":"Some of these terms are vague,"},{"Start":"04:30.320 ","End":"04:33.420","Text":"like what is short, what is the lot."},{"Start":"04:33.980 ","End":"04:36.865","Text":"Now, what we usually do,"},{"Start":"04:36.865 ","End":"04:41.420","Text":"say we\u0027re solving an exercise where we have to sketch the direction field"},{"Start":"04:41.420 ","End":"04:46.580","Text":"is we do a few of these line elements manually."},{"Start":"04:46.580 ","End":"04:47.870","Text":"I don\u0027t know how much is a few,"},{"Start":"04:47.870 ","End":"04:50.480","Text":"6, 8, 10 whatever."},{"Start":"04:50.480 ","End":"04:53.930","Text":"But to do a lot,"},{"Start":"04:53.930 ","End":"04:59.704","Text":"It\u0027s very tedious and so we usually bring in a computer sketch,"},{"Start":"04:59.704 ","End":"05:05.480","Text":"but we do a few manually just to show that we know what is going on."},{"Start":"05:05.480 ","End":"05:13.175","Text":"For example, I\u0027m going to choose 6 sample points and for each of these sample points,"},{"Start":"05:13.175 ","End":"05:21.060","Text":"x and y, we\u0027ll compute y\u0027 using f(x,y) which is x minus y."},{"Start":"05:21.060 ","End":"05:24.044","Text":"We\u0027ll have an x(y) and the slope."},{"Start":"05:24.044 ","End":"05:29.500","Text":"But the sketch, a slope is not as useful as an angle so"},{"Start":"05:29.500 ","End":"05:34.960","Text":"we\u0027ll use the arc-tangent of the slope to get the angle."},{"Start":"05:34.960 ","End":"05:40.100","Text":"Let\u0027s compute 6 points and then I\u0027ll graph them."},{"Start":"05:40.970 ","End":"05:47.510","Text":"I chose 3 points in the 1st quadrant."},{"Start":"05:47.510 ","End":"05:51.860","Text":"Then I chose a point in the 4th quadrant."},{"Start":"05:51.860 ","End":"05:56.610","Text":"Then I chose a point in the 3rd quadrant and a point in the 2nd quadrant."},{"Start":"05:56.610 ","End":"05:58.715","Text":"It doesn\u0027t really matter."},{"Start":"05:58.715 ","End":"06:03.425","Text":"We just want to show that you know how to do this manually."},{"Start":"06:03.425 ","End":"06:11.100","Text":"Now for each of these, we compute f(x,y) which is x minus y."},{"Start":"06:11.100 ","End":"06:13.590","Text":"1 minus 1 is 0,"},{"Start":"06:13.590 ","End":"06:16.425","Text":"1 minus 2 is negative 1,"},{"Start":"06:16.425 ","End":"06:18.660","Text":"2 minus 1 is 1,"},{"Start":"06:18.660 ","End":"06:21.495","Text":"this minus this is 2,"},{"Start":"06:21.495 ","End":"06:23.880","Text":"this minus this is 1,"},{"Start":"06:23.880 ","End":"06:27.145","Text":"and this minus this negative 3."},{"Start":"06:27.145 ","End":"06:32.325","Text":"Now, this is actually also equal to y\u0027,"},{"Start":"06:32.325 ","End":"06:34.140","Text":"which is the slope."},{"Start":"06:34.140 ","End":"06:36.560","Text":"To convert from slope to an angle,"},{"Start":"06:36.560 ","End":"06:42.680","Text":"we take the inverse tangent or arc-tangent and we\u0027ll work in degrees."},{"Start":"06:42.680 ","End":"06:47.660","Text":"The inverse tangent of 0 is just 0 degrees."},{"Start":"06:47.660 ","End":"06:49.565","Text":"For minus 1,"},{"Start":"06:49.565 ","End":"06:53.030","Text":"then we have minus 45 degrees."},{"Start":"06:53.030 ","End":"06:57.795","Text":"It\u0027s like a slope downwards, 45 degrees."},{"Start":"06:57.795 ","End":"07:04.370","Text":"Slope of 1 is exactly 45 degrees upwards."},{"Start":"07:04.370 ","End":"07:06.410","Text":"Slope of 2,"},{"Start":"07:06.410 ","End":"07:11.165","Text":"do the arc tangent comes out to about 63 degrees."},{"Start":"07:11.165 ","End":"07:14.502","Text":"Once again, we have 45 degrees."},{"Start":"07:14.502 ","End":"07:16.670","Text":"The minus 3 will be downwards."},{"Start":"07:16.670 ","End":"07:23.345","Text":"It\u0027ll be a negative and it happens to be minus 71 or 72 degrees."},{"Start":"07:23.345 ","End":"07:27.715","Text":"Here we have a piece of graph paper."},{"Start":"07:27.715 ","End":"07:33.230","Text":"We want to put these 6 line elements into the graph."},{"Start":"07:33.230 ","End":"07:38.060","Text":"I guess I better scroll a bit fast. Let\u0027s see."},{"Start":"07:38.060 ","End":"07:48.430","Text":"The first one is here at the point 1,1 with a slope of 0 degrees."},{"Start":"07:48.430 ","End":"07:52.933","Text":"The next one is 1,"},{"Start":"07:52.933 ","End":"07:57.140","Text":"2 with a slope of minus 45 degrees."},{"Start":"07:57.140 ","End":"08:02.610","Text":"Then we have 2,"},{"Start":"08:02.610 ","End":"08:07.275","Text":"1 with a slope of plus 45 degrees."},{"Start":"08:07.275 ","End":"08:12.540","Text":"Next, we have 1 minus 1."},{"Start":"08:12.540 ","End":"08:15.140","Text":"This will be 63 degrees."},{"Start":"08:15.140 ","End":"08:18.120","Text":"Well, obviously just estimating it."},{"Start":"08:18.920 ","End":"08:22.785","Text":"Then we want minus 1,"},{"Start":"08:22.785 ","End":"08:27.060","Text":"minus 2 with 45 degrees."},{"Start":"08:27.060 ","End":"08:31.200","Text":"Finally, minus 2,"},{"Start":"08:31.200 ","End":"08:36.850","Text":"1 with a slope of minus 71 degrees."},{"Start":"08:36.850 ","End":"08:41.410","Text":"I\u0027ve brought a computer picture."},{"Start":"08:41.410 ","End":"08:47.120","Text":"This is the direction field or parts of it for"},{"Start":"08:47.120 ","End":"08:54.830","Text":"the function f(x,y)=x minus y at each point x,y,"},{"Start":"08:54.830 ","End":"09:05.080","Text":"we compute x minus y and draw part of a line whose slope is x minus y."},{"Start":"09:05.080 ","End":"09:09.215","Text":"The software often puts arrows on."},{"Start":"09:09.215 ","End":"09:14.321","Text":"I guess they\u0027re not totally essential."},{"Start":"09:14.321 ","End":"09:19.930","Text":"Yeah, it\u0027s different than having just a point and a line sticking out,"},{"Start":"09:19.930 ","End":"09:23.270","Text":"put an arrow at the end is quite common."},{"Start":"09:23.310 ","End":"09:31.360","Text":"Then from here, we can either manually or computer-aided get some solutions."},{"Start":"09:31.360 ","End":"09:34.810","Text":"You get that basically by just following the arrows,"},{"Start":"09:34.810 ","End":"09:38.980","Text":"trying to get curves where these line elements,"},{"Start":"09:38.980 ","End":"09:41.095","Text":"we call them a tangent."},{"Start":"09:41.095 ","End":"09:43.930","Text":"Let\u0027s see if I can make an attempt,"},{"Start":"09:43.930 ","End":"09:47.870","Text":"starts something like this goes here,"},{"Start":"09:48.240 ","End":"09:53.215","Text":"something like that then maybe from here."},{"Start":"09:53.215 ","End":"09:58.060","Text":"I\u0027m just doing it really roughly just follow"},{"Start":"09:58.060 ","End":"10:05.110","Text":"the arrows and just giving you the general idea."},{"Start":"10:05.110 ","End":"10:12.650","Text":"From here maybe it looks like it\u0027s going this way, another one."},{"Start":"10:13.440 ","End":"10:19.880","Text":"It\u0027s a terrible job but that\u0027s why we have computer-aided."},{"Start":"10:20.040 ","End":"10:28.105","Text":"Here we have a much neater and more accurate picture."},{"Start":"10:28.105 ","End":"10:30.310","Text":"Here I messed up altogether,"},{"Start":"10:30.310 ","End":"10:34.070","Text":"you could see I should have been going downwards."},{"Start":"10:34.140 ","End":"10:43.360","Text":"Anyway, notice also that if I had an initial condition, for example,"},{"Start":"10:43.360 ","End":"10:49.450","Text":"if I said that y(0)=1,"},{"Start":"10:49.450 ","End":"10:51.820","Text":"that when x is 0,"},{"Start":"10:51.820 ","End":"10:53.275","Text":"y is 1,"},{"Start":"10:53.275 ","End":"10:56.690","Text":"then it passes through the point 0,"},{"Start":"10:56.690 ","End":"10:59.830","Text":"1, which is this point."},{"Start":"10:59.830 ","End":"11:09.590","Text":"Then the solution we want would be this particular solution here,"},{"Start":"11:11.790 ","End":"11:15.670","Text":"so that would be the solution that satisfies"},{"Start":"11:15.670 ","End":"11:19.630","Text":"the initial condition and then there\u0027s just 1 solution otherwise,"},{"Start":"11:19.630 ","End":"11:23.870","Text":"in general, we have a whole family of solutions."},{"Start":"11:23.970 ","End":"11:30.070","Text":"There are plenty of websites on the Internet which"},{"Start":"11:30.070 ","End":"11:35.650","Text":"will do this calculation,"},{"Start":"11:35.650 ","End":"11:42.880","Text":"free sites, you just feed in the information like the function is x minus y,"},{"Start":"11:42.880 ","End":"11:50.170","Text":"you tell it the range of x\u0027s and y\u0027s you want and various other parameters,"},{"Start":"11:50.170 ","End":"11:55.970","Text":"how big the arrows you want and so on and it does all this for you."},{"Start":"11:57.960 ","End":"12:02.275","Text":"If you want the general solution,"},{"Start":"12:02.275 ","End":"12:06.235","Text":"you could get a bunch of lines you can draw,"},{"Start":"12:06.235 ","End":"12:08.080","Text":"you usually just put"},{"Start":"12:08.080 ","End":"12:11.680","Text":"your cursor on a point that it will give the solution through that point,"},{"Start":"12:11.680 ","End":"12:18.520","Text":"you do a bunch of these but if you want just a particular solution,"},{"Start":"12:18.520 ","End":"12:21.145","Text":"initial value problem with initial condition,"},{"Start":"12:21.145 ","End":"12:24.160","Text":"then you would just say click at this point and it"},{"Start":"12:24.160 ","End":"12:28.120","Text":"would just draw this solution for you,"},{"Start":"12:28.120 ","End":"12:36.040","Text":"so it\u0027s both the general problem and the initial value problem done graphically."},{"Start":"12:36.040 ","End":"12:40.630","Text":"There\u0027s one more topic I\u0027d like to talk about."},{"Start":"12:40.630 ","End":"12:48.880","Text":"I want to show you what are isoclines and what they\u0027re good for."},{"Start":"12:48.880 ","End":"12:56.980","Text":"Mainly if you don\u0027t have access to computer-aided sketching of direction fields,"},{"Start":"12:56.980 ","End":"13:02.170","Text":"how would you go about other than tediously one-by-one,"},{"Start":"13:02.170 ","End":"13:05.815","Text":"sketching a whole bunch of little line elements everywhere?"},{"Start":"13:05.815 ","End":"13:11.950","Text":"There is a shortcut that we can use if we don\u0027t have the aid of a computer."},{"Start":"13:11.950 ","End":"13:16.300","Text":"Let me just go back a bit here,"},{"Start":"13:16.300 ","End":"13:18.580","Text":"let me tidy up a bit,"},{"Start":"13:18.580 ","End":"13:20.515","Text":"and move that there."},{"Start":"13:20.515 ","End":"13:23.035","Text":"Let me move this,"},{"Start":"13:23.035 ","End":"13:26.240","Text":"I\u0027ll put that over here."},{"Start":"13:28.080 ","End":"13:34.420","Text":"What we want to do for isoclines is to let"},{"Start":"13:34.420 ","End":"13:40.645","Text":"this f( x,y) equal a constant c. Now,"},{"Start":"13:40.645 ","End":"13:42.610","Text":"iso means the same, incline,"},{"Start":"13:42.610 ","End":"13:48.220","Text":"I guess comes from an incline or a slope so it means like same slope."},{"Start":"13:48.220 ","End":"13:56.070","Text":"I\u0027ll just write that close to what it means, Greek."},{"Start":"13:56.070 ","End":"14:05.320","Text":"If x minus y=c then everywhere along this graph will be a straight line, in fact,"},{"Start":"14:05.320 ","End":"14:13.090","Text":"we will have the same slope c. Let\u0027s see,"},{"Start":"14:13.090 ","End":"14:18.070","Text":"put y here let\u0027s see if their y=x minus c. Now,"},{"Start":"14:18.070 ","End":"14:22.480","Text":"we can take different values of c, so for example,"},{"Start":"14:22.480 ","End":"14:27.550","Text":"if I take c=0,"},{"Start":"14:27.550 ","End":"14:33.760","Text":"then I will get the line y=x."},{"Start":"14:33.760 ","End":"14:35.470","Text":"Here we are, y=x,"},{"Start":"14:35.470 ","End":"14:37.945","Text":"so this is the isoclines,"},{"Start":"14:37.945 ","End":"14:46.540","Text":"c=0 and that means that the slope is 0 everywhere along here."},{"Start":"14:46.540 ","End":"14:52.045","Text":"Notice that we actually have 1 line element already here with slope equals 0,"},{"Start":"14:52.045 ","End":"14:58.135","Text":"so let\u0027s put in some more like here and here"},{"Start":"14:58.135 ","End":"15:06.430","Text":"and here and a bunch more all along here,"},{"Start":"15:06.430 ","End":"15:14.545","Text":"so that gives us quite a few more line elements."},{"Start":"15:14.545 ","End":"15:20.920","Text":"Now, let\u0027s take another value of c. Let\u0027s take c=1,"},{"Start":"15:20.920 ","End":"15:22.240","Text":"what does that give us?"},{"Start":"15:22.240 ","End":"15:27.715","Text":"Y=x minus c, so y=x minus 1."},{"Start":"15:27.715 ","End":"15:29.935","Text":"Here\u0027s what it looks like."},{"Start":"15:29.935 ","End":"15:32.500","Text":"Here we have c=1."},{"Start":"15:32.500 ","End":"15:38.470","Text":"That means that the slope of the line elements,"},{"Start":"15:38.470 ","End":"15:42.370","Text":"1, we have a couple of them already here,"},{"Start":"15:42.370 ","End":"15:47.780","Text":"it\u0027s a bit sloppily drawn but we can now draw a few more."},{"Start":"15:48.120 ","End":"15:52.540","Text":"Here\u0027s a line element and here\u0027s the line element,"},{"Start":"15:52.540 ","End":"15:54.460","Text":"they don\u0027t have to start from the grid points."},{"Start":"15:54.460 ","End":"15:56.380","Text":"I could put a line element here,"},{"Start":"15:56.380 ","End":"15:58.930","Text":"and I could put a line element here,"},{"Start":"15:58.930 ","End":"16:03.550","Text":"I could put a line element here, and so on."},{"Start":"16:03.550 ","End":"16:07.375","Text":"Anyway, we want more get the idea."},{"Start":"16:07.375 ","End":"16:10.270","Text":"Now, let\u0027s take another value of c,"},{"Start":"16:10.270 ","End":"16:13.180","Text":"c equals minus 1,"},{"Start":"16:13.180 ","End":"16:18.040","Text":"that gives us y=x plus 1."},{"Start":"16:18.040 ","End":"16:22.990","Text":"On this isoclines, c equals minus 1 and that"},{"Start":"16:22.990 ","End":"16:26.380","Text":"means that the slope is minus 1 and"},{"Start":"16:26.380 ","End":"16:30.040","Text":"we actually have 1 here: That 1 came from the point 1,"},{"Start":"16:30.040 ","End":"16:33.280","Text":"2, which was this and we already see that,"},{"Start":"16:33.280 ","End":"16:38.185","Text":"so we can just go along and put bunch more."},{"Start":"16:38.185 ","End":"16:43.720","Text":"We could go all along and get a lot of these of slope minus"},{"Start":"16:43.720 ","End":"16:51.410","Text":"1 just as densely or as you like."},{"Start":"16:53.370 ","End":"16:55.810","Text":"We\u0027ll do a couple of others,"},{"Start":"16:55.810 ","End":"16:59.620","Text":"maybe just inspired by the points that we have."},{"Start":"16:59.620 ","End":"17:06.790","Text":"This point was 1 minus 1,"},{"Start":"17:06.790 ","End":"17:09.640","Text":"and its slope was 2,"},{"Start":"17:09.640 ","End":"17:12.895","Text":"so let\u0027s take c=2,"},{"Start":"17:12.895 ","End":"17:17.305","Text":"that gives us y=x minus 2;"},{"Start":"17:17.305 ","End":"17:21.205","Text":"let\u0027s mark it c=2."},{"Start":"17:21.205 ","End":"17:24.400","Text":"Then just like this: This was badly drawn,"},{"Start":"17:24.400 ","End":"17:27.760","Text":"should be a bit steeper, like 63 degrees,"},{"Start":"17:27.760 ","End":"17:30.550","Text":"so a bit steeper than that,"},{"Start":"17:30.550 ","End":"17:35.560","Text":"maybe it\u0027s hard to say what\u0027s exactly 60 degrees."},{"Start":"17:35.560 ","End":"17:40.060","Text":"We\u0027re doing this, remember without computer aid."},{"Start":"17:40.060 ","End":"17:45.175","Text":"Last one was this one here,"},{"Start":"17:45.175 ","End":"17:48.190","Text":"was from this point minus 2, 1,"},{"Start":"17:48.190 ","End":"17:52.150","Text":"we want a slope of minus 3."},{"Start":"17:52.150 ","End":"17:55.060","Text":"If c is minus 3,"},{"Start":"17:55.060 ","End":"18:00.850","Text":"the isoclines is y=x plus 3,"},{"Start":"18:00.850 ","End":"18:06.460","Text":"c equals minus 3 and on this isoclines,"},{"Start":"18:06.460 ","End":"18:07.870","Text":"all the slopes are minus 3,"},{"Start":"18:07.870 ","End":"18:11.785","Text":"which has an angle of minus 70 something degrees."},{"Start":"18:11.785 ","End":"18:18.420","Text":"We could just guess what is minus 70 something degrees,"},{"Start":"18:18.420 ","End":"18:26.495","Text":"we get a whole bunch more of these line elements."},{"Start":"18:26.495 ","End":"18:28.480","Text":"So you get the idea,"},{"Start":"18:28.480 ","End":"18:32.800","Text":"you can draw more isoclines on and you can populate the more"},{"Start":"18:32.800 ","End":"18:38.020","Text":"densely and that often saves you"},{"Start":"18:38.020 ","End":"18:42.950","Text":"from doing it computer-aided although usually"},{"Start":"18:42.950 ","End":"18:49.290","Text":"even this is too tedious and we do everything computer-aided."},{"Start":"18:49.530 ","End":"18:57.860","Text":"That was isoclines and that concludes the topic of direction fields."}],"ID":26146},{"Watched":false,"Name":"Exercise 1","Duration":"8m 48s","ChapterTopicVideoID":10484,"CourseChapterTopicPlaylistID":27385,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.280","Text":"In this exercise, we have a differential equation, y\u0027=x plus y."},{"Start":"00:06.280 ","End":"00:10.225","Text":"It\u0027s of the form y\u0027=f(x, y)."},{"Start":"00:10.225 ","End":"00:17.590","Text":"The idea here is just to do some manual computations to sketch some line elements,"},{"Start":"00:17.590 ","End":"00:20.004","Text":"for the points x, y,"},{"Start":"00:20.004 ","End":"00:26.850","Text":"where x and y can be anything from minus 2-2 whole numbers."},{"Start":"00:26.850 ","End":"00:31.980","Text":"There\u0027s 5 times 5 possibilities as 25 points altogether."},{"Start":"00:31.980 ","End":"00:35.290","Text":"After we\u0027ve sketched those 25 line elements"},{"Start":"00:35.290 ","End":"00:40.205","Text":"then compare with a computer aided sketch of the direction field."},{"Start":"00:40.205 ","End":"00:42.525","Text":"I made a table."},{"Start":"00:42.525 ","End":"00:45.465","Text":"This is x and this is y,"},{"Start":"00:45.465 ","End":"00:49.425","Text":"each of them from minus 2-2,"},{"Start":"00:49.425 ","End":"00:54.010","Text":"like here, 25 combinations."},{"Start":"00:54.590 ","End":"01:01.335","Text":"Here I put x plus y minus 2 and minus 2 is minus 4,"},{"Start":"01:01.335 ","End":"01:05.445","Text":"minus 1, and minus 2 is minus 3, and so on."},{"Start":"01:05.445 ","End":"01:10.595","Text":"Like here, 0 and 0 is 0 or 1 and 1 is 2, and so on."},{"Start":"01:10.595 ","End":"01:11.870","Text":"I just filled it all in."},{"Start":"01:11.870 ","End":"01:13.160","Text":"It\u0027s just routine."},{"Start":"01:13.160 ","End":"01:15.740","Text":"Just put in x plus y in each square."},{"Start":"01:15.740 ","End":"01:19.520","Text":"Now, this is y\u0027 which is the slope."},{"Start":"01:19.520 ","End":"01:21.410","Text":"But when we sketch,"},{"Start":"01:21.410 ","End":"01:26.190","Text":"we prefer to have an angle in degrees."},{"Start":"01:26.190 ","End":"01:28.015","Text":"It\u0027s easier to sketch."},{"Start":"01:28.015 ","End":"01:30.860","Text":"What we do is for each of these lookup,"},{"Start":"01:30.860 ","End":"01:34.320","Text":"the inverse tangent or arc tangent."},{"Start":"01:34.320 ","End":"01:40.160","Text":"The arc tangent of minus 4 comes out around it off to the nearest degree,"},{"Start":"01:40.160 ","End":"01:43.700","Text":"and that\u0027s enough precision for us."},{"Start":"01:43.700 ","End":"01:49.085","Text":"Arc tangent of minus 3 is minus 72 degrees."},{"Start":"01:49.085 ","End":"01:51.620","Text":"Here we have minus 63 degrees."},{"Start":"01:51.620 ","End":"01:57.905","Text":"Minus 1 is minus 45 degrees on for 0 it\u0027s just 0."},{"Start":"01:57.905 ","End":"01:59.660","Text":"Now some of them repeat."},{"Start":"01:59.660 ","End":"02:01.820","Text":"I can do this 0,"},{"Start":"02:01.820 ","End":"02:05.970","Text":"this 0, this 0 and this 0."},{"Start":"02:08.030 ","End":"02:16.215","Text":"Also I see that the minus 2 here and here."},{"Start":"02:16.215 ","End":"02:20.720","Text":"In fact we can also do it for 2 because it\u0027s just the negative."},{"Start":"02:20.720 ","End":"02:23.780","Text":"So here we have 63 and 63 degrees."},{"Start":"02:23.780 ","End":"02:26.850","Text":"No need to go to the calculator again."},{"Start":"02:27.470 ","End":"02:33.350","Text":"Where we have a minus 4 is minus 76 degrees,"},{"Start":"02:33.350 ","End":"02:36.005","Text":"so here we have 76 degrees."},{"Start":"02:36.005 ","End":"02:41.225","Text":"Minus 3 is 72 degrees, minus."},{"Start":"02:41.225 ","End":"02:43.250","Text":"Here would be with a plus,"},{"Start":"02:43.250 ","End":"02:47.330","Text":"and here it would be with a plus."},{"Start":"02:47.330 ","End":"02:49.850","Text":"I think that\u0027s all the 3\u0027s."},{"Start":"02:49.850 ","End":"02:53.800","Text":"Minus 1 is minus 45 degrees,"},{"Start":"02:53.800 ","End":"03:00.090","Text":"and plus 1 inverse tangent is 45 degrees."},{"Start":"03:00.090 ","End":"03:06.710","Text":"Here, 45 degrees and 2 is 63."},{"Start":"03:06.710 ","End":"03:17.490","Text":"Here we have another 1 which is 45 degrees and a minus 45 degrees. That\u0027s it."},{"Start":"03:19.400 ","End":"03:23.295","Text":"Next I want a bit of graph paper."},{"Start":"03:23.295 ","End":"03:26.280","Text":"Here\u0027s a bit of graph."},{"Start":"03:26.280 ","End":"03:34.360","Text":"That\u0027s the y and that\u0027s the x."},{"Start":"03:34.360 ","End":"03:38.840","Text":"Let\u0027s take for example, this point here,"},{"Start":"03:38.840 ","End":"03:40.505","Text":"minus 1, 1,"},{"Start":"03:40.505 ","End":"03:44.195","Text":"x is minus 1, y is 1."},{"Start":"03:44.195 ","End":"03:47.090","Text":"I see it\u0027s 0 degrees."},{"Start":"03:47.090 ","End":"03:51.100","Text":"Now we learned to put an arrow,"},{"Start":"03:51.620 ","End":"04:01.535","Text":"like we mark the point and put an arrow at 0 degrees."},{"Start":"04:01.535 ","End":"04:03.835","Text":"What I want to say is that,"},{"Start":"04:03.835 ","End":"04:07.565","Text":"it\u0027s not quite standard the way we draw this."},{"Start":"04:07.565 ","End":"04:12.035","Text":"Sometimes we put the arrow symmetrically."},{"Start":"04:12.035 ","End":"04:16.770","Text":"You could put the segment here."},{"Start":"04:16.910 ","End":"04:21.680","Text":"You could draw it like this and the arrow itself is also optional."},{"Start":"04:21.680 ","End":"04:24.230","Text":"Usually when the computer draws it,"},{"Start":"04:24.230 ","End":"04:27.905","Text":"it puts it symmetrically."},{"Start":"04:27.905 ","End":"04:34.325","Text":"It also doesn\u0027t show the point, possibly like this."},{"Start":"04:34.325 ","End":"04:39.034","Text":"Here I adjusted down some of the variations for line elements."},{"Start":"04:39.034 ","End":"04:43.820","Text":"It could be with the point at the end or the point in the middle."},{"Start":"04:43.820 ","End":"04:45.800","Text":"The point itself is optional."},{"Start":"04:45.800 ","End":"04:47.300","Text":"You could throw it out."},{"Start":"04:47.300 ","End":"04:49.850","Text":"The arrow is also optional."},{"Start":"04:49.850 ","End":"04:51.380","Text":"This is what I just said,"},{"Start":"04:51.380 ","End":"04:52.850","Text":"middle or end point."},{"Start":"04:52.850 ","End":"04:55.220","Text":"I think I\u0027ll continue doing them this way with"},{"Start":"04:55.220 ","End":"04:59.045","Text":"the arrow and without the point and in the middle."},{"Start":"04:59.045 ","End":"05:01.565","Text":"Let\u0027s add a bit of color."},{"Start":"05:01.565 ","End":"05:06.305","Text":"What I\u0027m going to do now is the first row here,"},{"Start":"05:06.305 ","End":"05:11.290","Text":"where x is minus 2 and y goes from minus 2 to 2,"},{"Start":"05:11.290 ","End":"05:12.850","Text":"so we\u0027re going up here,"},{"Start":"05:12.850 ","End":"05:16.975","Text":"so we have minus 76 degrees."},{"Start":"05:16.975 ","End":"05:19.610","Text":"It\u0027s hard to say exactly,"},{"Start":"05:19.610 ","End":"05:28.469","Text":"but let\u0027s say minus 76 degrees then minus 72 degrees, not as steep."},{"Start":"05:28.600 ","End":"05:33.110","Text":"Hard to say, minus 63 degrees,"},{"Start":"05:33.110 ","End":"05:34.775","Text":"a bit less deep,"},{"Start":"05:34.775 ","End":"05:40.820","Text":"minus 45 degrees, so that\u0027s something like that."},{"Start":"05:40.820 ","End":"05:45.620","Text":"Then 0 degrees, so that\u0027s horizontal."},{"Start":"05:45.620 ","End":"05:50.330","Text":"Then let\u0027s go for the second row,"},{"Start":"05:50.330 ","End":"05:52.205","Text":"which will be the minus 1,"},{"Start":"05:52.205 ","End":"05:54.275","Text":"working our way up."},{"Start":"05:54.275 ","End":"06:01.980","Text":"Minus 72 degrees, minus 63 degrees."},{"Start":"06:01.980 ","End":"06:05.320","Text":"It really hard to guess."},{"Start":"06:05.600 ","End":"06:09.570","Text":"Minus 45, 0 and 45,"},{"Start":"06:09.570 ","End":"06:12.120","Text":"so there\u0027s minus 45,"},{"Start":"06:12.120 ","End":"06:13.905","Text":"this one we had already,"},{"Start":"06:13.905 ","End":"06:16.725","Text":"and then 45 degrees."},{"Start":"06:16.725 ","End":"06:22.565","Text":"Then the next one where x is 0,"},{"Start":"06:22.565 ","End":"06:26.330","Text":"we\u0027ve got minus 63 degrees,"},{"Start":"06:26.330 ","End":"06:34.560","Text":"minus 45 degrees, 0,"},{"Start":"06:34.560 ","End":"06:40.500","Text":"then 45 degrees, and then 63 degrees."},{"Start":"06:40.500 ","End":"06:43.110","Text":"The next row."},{"Start":"06:43.110 ","End":"06:46.170","Text":"Let\u0027s say that\u0027s our x is 1."},{"Start":"06:46.170 ","End":"06:48.555","Text":"We\u0027re working our way up these 5 points,"},{"Start":"06:48.555 ","End":"06:55.470","Text":"and we start from minus 45 degrees,"},{"Start":"06:55.470 ","End":"07:03.030","Text":"then 0,"},{"Start":"07:03.030 ","End":"07:12.070","Text":"then plus 45 degrees, 63 degrees, 70."},{"Start":"07:17.870 ","End":"07:23.845","Text":"This is just to give you an idea of how tedious it is to do it manually."},{"Start":"07:23.845 ","End":"07:25.560","Text":"That\u0027s why we have computer, I think."},{"Start":"07:25.560 ","End":"07:27.465","Text":"We got one more route to go."},{"Start":"07:27.465 ","End":"07:32.575","Text":"That\u0027s this one where we start off here at 0 degrees,"},{"Start":"07:32.575 ","End":"07:34.300","Text":"then 45 degrees,"},{"Start":"07:34.300 ","End":"07:39.610","Text":"I\u0027m reading these, 63 degrees,"},{"Start":"07:39.610 ","End":"07:45.085","Text":"72 degrees, 76 degrees."},{"Start":"07:45.085 ","End":"07:48.680","Text":"That\u0027s the manual part."},{"Start":"07:49.290 ","End":"07:54.380","Text":"Here\u0027s the computer aided."},{"Start":"07:54.590 ","End":"07:57.910","Text":"In my program I can set the length of the arrows."},{"Start":"07:57.910 ","End":"08:01.240","Text":"I should have made them longer to make it look more like this."},{"Start":"08:01.240 ","End":"08:06.290","Text":"But you can see that we generally got the same thing."},{"Start":"08:06.290 ","End":"08:10.405","Text":"To tell the truth my compute aided sketch"},{"Start":"08:10.405 ","End":"08:20.215","Text":"actually was like this and I just superimposed it on the graph paper."},{"Start":"08:20.215 ","End":"08:24.220","Text":"I didn\u0027t do another sketch though."},{"Start":"08:24.220 ","End":"08:31.675","Text":"When I increased the number of arrows and then I got something like this."},{"Start":"08:31.675 ","End":"08:36.380","Text":"We can get much more dense or more precision."},{"Start":"08:36.380 ","End":"08:37.870","Text":"If your computer aided,"},{"Start":"08:37.870 ","End":"08:42.590","Text":"why just do 25 points when you can get a lot more?"},{"Start":"08:44.010 ","End":"08:48.320","Text":"I think that\u0027s it."}],"ID":10848},{"Watched":false,"Name":"Exercise 2","Duration":"5m 25s","ChapterTopicVideoID":10485,"CourseChapterTopicPlaylistID":27385,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.115","Text":"Here we have an exercise, it\u0027s about isoclines."},{"Start":"00:05.115 ","End":"00:07.860","Text":"We have the differential equation,"},{"Start":"00:07.860 ","End":"00:11.550","Text":"I believe this was the same equation as in the previous exercise."},{"Start":"00:11.550 ","End":"00:17.550","Text":"To describe the shape or the equation of a general isocline"},{"Start":"00:17.550 ","End":"00:24.750","Text":"associated with this differential equation and to sketch a few isoclines, say 3 of them."},{"Start":"00:24.750 ","End":"00:32.110","Text":"In part b, we have to draw a few line elements on each of these isoclines."},{"Start":"00:32.620 ","End":"00:36.450","Text":"Then just to compare what we got with"},{"Start":"00:36.450 ","End":"00:42.090","Text":"the computer-aided sketch of the direction field to see that we are on the right track."},{"Start":"00:43.490 ","End":"00:47.060","Text":"Now an isoclines is where the slope,"},{"Start":"00:47.060 ","End":"00:49.985","Text":"this y prime is constant."},{"Start":"00:49.985 ","End":"00:59.255","Text":"We just replace y prime by a constant c. This is the general form of an isocline."},{"Start":"00:59.255 ","End":"01:01.580","Text":"Now we want to sketch,"},{"Start":"01:01.580 ","End":"01:03.905","Text":"say, 3 of these."},{"Start":"01:03.905 ","End":"01:07.140","Text":"Let\u0027s choose, say,"},{"Start":"01:07.140 ","End":"01:09.510","Text":"c=0, then 1,"},{"Start":"01:09.510 ","End":"01:11.995","Text":"and then minus 1."},{"Start":"01:11.995 ","End":"01:15.830","Text":"That gives us x plus y=0,"},{"Start":"01:15.830 ","End":"01:18.350","Text":"then 1 and then minus 1."},{"Start":"01:18.350 ","End":"01:22.670","Text":"We could put y in terms of x. I don\u0027t think there\u0027s any need to."},{"Start":"01:22.670 ","End":"01:26.749","Text":"I think it\u0027s fairly straightforward to sketch each of these."},{"Start":"01:26.749 ","End":"01:30.665","Text":"The first one, x plus y=0,"},{"Start":"01:30.665 ","End":"01:34.580","Text":"is y equals minus x."},{"Start":"01:34.580 ","End":"01:37.720","Text":"You can just mentally do a few points."},{"Start":"01:37.720 ","End":"01:40.050","Text":"When x is 1, y is minus 1,"},{"Start":"01:40.050 ","End":"01:43.240","Text":"when x is 2, y is minus 2 and so on."},{"Start":"01:43.240 ","End":"01:45.950","Text":"This is the line,"},{"Start":"01:45.950 ","End":"01:48.130","Text":"it passes through the origin."},{"Start":"01:48.130 ","End":"01:52.915","Text":"The next one, x plus y=1."},{"Start":"01:52.915 ","End":"01:54.710","Text":"This one we could do with intercepts."},{"Start":"01:54.710 ","End":"01:58.880","Text":"When x is 0, y is 1 and when y is 0,"},{"Start":"01:58.880 ","End":"02:02.705","Text":"x is 1. Here it is."},{"Start":"02:02.705 ","End":"02:03.770","Text":"Oh, I should have labeled them."},{"Start":"02:03.770 ","End":"02:06.065","Text":"This is where c is 0."},{"Start":"02:06.065 ","End":"02:09.125","Text":"This is where c=1."},{"Start":"02:09.125 ","End":"02:13.100","Text":"Next we need where c is minus 1."},{"Start":"02:13.100 ","End":"02:17.300","Text":"Here we are again using the intercepts method when x is 0,"},{"Start":"02:17.300 ","End":"02:19.790","Text":"y is minus 1 and vice versa."},{"Start":"02:19.790 ","End":"02:24.125","Text":"This is where c is minus 1."},{"Start":"02:24.125 ","End":"02:32.870","Text":"Now, c would be the slope of the line elements along the isocline."},{"Start":"02:32.870 ","End":"02:35.270","Text":"But we don\u0027t want the slope,"},{"Start":"02:35.270 ","End":"02:37.280","Text":"we want the angle."},{"Start":"02:37.280 ","End":"02:41.715","Text":"It\u0027s easier to imagine to sketch."},{"Start":"02:41.715 ","End":"02:46.430","Text":"We do that with the inverse tangent or arc tangent of the slope,"},{"Start":"02:46.430 ","End":"02:48.110","Text":"we get the angle."},{"Start":"02:48.110 ","End":"02:51.410","Text":"For 0, we get 0 degrees."},{"Start":"02:51.410 ","End":"02:55.980","Text":"Slope of 1 is 45 degrees."},{"Start":"02:55.990 ","End":"03:02.405","Text":"The inverse tangent of minus 1 would be minus 45 degrees."},{"Start":"03:02.405 ","End":"03:04.520","Text":"Let\u0027s draw some line elements."},{"Start":"03:04.520 ","End":"03:09.894","Text":"The first one, this is the one slope 0."},{"Start":"03:09.894 ","End":"03:18.140","Text":"Angle 0, which is positive x-direction here and here."},{"Start":"03:18.140 ","End":"03:21.300","Text":"I won\u0027t bother with the arrow."},{"Start":"03:23.900 ","End":"03:26.780","Text":"We can put some in the middle also,"},{"Start":"03:26.780 ","End":"03:32.160","Text":"they\u0027re all going to be slope 0, which means flat."},{"Start":"03:35.150 ","End":"03:41.625","Text":"That will do. Change your mind put some arrows on."},{"Start":"03:41.625 ","End":"03:48.300","Text":"C=1 slope of 1 angle of 45 degrees."},{"Start":"03:48.300 ","End":"03:53.330","Text":"Maybe like this, and like this,"},{"Start":"03:53.330 ","End":"03:57.275","Text":"this just however many."},{"Start":"03:57.275 ","End":"04:03.165","Text":"Last one and added some arrows."},{"Start":"04:03.165 ","End":"04:06.120","Text":"Minus 1 slope,"},{"Start":"04:06.120 ","End":"04:09.989","Text":"it means sloping downwards 45 degrees."},{"Start":"04:09.989 ","End":"04:18.330","Text":"In this case, it actually comes out parallel to the isocline itself. A few more."},{"Start":"04:21.250 ","End":"04:26.540","Text":"That will do. Here we are,"},{"Start":"04:26.540 ","End":"04:29.150","Text":"we added some arrows at the end."},{"Start":"04:29.150 ","End":"04:31.265","Text":"These are 3 isoclines."},{"Start":"04:31.265 ","End":"04:40.295","Text":"Now let\u0027s just see that this looks something like the computer-aided direction field."},{"Start":"04:40.295 ","End":"04:43.910","Text":"Here we are. Yes, it makes sense."},{"Start":"04:43.910 ","End":"04:47.120","Text":"Let\u0027s take the isocline c=0."},{"Start":"04:47.120 ","End":"04:48.380","Text":"The slope is 0."},{"Start":"04:48.380 ","End":"04:51.020","Text":"Look this arrow, this arrow,"},{"Start":"04:51.020 ","End":"04:56.255","Text":"this arrow, this arrow and this arrow, they correspond."},{"Start":"04:56.255 ","End":"05:02.440","Text":"Let\u0027s take the isoclines c=1."},{"Start":"05:02.440 ","End":"05:05.930","Text":"You can see that we\u0027ve got this, this,"},{"Start":"05:05.930 ","End":"05:12.200","Text":"this, this that matches."},{"Start":"05:12.200 ","End":"05:16.130","Text":"Also on the isoclines where the slope is minus 1,"},{"Start":"05:16.130 ","End":"05:18.350","Text":"we have this, we have this,"},{"Start":"05:18.350 ","End":"05:20.210","Text":"we have this, we have this."},{"Start":"05:20.210 ","End":"05:23.135","Text":"It looks like it corresponds."},{"Start":"05:23.135 ","End":"05:25.620","Text":"We\u0027re done."}],"ID":10849},{"Watched":false,"Name":"Exercise 3","Duration":"3m 12s","ChapterTopicVideoID":10486,"CourseChapterTopicPlaylistID":27385,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.320 ","End":"00:07.920","Text":"In this exercise, we have the familiar differential equation, y\u0027=x plus y."},{"Start":"00:07.920 ","End":"00:11.970","Text":"We\u0027ve been using this in previous exercises."},{"Start":"00:11.970 ","End":"00:19.410","Text":"First part is to bring in a computer aided sketch of the direction field."},{"Start":"00:19.410 ","End":"00:22.440","Text":"Here it is already."},{"Start":"00:22.440 ","End":"00:24.630","Text":"In the second part,"},{"Start":"00:24.630 ","End":"00:28.545","Text":"we have to manually sketch solutions."},{"Start":"00:28.545 ","End":"00:35.235","Text":"There\u0027s going to be 3 of them passing respectively through these 3 points."},{"Start":"00:35.235 ","End":"00:40.185","Text":"The points are the origins, 0,0 is here."},{"Start":"00:40.185 ","End":"00:43.485","Text":"Next one is 0,1,"},{"Start":"00:43.485 ","End":"00:45.945","Text":"Which would be here."},{"Start":"00:45.945 ","End":"00:49.845","Text":"The third one 0,"},{"Start":"00:49.845 ","End":"00:52.915","Text":"minus 1 is here."},{"Start":"00:52.915 ","End":"00:58.580","Text":"Now, if we compute the slopes at each of these,"},{"Start":"00:58.580 ","End":"01:01.190","Text":"remember it was x plus y."},{"Start":"01:01.190 ","End":"01:04.550","Text":"0 plus 0 is 0."},{"Start":"01:04.550 ","End":"01:06.738","Text":"Here, 0 plus 1 is 1, 0 plus,"},{"Start":"01:06.738 ","End":"01:11.970","Text":"minus 1 is minus 1."},{"Start":"01:11.970 ","End":"01:15.490","Text":"If we do that in degrees,"},{"Start":"01:15.490 ","End":"01:20.060","Text":"then this one comes out to be 0 degrees."},{"Start":"01:20.060 ","End":"01:24.110","Text":"Arc tangent of 1 is 45 degrees."},{"Start":"01:24.110 ","End":"01:28.710","Text":"For minus 1 it\u0027s minus 45 degrees."},{"Start":"01:28.960 ","End":"01:31.250","Text":"We don\u0027t have to do this part,"},{"Start":"01:31.250 ","End":"01:38.240","Text":"but I just find it helpful to just to put line segments through these initial points."},{"Start":"01:38.240 ","End":"01:44.370","Text":"0 degrees is horizontal,"},{"Start":"01:44.830 ","End":"01:50.735","Text":"45 degrees is diagonal like this,"},{"Start":"01:50.735 ","End":"01:55.340","Text":"and minus 45 degrees like this."},{"Start":"01:55.340 ","End":"01:57.335","Text":"Just to get an idea,"},{"Start":"01:57.335 ","End":"02:02.750","Text":"so let\u0027s try to do this manually and it\u0027s not going to be exact, of course."},{"Start":"02:02.750 ","End":"02:06.660","Text":"Let\u0027s see if we can follow the arrows here."},{"Start":"02:06.760 ","End":"02:14.585","Text":"This may be then on the other side back like this, something like that."},{"Start":"02:14.585 ","End":"02:20.465","Text":"Here we start horizontally but we start going upwards."},{"Start":"02:20.465 ","End":"02:24.500","Text":"Here, something like this."},{"Start":"02:24.500 ","End":"02:33.995","Text":"Here, it looks like it\u0027s going to be 45 degrees negative all the way."},{"Start":"02:33.995 ","End":"02:40.960","Text":"So perhaps something like that."},{"Start":"02:41.410 ","End":"02:45.395","Text":"As far as the rough sketch that will do."},{"Start":"02:45.395 ","End":"02:47.240","Text":"But just to show you,"},{"Start":"02:47.240 ","End":"02:51.259","Text":"I also brought in the computer aided."},{"Start":"02:51.259 ","End":"02:59.480","Text":"There are computer aided calculators on the web that not only give you a direction field,"},{"Start":"02:59.480 ","End":"03:02.420","Text":"but if you touch at a given point,"},{"Start":"03:02.420 ","End":"03:03.560","Text":"like here, here or here,"},{"Start":"03:03.560 ","End":"03:08.165","Text":"it gives you the solution that goes through that point."},{"Start":"03:08.165 ","End":"03:13.050","Text":"So we didn\u0027t do too badly. That\u0027s it."}],"ID":10850},{"Watched":false,"Name":"Exercise 4","Duration":"8m 28s","ChapterTopicVideoID":10487,"CourseChapterTopicPlaylistID":27385,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.290 ","End":"00:05.625","Text":"In this exercise, we\u0027re given a differential equation,"},{"Start":"00:05.625 ","End":"00:08.580","Text":"y\u0027 is some function of x and y,"},{"Start":"00:08.580 ","End":"00:12.645","Text":"in this case, x^2 plus y^2 minus 2."},{"Start":"00:12.645 ","End":"00:14.310","Text":"There are several parts, first of all,"},{"Start":"00:14.310 ","End":"00:17.670","Text":"to describe the isoclines associated with"},{"Start":"00:17.670 ","End":"00:21.900","Text":"this differential equation and Sketch 3 of them."},{"Start":"00:21.900 ","End":"00:23.910","Text":"The ones with slope minus 1,"},{"Start":"00:23.910 ","End":"00:26.790","Text":"slope 0, and slope 2."},{"Start":"00:26.790 ","End":"00:32.985","Text":"On these isoclines, we want to draw a few line elements."},{"Start":"00:32.985 ","End":"00:35.655","Text":"How many is a few is up to you,"},{"Start":"00:35.655 ","End":"00:39.635","Text":"and now we want to compare"},{"Start":"00:39.635 ","End":"00:42.080","Text":"these line elements which we manually drew with"},{"Start":"00:42.080 ","End":"00:44.750","Text":"a computer aided sketch of the direction field,"},{"Start":"00:44.750 ","End":"00:49.010","Text":"and see that it\u0027s compatible with this, that it\u0027s part of this."},{"Start":"00:49.010 ","End":"00:55.340","Text":"Finally, to use the direction field, that\u0027s the DF,"},{"Start":"00:55.340 ","End":"01:03.720","Text":"to manually sketch the solutions of the equation passing through these 3 points."},{"Start":"01:04.240 ","End":"01:07.085","Text":"Let\u0027s get started."},{"Start":"01:07.085 ","End":"01:13.430","Text":"Now an isocline is obtained when we let this y\u0027 be some constant."},{"Start":"01:13.430 ","End":"01:18.710","Text":"We want the slope of the direction field to be a constant,"},{"Start":"01:18.710 ","End":"01:20.060","Text":"call it c,"},{"Start":"01:20.060 ","End":"01:23.225","Text":"and different c\u0027s will give us different isoclines."},{"Start":"01:23.225 ","End":"01:28.195","Text":"Now here we\u0027re told which to pick, so these 3."},{"Start":"01:28.195 ","End":"01:34.165","Text":"Now if c is 0, then we get x^2 plus y^2=2."},{"Start":"01:34.165 ","End":"01:37.235","Text":"Actually we could have just brought the 2 over and said x^2 plus"},{"Start":"01:37.235 ","End":"01:40.985","Text":"y^2=2 plus c. When c is 0, it\u0027s 2."},{"Start":"01:40.985 ","End":"01:43.490","Text":"When c is 2, it\u0027s 2 plus 2 is 4,"},{"Start":"01:43.490 ","End":"01:45.905","Text":"2 plus minus 1 is 1."},{"Start":"01:45.905 ","End":"01:48.515","Text":"But in each of these 3 cases,"},{"Start":"01:48.515 ","End":"01:53.220","Text":"it\u0027s going to be a circle centered at the origin."},{"Start":"01:53.220 ","End":"01:55.414","Text":"Before we start sketching,"},{"Start":"01:55.414 ","End":"02:00.755","Text":"we would like to have the angle rather than a slope."},{"Start":"02:00.755 ","End":"02:02.934","Text":"To get the angle,"},{"Start":"02:02.934 ","End":"02:04.340","Text":"once we\u0027re given the slope,"},{"Start":"02:04.340 ","End":"02:07.535","Text":"we just take the inverse tangent or arc tangent."},{"Start":"02:07.535 ","End":"02:09.470","Text":"If we do it for 0, 2,"},{"Start":"02:09.470 ","End":"02:11.315","Text":"and minus 1,"},{"Start":"02:11.315 ","End":"02:13.370","Text":"these are the 3 angles we get."},{"Start":"02:13.370 ","End":"02:14.929","Text":"Well, the first and last, you should know,"},{"Start":"02:14.929 ","End":"02:21.020","Text":"0 is 0 and the angle whose tangent is minus 1 is minus 45 degrees."},{"Start":"02:21.020 ","End":"02:23.150","Text":"For 2, you\u0027d need a calculator."},{"Start":"02:23.150 ","End":"02:25.520","Text":"I rounded it off to the nearest degree,"},{"Start":"02:25.520 ","End":"02:27.380","Text":"even that\u0027s too accurate."},{"Start":"02:27.380 ","End":"02:30.769","Text":"Now we want to make some sketches."},{"Start":"02:30.769 ","End":"02:34.290","Text":"I\u0027ll need a bit of graph paper."},{"Start":"02:34.520 ","End":"02:39.135","Text":"Here\u0027s the first isocline,"},{"Start":"02:39.135 ","End":"02:41.430","Text":"x^2 plus y^2 is 2,"},{"Start":"02:41.430 ","End":"02:44.705","Text":"that corresponds to c=0."},{"Start":"02:44.705 ","End":"02:51.740","Text":"The way I knew to sketch this was I noticed that it goes through the 0.11,"},{"Start":"02:51.740 ","End":"02:55.235","Text":"when x is 1, y is 1 or plus or minus,"},{"Start":"02:55.235 ","End":"02:58.639","Text":"and so this is the circle."},{"Start":"02:58.639 ","End":"03:04.190","Text":"The next one we\u0027ll just have radius 2 and this one will have radius 1."},{"Start":"03:04.190 ","End":"03:08.495","Text":"This one is where c=2,"},{"Start":"03:08.495 ","End":"03:16.020","Text":"and this one is where c=negative 1."},{"Start":"03:16.020 ","End":"03:20.420","Text":"The 0 one would be easiest because that\u0027s 0 degrees,"},{"Start":"03:20.420 ","End":"03:22.220","Text":"in other words, horizontal,"},{"Start":"03:22.220 ","End":"03:32.940","Text":"so we could just put a bunch of arrows everywhere along this circle."},{"Start":"03:34.600 ","End":"03:43.770","Text":"Just put a few. Next one is minus 1,"},{"Start":"03:43.770 ","End":"03:45.705","Text":"which we look up."},{"Start":"03:45.705 ","End":"03:50.310","Text":"That\u0027s minus 45 degrees,"},{"Start":"03:50.310 ","End":"03:53.285","Text":"so c diagonally down."},{"Start":"03:53.285 ","End":"03:57.899","Text":"Depends, I\u0027m using the endpoint or the middle of the segment."},{"Start":"03:57.899 ","End":"04:00.545","Text":"Let\u0027s do the middle here,"},{"Start":"04:00.545 ","End":"04:06.380","Text":"here, here, here."},{"Start":"04:06.380 ","End":"04:08.450","Text":"I\u0027m not going to do it very accurately,"},{"Start":"04:08.450 ","End":"04:09.710","Text":"but you get the idea."},{"Start":"04:09.710 ","End":"04:13.055","Text":"Put sloping 45 degrees down,"},{"Start":"04:13.055 ","End":"04:21.710","Text":"and those are the directions for the middle isocline."},{"Start":"04:21.710 ","End":"04:23.540","Text":"Now the outer one,"},{"Start":"04:23.540 ","End":"04:31.140","Text":"we need 63 degrees from the positive x-direction."},{"Start":"04:31.140 ","End":"04:33.695","Text":"It\u0027s steeper and upwards."},{"Start":"04:33.695 ","End":"04:38.465","Text":"Something like, I don\u0027t know if this is 63 degrees."},{"Start":"04:38.465 ","End":"04:42.150","Text":"We\u0027ll do a few of those,"},{"Start":"04:42.150 ","End":"04:44.570","Text":"just to get the idea."},{"Start":"04:44.570 ","End":"04:50.465","Text":"63 degrees upwards, 63 degrees."},{"Start":"04:50.465 ","End":"04:53.300","Text":"You could mark the point on the circle."},{"Start":"04:53.300 ","End":"04:57.680","Text":"I\u0027m not bothering, because the computer sketch doesn\u0027t do that."},{"Start":"04:57.680 ","End":"05:00.860","Text":"63 degrees upwards, 63,"},{"Start":"05:00.860 ","End":"05:06.075","Text":"and here, I\u0027ll do one more over here."},{"Start":"05:06.075 ","End":"05:08.142","Text":"You get the idea."},{"Start":"05:08.142 ","End":"05:11.980","Text":"This is just manual sketch."},{"Start":"05:12.230 ","End":"05:16.074","Text":"Now, here\u0027s the computer aided."},{"Start":"05:16.074 ","End":"05:20.000","Text":"What I think I should do is put these circles on here too,"},{"Start":"05:20.000 ","End":"05:24.165","Text":"then it\u0027d be easier to compare. Here we are."},{"Start":"05:24.165 ","End":"05:26.690","Text":"I guess if we go along the,"},{"Start":"05:26.690 ","End":"05:28.355","Text":"let\u0027s say the middle circle,"},{"Start":"05:28.355 ","End":"05:33.260","Text":"we can see that the arrows are pretty much flat,"},{"Start":"05:33.260 ","End":"05:36.215","Text":"horizontal and to the right, this one, this one."},{"Start":"05:36.215 ","End":"05:40.369","Text":"Not exactly because these arrows are not exactly"},{"Start":"05:40.369 ","End":"05:45.515","Text":"pinned to the circle but roughly these are horizontal arrows here."},{"Start":"05:45.515 ","End":"05:47.480","Text":"If we go along the middle circle,"},{"Start":"05:47.480 ","End":"05:49.690","Text":"we should see diagonally,"},{"Start":"05:49.690 ","End":"05:53.770","Text":"I don\u0027t know what you would call it, Southeast, let\u0027s say."},{"Start":"05:54.170 ","End":"05:56.480","Text":"Here if we go along,"},{"Start":"05:56.480 ","End":"06:01.805","Text":"we see that these arrows that we encounter are south easterly."},{"Start":"06:01.805 ","End":"06:04.220","Text":"On the outer one,"},{"Start":"06:04.220 ","End":"06:11.280","Text":"it\u0027s going to be steeper than 45 degrees, steeper than Northeast."},{"Start":"06:11.280 ","End":"06:13.669","Text":"If we look at the arrows along,"},{"Start":"06:13.669 ","End":"06:16.710","Text":"as we go along here, it looks about right."},{"Start":"06:19.400 ","End":"06:24.305","Text":"The next thing we were asked to do was to draw some solution curves."},{"Start":"06:24.305 ","End":"06:26.570","Text":"We were given 3 points."},{"Start":"06:26.570 ","End":"06:28.640","Text":"0, 0,"},{"Start":"06:28.640 ","End":"06:30.980","Text":"that would be the origin."},{"Start":"06:30.980 ","End":"06:33.580","Text":"0, negative 2,"},{"Start":"06:33.580 ","End":"06:37.005","Text":"that would be this one,"},{"Start":"06:37.005 ","End":"06:38.820","Text":"and 0, 2,"},{"Start":"06:38.820 ","End":"06:41.915","Text":"which just happened to be on this outer circle."},{"Start":"06:41.915 ","End":"06:44.240","Text":"This would be 0, 0,"},{"Start":"06:44.240 ","End":"06:45.710","Text":"0, 2,"},{"Start":"06:45.710 ","End":"06:48.600","Text":"and 0, minus 2."},{"Start":"06:49.340 ","End":"06:55.649","Text":"Ignore these orange circles now."},{"Start":"06:55.649 ","End":"06:57.290","Text":"Now we\u0027re just going to do a rough sketch."},{"Start":"06:57.290 ","End":"06:58.820","Text":"The one for here,"},{"Start":"06:58.820 ","End":"07:00.080","Text":"let\u0027s go first of all,"},{"Start":"07:00.080 ","End":"07:02.000","Text":"backwards with the arrows,"},{"Start":"07:02.000 ","End":"07:09.009","Text":"it starts to go up and then somehow down like this."},{"Start":"07:09.009 ","End":"07:10.770","Text":"On the other side,"},{"Start":"07:10.770 ","End":"07:15.635","Text":"going further down and following the arrows."},{"Start":"07:15.635 ","End":"07:18.380","Text":"Really just a rough idea."},{"Start":"07:18.380 ","End":"07:20.225","Text":"For this point,"},{"Start":"07:20.225 ","End":"07:23.689","Text":"we see where the arrows are going here."},{"Start":"07:23.689 ","End":"07:26.510","Text":"That would be like at 63 degrees,"},{"Start":"07:26.510 ","End":"07:29.150","Text":"but then it gets steeper."},{"Start":"07:29.150 ","End":"07:31.940","Text":"Here, seems to get shallower."},{"Start":"07:31.940 ","End":"07:35.374","Text":"It almost seems to go very close to this."},{"Start":"07:35.374 ","End":"07:39.200","Text":"From here, let\u0027s see."},{"Start":"07:39.200 ","End":"07:42.560","Text":"Again, we\u0027re hitting at 63 degrees coming from,"},{"Start":"07:42.560 ","End":"07:44.165","Text":"I don\u0027t know, here."},{"Start":"07:44.165 ","End":"07:50.610","Text":"Then flattening out and coming up something like this."},{"Start":"07:51.200 ","End":"07:56.460","Text":"That\u0027s as good as you could do freehand."},{"Start":"07:56.460 ","End":"07:59.775","Text":"What I\u0027ll do now is I\u0027ll show you,"},{"Start":"07:59.775 ","End":"08:03.230","Text":"I don\u0027t seem to have quite enough space."},{"Start":"08:03.230 ","End":"08:13.170","Text":"But here below, I put the one that was computer sketched,"},{"Start":"08:13.170 ","End":"08:19.865","Text":"and I think you can get the idea that we didn\u0027t do too badly."},{"Start":"08:19.865 ","End":"08:23.240","Text":"If you go back and see what we did for our 3 lines,"},{"Start":"08:23.240 ","End":"08:28.620","Text":"is close. That\u0027s it."}],"ID":10851}],"Thumbnail":null,"ID":27385},{"Name":"Numerical Methods","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Euler\u0027s Method","Duration":"13m 56s","ChapterTopicVideoID":25334,"CourseChapterTopicPlaylistID":153003,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"Now we come to Euler\u0027s method."},{"Start":"00:02.970 ","End":"00:09.435","Text":"It\u0027s a numerical method for approximating the solution to an initial value problem."},{"Start":"00:09.435 ","End":"00:12.120","Text":"This is the initial value problem."},{"Start":"00:12.120 ","End":"00:13.980","Text":"Here is the differential equation,"},{"Start":"00:13.980 ","End":"00:15.930","Text":"y\u0027 is f(x, y),"},{"Start":"00:15.930 ","End":"00:21.310","Text":"with an initial condition that the solution passes through x_0, y_0."},{"Start":"00:21.310 ","End":"00:24.885","Text":"We\u0027ll take a particular example."},{"Start":"00:24.885 ","End":"00:29.190","Text":"It\u0027s the same example we had before with direction fields,"},{"Start":"00:29.190 ","End":"00:31.110","Text":"y\u0027 is x minus y,"},{"Start":"00:31.110 ","End":"00:36.070","Text":"only here we have the extra initial condition."},{"Start":"00:36.230 ","End":"00:38.580","Text":"Because of the initial condition,"},{"Start":"00:38.580 ","End":"00:40.635","Text":"we\u0027ll only have one solution,"},{"Start":"00:40.635 ","End":"00:45.185","Text":"and what we\u0027re asked to do is to approximate it on an interval."},{"Start":"00:45.185 ","End":"00:50.490","Text":"The interval will start from the x that we\u0027re given,"},{"Start":"00:50.490 ","End":"00:53.510","Text":"and up to 1; 1 is arbitrary."},{"Start":"00:53.510 ","End":"00:58.020","Text":"I could have said to approximate it on the interval from 0-3."},{"Start":"00:58.540 ","End":"01:04.729","Text":"Sometimes a variation of this problem is to"},{"Start":"01:04.729 ","End":"01:11.825","Text":"approximate the value of y at a particular point in this case 1,"},{"Start":"01:11.825 ","End":"01:13.730","Text":"rather than over the whole interval."},{"Start":"01:13.730 ","End":"01:16.970","Text":"Sometimes this is a common problem."},{"Start":"01:16.970 ","End":"01:23.260","Text":"Given this, approximate y(1)."},{"Start":"01:23.260 ","End":"01:26.310","Text":"The first thing we do in Euler\u0027s method,"},{"Start":"01:26.310 ","End":"01:33.500","Text":"is choose a step-size h. You\u0027ll soon see what this means."},{"Start":"01:33.500 ","End":"01:37.655","Text":"For example, h=0.25."},{"Start":"01:37.655 ","End":"01:40.430","Text":"Now, typically, but not always,"},{"Start":"01:40.430 ","End":"01:43.309","Text":"we take the width of the interval,"},{"Start":"01:43.309 ","End":"01:44.540","Text":"in this case it\u0027s 1,"},{"Start":"01:44.540 ","End":"01:46.130","Text":"and divide it by a whole number,"},{"Start":"01:46.130 ","End":"01:48.110","Text":"here we divide it into 4."},{"Start":"01:48.110 ","End":"01:50.750","Text":"We could have divided it into 5,"},{"Start":"01:50.750 ","End":"01:53.030","Text":"and got h=0.2,"},{"Start":"01:53.030 ","End":"01:55.030","Text":"or divided it into 10,"},{"Start":"01:55.030 ","End":"01:58.150","Text":"or even into 20."},{"Start":"01:58.150 ","End":"02:04.855","Text":"Now, what are the advantages of choosing h larger or smaller?"},{"Start":"02:04.855 ","End":"02:07.245","Text":"Well, in one sense,"},{"Start":"02:07.245 ","End":"02:08.520","Text":"the smaller the better,"},{"Start":"02:08.520 ","End":"02:10.480","Text":"because the smaller the step-size,"},{"Start":"02:10.480 ","End":"02:13.505","Text":"the more accurate the approximation."},{"Start":"02:13.505 ","End":"02:20.240","Text":"But you have to pay for it because there\u0027s more computation work involved,"},{"Start":"02:20.240 ","End":"02:21.830","Text":"usually a lot more."},{"Start":"02:21.830 ","End":"02:24.005","Text":"It\u0027s a trade-off."},{"Start":"02:24.005 ","End":"02:27.905","Text":"It\u0027s a compromise between the accuracy,"},{"Start":"02:27.905 ","End":"02:29.930","Text":"meaning how good the approximation is,"},{"Start":"02:29.930 ","End":"02:33.510","Text":"and how much computation work or effort."},{"Start":"02:34.670 ","End":"02:37.830","Text":"This is by experience."},{"Start":"02:37.830 ","End":"02:40.900","Text":"Sometimes you choose one value of h,"},{"Start":"02:40.900 ","End":"02:47.225","Text":"and then you try a smaller one to see if you get much improvement in accuracy."},{"Start":"02:47.225 ","End":"02:50.120","Text":"There are various techniques, it\u0027s a naught."},{"Start":"02:50.120 ","End":"02:52.490","Text":"But often it will be given to you in the problem."},{"Start":"02:52.490 ","End":"02:55.145","Text":"They\u0027ll say, using h equals,"},{"Start":"02:55.145 ","End":"02:57.280","Text":"and give you its value."},{"Start":"02:57.280 ","End":"03:02.330","Text":"The general idea is to use an iterative process."},{"Start":"03:02.330 ","End":"03:05.000","Text":"We only have one point, x_0,"},{"Start":"03:05.000 ","End":"03:09.485","Text":"y_0 and using h and the formula,"},{"Start":"03:09.485 ","End":"03:13.635","Text":"we iteratively compute new points,"},{"Start":"03:13.635 ","End":"03:15.090","Text":"x_1, y_1,"},{"Start":"03:15.090 ","End":"03:18.150","Text":"then x_2, y_2, x_3, y_3."},{"Start":"03:18.150 ","End":"03:20.795","Text":"We\u0027ll get a set of points,"},{"Start":"03:20.795 ","End":"03:24.845","Text":"and then we\u0027ll join those points with a polygonal path;"},{"Start":"03:24.845 ","End":"03:27.230","Text":"just connect the dots,"},{"Start":"03:27.230 ","End":"03:31.920","Text":"and get an approximation to the solution."},{"Start":"03:32.030 ","End":"03:35.870","Text":"We do this using the formulas."},{"Start":"03:35.870 ","End":"03:37.865","Text":"These are recursive formulas,"},{"Start":"03:37.865 ","End":"03:41.300","Text":"there\u0027s one for x, and one for y."},{"Start":"03:41.300 ","End":"03:46.100","Text":"Each point is obtained in terms of the previous point."},{"Start":"03:46.100 ","End":"03:49.250","Text":"We get x_n plus 1 in terms of x_n,"},{"Start":"03:49.250 ","End":"03:55.345","Text":"and y_n plus 1 in terms of y_n, and also x_n."},{"Start":"03:55.345 ","End":"03:59.780","Text":"These are the 2 formulas that we use to give"},{"Start":"03:59.780 ","End":"04:05.630","Text":"us our approximation and we will see this demonstrated in a moment."},{"Start":"04:05.630 ","End":"04:10.100","Text":"There is a variation of this second formula that you will often"},{"Start":"04:10.100 ","End":"04:14.165","Text":"see so I\u0027m mentioning it though I prefer it this way."},{"Start":"04:14.165 ","End":"04:19.610","Text":"But sometimes we take this f(x_n and y_n)."},{"Start":"04:19.610 ","End":"04:22.905","Text":"This is simply an approximation,"},{"Start":"04:22.905 ","End":"04:24.510","Text":"or if you like,"},{"Start":"04:24.510 ","End":"04:30.360","Text":"it\u0027s the slope that passes through x_n, y_n."},{"Start":"04:30.360 ","End":"04:34.090","Text":"Often this is called y_n\u0027,"},{"Start":"04:34.090 ","End":"04:37.730","Text":"it\u0027s like the derivative at the nth point."},{"Start":"04:37.730 ","End":"04:42.660","Text":"Then this formula becomes simpler,"},{"Start":"04:43.150 ","End":"04:47.730","Text":"y_n plus 1, is y_n plus h times y_n\u0027."},{"Start":"04:47.730 ","End":"04:48.870","Text":"But it\u0027s not really simple,"},{"Start":"04:48.870 ","End":"04:53.055","Text":"because you have to remember that y_n\u0027 is this."},{"Start":"04:53.055 ","End":"04:54.705","Text":"Anyway, that\u0027s a variation,"},{"Start":"04:54.705 ","End":"04:57.040","Text":"I will stick with these."},{"Start":"04:57.700 ","End":"05:02.435","Text":"Here\u0027s an illustration of the iterative process."},{"Start":"05:02.435 ","End":"05:07.310","Text":"We start off at point x_0, y_0,"},{"Start":"05:07.310 ","End":"05:09.920","Text":"and this point is going to be accurate,"},{"Start":"05:09.920 ","End":"05:11.240","Text":"not an approximation,"},{"Start":"05:11.240 ","End":"05:13.085","Text":"because that\u0027s where we start from."},{"Start":"05:13.085 ","End":"05:20.270","Text":"The blue line here is the exact solution,"},{"Start":"05:20.270 ","End":"05:23.030","Text":"which we don\u0027t know how to compute."},{"Start":"05:23.030 ","End":"05:28.793","Text":"Well, sometimes we can solve such an IVP and get a formula,"},{"Start":"05:28.793 ","End":"05:33.345","Text":"but in principle it exists, the exact solution."},{"Start":"05:33.345 ","End":"05:34.840","Text":"What we do,"},{"Start":"05:34.840 ","End":"05:37.505","Text":"is we take steps."},{"Start":"05:37.505 ","End":"05:39.680","Text":"Each step here is h,"},{"Start":"05:39.680 ","End":"05:44.150","Text":"and each time we compute a new x_n, y_n,"},{"Start":"05:44.150 ","End":"05:45.680","Text":"we get x_1, y_1,"},{"Start":"05:45.680 ","End":"05:49.475","Text":"x_2, y_2, x_3, y_3."},{"Start":"05:49.475 ","End":"05:58.800","Text":"Until we pass the point we need to get up to which is in this case 1,"},{"Start":"05:58.880 ","End":"06:08.135","Text":"this second illustration actually shows how we get this formula or the pair of formulas."},{"Start":"06:08.135 ","End":"06:10.805","Text":"We have x_n,"},{"Start":"06:10.805 ","End":"06:13.455","Text":"y_n, that would be one of these."},{"Start":"06:13.455 ","End":"06:17.220","Text":"How do we get the next point?"},{"Start":"06:17.220 ","End":"06:23.290","Text":"What we do is we add h to the x."},{"Start":"06:23.510 ","End":"06:26.720","Text":"In this case, each of these steps is equal,"},{"Start":"06:26.720 ","End":"06:28.550","Text":"and each of them is equal to h,"},{"Start":"06:28.550 ","End":"06:33.740","Text":"so this would also be like h. That gives us the first formula,"},{"Start":"06:33.740 ","End":"06:34.880","Text":"that x_n plus 1,"},{"Start":"06:34.880 ","End":"06:37.320","Text":"is x_n plus h. Now,"},{"Start":"06:37.320 ","End":"06:40.710","Text":"what happens to y?"},{"Start":"06:40.710 ","End":"06:47.030","Text":"Well, we take the tangent line which starts from here and has a"},{"Start":"06:47.030 ","End":"06:53.375","Text":"slope m. The slope is just this function,"},{"Start":"06:53.375 ","End":"06:57.215","Text":"that\u0027s the y\u0027, it\u0027s f(x_n, y_n)."},{"Start":"06:57.215 ","End":"07:06.110","Text":"Now, Delta y over h is the slope."},{"Start":"07:06.110 ","End":"07:08.330","Text":"The slope is the rise overrun."},{"Start":"07:08.330 ","End":"07:14.120","Text":"If you like, Delta y is equal to h times"},{"Start":"07:14.120 ","End":"07:20.425","Text":"the slope which is f(x_n, y_n)."},{"Start":"07:20.425 ","End":"07:23.010","Text":"Then you can see what this is,"},{"Start":"07:23.010 ","End":"07:25.245","Text":"because y_n plus 1,"},{"Start":"07:25.245 ","End":"07:33.750","Text":"is just y _n plus Delta y; the increase."},{"Start":"07:33.750 ","End":"07:36.760","Text":"Delta y is this."},{"Start":"07:38.090 ","End":"07:40.925","Text":"Sorry, n plus 1 here."},{"Start":"07:40.925 ","End":"07:44.750","Text":"You can see that we get this formula."},{"Start":"07:44.750 ","End":"07:46.230","Text":"Although you don\u0027t have to derive it,"},{"Start":"07:46.230 ","End":"07:51.065","Text":"this is just an illustration in case you\u0027d like to see what this is all about."},{"Start":"07:51.065 ","End":"07:57.345","Text":"On the next page I\u0027ll continue with our example."},{"Start":"07:57.345 ","End":"08:03.879","Text":"Before I move on, I just wanted to point out that you can see that the discrepancy;"},{"Start":"08:03.879 ","End":"08:12.300","Text":"the difference between the approximation and the actual solution in this case,"},{"Start":"08:12.300 ","End":"08:13.560","Text":"at least gets larger,"},{"Start":"08:13.560 ","End":"08:14.995","Text":"and this is typical."},{"Start":"08:14.995 ","End":"08:17.890","Text":"If we take a smaller h,"},{"Start":"08:17.890 ","End":"08:19.600","Text":"a smaller step-size,"},{"Start":"08:19.600 ","End":"08:25.490","Text":"then this sticks closer to this actual solution."},{"Start":"08:25.490 ","End":"08:29.410","Text":"Like we said before, smaller step sizes are better for accuracy,"},{"Start":"08:29.410 ","End":"08:31.285","Text":"but they involve more work,"},{"Start":"08:31.285 ","End":"08:33.800","Text":"and so it\u0027s a trade-off."},{"Start":"08:35.000 ","End":"08:37.370","Text":"Here we are, on the new page,"},{"Start":"08:37.370 ","End":"08:44.705","Text":"and I copied the formulas for the iterative generation of the x_n\u0027s and the y_n\u0027s."},{"Start":"08:44.705 ","End":"08:49.800","Text":"Remember that we chose h being a quarter,"},{"Start":"08:49.800 ","End":"08:56.220","Text":"and we also have the initial condition that our solution starts from 0,"},{"Start":"08:56.220 ","End":"09:00.370","Text":"1, when x is 0, y is 1."},{"Start":"09:00.370 ","End":"09:05.430","Text":"Now we start applying Euler\u0027s method iteratively."},{"Start":"09:05.430 ","End":"09:07.485","Text":"We have x_0, x_1,"},{"Start":"09:07.485 ","End":"09:10.970","Text":"so we let n=0 in the beginning."},{"Start":"09:10.970 ","End":"09:13.820","Text":"That will give us that from here,"},{"Start":"09:13.820 ","End":"09:21.860","Text":"x_1 is x_0 plus h, which is 0.25."},{"Start":"09:21.860 ","End":"09:24.440","Text":"I\u0027m putting the values I collected in red,"},{"Start":"09:24.440 ","End":"09:26.465","Text":"it will be easy to see later."},{"Start":"09:26.465 ","End":"09:28.820","Text":"Y_1 from this formula,"},{"Start":"09:28.820 ","End":"09:34.410","Text":"is y naught plus h times f(x_0, y_0)."},{"Start":"09:35.300 ","End":"09:38.555","Text":"I should have repeated that,"},{"Start":"09:38.555 ","End":"09:41.225","Text":"f(x, y),"},{"Start":"09:41.225 ","End":"09:44.000","Text":"is x minus y."},{"Start":"09:44.000 ","End":"09:53.115","Text":"So f(x_0, y_0) is 0 minus 1."},{"Start":"09:53.115 ","End":"09:55.440","Text":"When you do the computation,"},{"Start":"09:55.440 ","End":"10:00.765","Text":"you get that y_1 is 0.75."},{"Start":"10:00.765 ","End":"10:03.880","Text":"Although this is a numerical method, not a graphical,"},{"Start":"10:03.880 ","End":"10:07.080","Text":"I\u0027d like to start plotting a little bit."},{"Start":"10:07.300 ","End":"10:11.510","Text":"I brought back a sketch we had earlier."},{"Start":"10:11.510 ","End":"10:14.150","Text":"We actually had several solutions, and if you remember,"},{"Start":"10:14.150 ","End":"10:19.580","Text":"we had one that went through the point 0, 1. This is 0."},{"Start":"10:19.580 ","End":"10:22.060","Text":"I also need the 1,"},{"Start":"10:22.060 ","End":"10:25.340","Text":"because we\u0027re going to approximate on the integral from 0-1."},{"Start":"10:25.340 ","End":"10:30.230","Text":"In general, the actual solution would be"},{"Start":"10:30.230 ","End":"10:36.670","Text":"this part here up to here."},{"Start":"10:36.680 ","End":"10:41.874","Text":"We\u0027ll see how the approximation goes."},{"Start":"10:41.874 ","End":"10:47.285","Text":"The first point in the iteration is spot on."},{"Start":"10:47.285 ","End":"10:49.460","Text":"We start from the exact point."},{"Start":"10:49.460 ","End":"10:52.090","Text":"I\u0027ll do the approximation in red,"},{"Start":"10:52.090 ","End":"10:54.295","Text":"so that one spot on."},{"Start":"10:54.295 ","End":"10:59.285","Text":"The next one, x is 0.25."},{"Start":"10:59.285 ","End":"11:02.105","Text":"That\u0027s halfway from here to here,"},{"Start":"11:02.105 ","End":"11:09.565","Text":"and 0.75, which would be somewhere here."},{"Start":"11:09.565 ","End":"11:13.305","Text":"Let\u0027s go for another iteration."},{"Start":"11:13.305 ","End":"11:19.970","Text":"When n is 1, we get that the next x is going to be 0.5."},{"Start":"11:19.970 ","End":"11:21.950","Text":"We could do all the x\u0027s at once."},{"Start":"11:21.950 ","End":"11:23.420","Text":"We start from here,"},{"Start":"11:23.420 ","End":"11:25.610","Text":"then we add 0.25,"},{"Start":"11:25.610 ","End":"11:28.880","Text":"and 0.25, another 0.25,"},{"Start":"11:28.880 ","End":"11:31.580","Text":"and then we get the last one here."},{"Start":"11:31.580 ","End":"11:38.240","Text":"It\u0027s just basically dividing this interval into 4 bits equal. That\u0027s the x_2."},{"Start":"11:38.240 ","End":"11:40.530","Text":"The y_2, well,"},{"Start":"11:40.530 ","End":"11:42.615","Text":"the formula gets tedious."},{"Start":"11:42.615 ","End":"11:45.270","Text":"I\u0027ll leave you to check the calculations,"},{"Start":"11:45.270 ","End":"11:50.310","Text":"and we got the y_2 is 0.625,"},{"Start":"11:50.310 ","End":"11:52.480","Text":"which is like 5/8."},{"Start":"11:54.200 ","End":"11:57.210","Text":"That comes out somewhere here,"},{"Start":"11:57.210 ","End":"12:01.815","Text":"and looking back, this is incorrect."},{"Start":"12:01.815 ","End":"12:04.130","Text":"This point it should be below."},{"Start":"12:04.130 ","End":"12:06.815","Text":"That\u0027s better. This is not something you\u0027d"},{"Start":"12:06.815 ","End":"12:09.500","Text":"be asked to do because this is a numerical method,"},{"Start":"12:09.500 ","End":"12:13.535","Text":"but I just thought that the graph would maybe help."},{"Start":"12:13.535 ","End":"12:20.400","Text":"Let\u0027s do one more iteration."},{"Start":"12:21.140 ","End":"12:24.840","Text":"I\u0027ll leave you to check the computations."},{"Start":"12:24.840 ","End":"12:28.660","Text":"x_3 is 0.75, which is here, and y_3,"},{"Start":"12:28.660 ","End":"12:34.745","Text":"using the formula gives us roughly 0.6,"},{"Start":"12:34.745 ","End":"12:40.725","Text":"which would be roughly here."},{"Start":"12:40.725 ","End":"12:44.495","Text":"Then the last one, when n=3."},{"Start":"12:44.495 ","End":"12:46.540","Text":"Here\u0027s the computation."},{"Start":"12:46.540 ","End":"12:47.800","Text":"I\u0027ll leave you to check it."},{"Start":"12:47.800 ","End":"12:51.365","Text":"Just notice that, x_4 is 1,"},{"Start":"12:51.365 ","End":"12:54.550","Text":"and y_4 comes out to be this,"},{"Start":"12:54.550 ","End":"12:56.980","Text":"which is actually higher than this,"},{"Start":"12:56.980 ","End":"13:00.200","Text":"and it comes out somewhere around here."},{"Start":"13:00.480 ","End":"13:05.785","Text":"We don\u0027t do any more iterations because we\u0027ve got the value of x,"},{"Start":"13:05.785 ","End":"13:10.660","Text":"which is equal to the right limit of the interval;"},{"Start":"13:10.660 ","End":"13:12.070","Text":"the interval of 0-1,"},{"Start":"13:12.070 ","End":"13:13.360","Text":"and we\u0027ve reached 1,"},{"Start":"13:13.360 ","End":"13:17.050","Text":"so there\u0027s no need to go on anymore."},{"Start":"13:17.050 ","End":"13:20.810","Text":"That\u0027s what I wrote here."},{"Start":"13:21.750 ","End":"13:28.170","Text":"That in essence is the method."},{"Start":"13:28.170 ","End":"13:35.050","Text":"You could join these with a line."},{"Start":"13:35.360 ","End":"13:37.950","Text":"I didn\u0027t do this very well,"},{"Start":"13:37.950 ","End":"13:41.630","Text":"it may be best not to draw the line."},{"Start":"13:41.630 ","End":"13:47.390","Text":"But that\u0027s the approximation and we\u0027re done for the tutorial."},{"Start":"13:47.390 ","End":"13:56.760","Text":"There\u0027ll be a bit more in the solved exercises following this. There we are."}],"ID":26151},{"Watched":false,"Name":"The Picard Iterative Process","Duration":"3m 5s","ChapterTopicVideoID":25333,"CourseChapterTopicPlaylistID":153003,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.170","Text":"In this clip, we\u0027ll learn another way of finding solutions to differential equations,"},{"Start":"00:07.170 ","End":"00:11.460","Text":"specifically certain initial value problem."},{"Start":"00:11.460 ","End":"00:14.160","Text":"I\u0027ll show you what problem that is."},{"Start":"00:14.160 ","End":"00:17.700","Text":"It\u0027s a first-order differential equation where we have on the left"},{"Start":"00:17.700 ","End":"00:21.900","Text":"y\u0027 and on the right a function of x and y,"},{"Start":"00:21.900 ","End":"00:27.255","Text":"and the initial condition y(x_0) is y_0."},{"Start":"00:27.255 ","End":"00:30.110","Text":"There is an integral form of this."},{"Start":"00:30.110 ","End":"00:32.029","Text":"If you integrate both sides here,"},{"Start":"00:32.029 ","End":"00:35.480","Text":"we get y is the integral of f(x and y)."},{"Start":"00:35.480 ","End":"00:39.785","Text":"Everything is a function of x really because y is also a function of x."},{"Start":"00:39.785 ","End":"00:46.850","Text":"We can just take the antiderivative which is the integral and we replace x by t,"},{"Start":"00:46.850 ","End":"00:51.335","Text":"and then we take the integral from x_0 minus x."},{"Start":"00:51.335 ","End":"00:54.305","Text":"That will solve this part."},{"Start":"00:54.305 ","End":"00:56.435","Text":"But when x is x_0,"},{"Start":"00:56.435 ","End":"00:57.780","Text":"this bid gives us 0."},{"Start":"00:57.780 ","End":"01:01.990","Text":"We add the y_0 to match the initial condition."},{"Start":"01:01.990 ","End":"01:07.100","Text":"This way of writing it as an integral is the basis for an iterative process;"},{"Start":"01:07.100 ","End":"01:13.430","Text":"it\u0027s a numerical process that successively gets closer and closer to a solution."},{"Start":"01:13.430 ","End":"01:18.379","Text":"The general idea as we start off with some initial guess for y,"},{"Start":"01:18.379 ","End":"01:21.665","Text":"then we plug it in here and we get an output,"},{"Start":"01:21.665 ","End":"01:23.140","Text":"and that will be our next y,"},{"Start":"01:23.140 ","End":"01:26.090","Text":"and then we feed it back in here and get another y out."},{"Start":"01:26.090 ","End":"01:28.025","Text":"We just keep doing this."},{"Start":"01:28.025 ","End":"01:30.845","Text":"We get closer and closer to an actual solution."},{"Start":"01:30.845 ","End":"01:34.790","Text":"Sometimes, we can take the limit and really get an exact solution."},{"Start":"01:34.790 ","End":"01:36.530","Text":"It has another name,"},{"Start":"01:36.530 ","End":"01:39.635","Text":"the Picard\u0027s method of successive approximations."},{"Start":"01:39.635 ","End":"01:41.540","Text":"There are other variations,"},{"Start":"01:41.540 ","End":"01:44.765","Text":"but it\u0027s iterative or successive, that\u0027s the idea."},{"Start":"01:44.765 ","End":"01:46.960","Text":"There are 3 main steps."},{"Start":"01:46.960 ","End":"01:49.880","Text":"The general thing is that we build a sequence of"},{"Start":"01:49.880 ","End":"01:52.790","Text":"functions and we\u0027ll give them subscripts,"},{"Start":"01:52.790 ","End":"01:55.100","Text":"so we have y_0, y_1, y_2,"},{"Start":"01:55.100 ","End":"01:58.040","Text":"and so on, an infinite sequence."},{"Start":"01:58.040 ","End":"02:05.565","Text":"The approximations y_n(x) tend to the solution y(x) as n goes to infinity."},{"Start":"02:05.565 ","End":"02:09.200","Text":"The first step is to start with a constant function."},{"Start":"02:09.200 ","End":"02:13.415","Text":"We just let y_0(x) be the constant y_0."},{"Start":"02:13.415 ","End":"02:18.075","Text":"The notation is not great here because y(0) is a bit ambiguous."},{"Start":"02:18.075 ","End":"02:22.490","Text":"The one hand is the first function or the 0_th function in the approximation,"},{"Start":"02:22.490 ","End":"02:26.060","Text":"on the other hand, this is the y_0 from the initial condition."},{"Start":"02:26.060 ","End":"02:28.000","Text":"I have explained that here."},{"Start":"02:28.000 ","End":"02:30.560","Text":"But notation, but that\u0027s how it\u0027s done."},{"Start":"02:30.560 ","End":"02:35.690","Text":"The second step is to define a sequence recursively,"},{"Start":"02:35.690 ","End":"02:38.210","Text":"where we have the 0s sequence,"},{"Start":"02:38.210 ","End":"02:46.090","Text":"and the following n is gotten from the current n by substituting it in this formula."},{"Start":"02:46.090 ","End":"02:51.215","Text":"We get a sequence, and the nth term is called the nth Picard approximation."},{"Start":"02:51.215 ","End":"02:54.800","Text":"Step 3 is not always possible to get"},{"Start":"02:54.800 ","End":"03:00.050","Text":"the actual exact solution by taking the limit of these y ends."},{"Start":"03:00.050 ","End":"03:03.590","Text":"But if not, do as many as practical,"},{"Start":"03:03.590 ","End":"03:06.810","Text":"and that will be your nth approximation."}],"ID":26150},{"Watched":false,"Name":"Exercise 1","Duration":"9m 25s","ChapterTopicVideoID":25331,"CourseChapterTopicPlaylistID":153003,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.565","Text":"This is an exercise in which we are going to use Euler\u0027s method."},{"Start":"00:05.565 ","End":"00:09.300","Text":"Our task is, given this problem,"},{"Start":"00:09.300 ","End":"00:12.660","Text":"y\u0027 is some function of x and y, in this case,"},{"Start":"00:12.660 ","End":"00:21.000","Text":"x plus y, we\u0027re given an initial condition that y at 0 is 0."},{"Start":"00:21.000 ","End":"00:27.375","Text":"The first step is to estimate y(1) using Euler\u0027s method"},{"Start":"00:27.375 ","End":"00:34.230","Text":"with step-size of h=1/4 or 0.25,"},{"Start":"00:34.230 ","End":"00:36.765","Text":"that\u0027s a manual computation."},{"Start":"00:36.765 ","End":"00:40.875","Text":"Then we\u0027re going to decrease the step-size,"},{"Start":"00:40.875 ","End":"00:44.600","Text":"but we\u0027ll do it computer aided because then it gets tedious,"},{"Start":"00:44.600 ","End":"00:47.840","Text":"and then we\u0027ll decrease the step-size still more,"},{"Start":"00:47.840 ","End":"00:57.755","Text":"and then we\u0027re actually going to produce the solution to the initial value problem."},{"Start":"00:57.755 ","End":"00:59.240","Text":"We can\u0027t always do that,"},{"Start":"00:59.240 ","End":"01:00.500","Text":"but in this case we can."},{"Start":"01:00.500 ","End":"01:04.430","Text":"Then we\u0027ll check the accurate result with"},{"Start":"01:04.430 ","End":"01:08.694","Text":"the estimations that we had at the 3 decimal places."},{"Start":"01:08.694 ","End":"01:11.695","Text":"That\u0027s the idea of this exercise."},{"Start":"01:11.695 ","End":"01:17.810","Text":"Here I brought you the formulas for Euler\u0027s method, it\u0027s iterative."},{"Start":"01:17.810 ","End":"01:23.280","Text":"We actually have x_0 and y_0."},{"Start":"01:23.420 ","End":"01:26.950","Text":"The initial condition I got from here,"},{"Start":"01:26.950 ","End":"01:32.590","Text":"it was given that y(0) is 0."},{"Start":"01:32.590 ","End":"01:39.320","Text":"Now, we\u0027re going to use Euler\u0027s method with this step-size."},{"Start":"01:39.320 ","End":"01:43.695","Text":"Just remember that this is h,"},{"Start":"01:43.695 ","End":"01:45.540","Text":"and we have f(x,"},{"Start":"01:45.540 ","End":"01:48.010","Text":"y) is x plus y."},{"Start":"01:48.010 ","End":"01:53.170","Text":"Now we can apply this, starting with n=0."},{"Start":"01:53.170 ","End":"01:55.690","Text":"This will give us the new x,"},{"Start":"01:55.690 ","End":"01:57.130","Text":"this will give us the new y."},{"Start":"01:57.130 ","End":"02:01.480","Text":"We\u0027ll get x_1, each time increasing by the step-size."},{"Start":"02:01.480 ","End":"02:05.905","Text":"Zero plus 0.25 is 0.25."},{"Start":"02:05.905 ","End":"02:09.230","Text":"As for y, we take the previous y,"},{"Start":"02:09.230 ","End":"02:15.470","Text":"or the current y plus h times f of the previous point,"},{"Start":"02:15.470 ","End":"02:17.994","Text":"f(0,0) is 0 plus 0."},{"Start":"02:17.994 ","End":"02:19.640","Text":"Anyway, comes out 0,"},{"Start":"02:19.640 ","End":"02:22.585","Text":"so that\u0027s x_1, y_1."},{"Start":"02:22.585 ","End":"02:25.695","Text":"Onto the next point."},{"Start":"02:25.695 ","End":"02:27.150","Text":"We\u0027re going to get x_2,"},{"Start":"02:27.150 ","End":"02:30.855","Text":"y_2 from this formula with n=1."},{"Start":"02:30.855 ","End":"02:33.600","Text":"We\u0027ve got x_2 in terms of x_1,"},{"Start":"02:33.600 ","End":"02:37.140","Text":"and y_2 in terms of y_1 and x_1."},{"Start":"02:37.140 ","End":"02:44.045","Text":"I\u0027ll leave you to check this calculation because it gets tedious after a while."},{"Start":"02:44.045 ","End":"02:46.885","Text":"Then we\u0027ll go to the next point."},{"Start":"02:46.885 ","End":"02:49.470","Text":"We let n=2."},{"Start":"02:49.470 ","End":"02:53.270","Text":"X increases each time you see by 0.25,"},{"Start":"02:53.270 ","End":"02:55.640","Text":"0, 0.25, 0.5,"},{"Start":"02:55.640 ","End":"02:57.710","Text":"0.75, and the next one will be 1,"},{"Start":"02:57.710 ","End":"02:59.195","Text":"that will be the last one."},{"Start":"02:59.195 ","End":"03:02.090","Text":"Again, using this formula,"},{"Start":"03:02.090 ","End":"03:05.060","Text":"we\u0027re using the previous y here and here,"},{"Start":"03:05.060 ","End":"03:13.535","Text":"and we\u0027re using the previous x, which is here."},{"Start":"03:13.535 ","End":"03:16.540","Text":"This is the computation."},{"Start":"03:16.540 ","End":"03:21.545","Text":"When I said with calculator,"},{"Start":"03:21.545 ","End":"03:23.720","Text":"use a hand calculator,"},{"Start":"03:23.720 ","End":"03:27.510","Text":"but that\u0027s not the same as computer aided."},{"Start":"03:29.090 ","End":"03:31.470","Text":"One more iteration,"},{"Start":"03:31.470 ","End":"03:36.060","Text":"when n=3, then we have our new x,"},{"Start":"03:36.060 ","End":"03:37.920","Text":"which is x_4, is 1,"},{"Start":"03:37.920 ","End":"03:40.220","Text":"and we know that this is going to be the last one because"},{"Start":"03:40.220 ","End":"03:43.100","Text":"we want the value of y when x is 1."},{"Start":"03:43.100 ","End":"03:45.980","Text":"The fourth iteration of y, again,"},{"Start":"03:45.980 ","End":"03:47.480","Text":"the same formula,"},{"Start":"03:47.480 ","End":"03:49.160","Text":"we take this number here,"},{"Start":"03:49.160 ","End":"03:57.170","Text":"the old y plus the step-size times the function x plus y with this x and this y,"},{"Start":"03:57.170 ","End":"03:59.850","Text":"and this is what we get."},{"Start":"04:00.160 ","End":"04:03.350","Text":"This concludes Part A."},{"Start":"04:03.350 ","End":"04:07.710","Text":"We have an estimate for y(1)."},{"Start":"04:07.940 ","End":"04:10.310","Text":"Here we are in a new page."},{"Start":"04:10.310 ","End":"04:12.965","Text":"I copied the initial value problem,"},{"Start":"04:12.965 ","End":"04:15.440","Text":"and our result of Part A,"},{"Start":"04:15.440 ","End":"04:18.080","Text":"that when the step-size is 0.25,"},{"Start":"04:18.080 ","End":"04:21.400","Text":"our estimate for y(1) is this."},{"Start":"04:21.400 ","End":"04:24.390","Text":"Now let\u0027s move on to Part B."},{"Start":"04:24.390 ","End":"04:26.750","Text":"Here, I just looked it up,"},{"Start":"04:26.750 ","End":"04:31.685","Text":"there are online various Euler method calculators."},{"Start":"04:31.685 ","End":"04:33.860","Text":"Some of them give you the steps in the middle,"},{"Start":"04:33.860 ","End":"04:37.055","Text":"or you can just say you want the final answer."},{"Start":"04:37.055 ","End":"04:39.725","Text":"You feed in the function,"},{"Start":"04:39.725 ","End":"04:43.010","Text":"the initial condition and the step-size,"},{"Start":"04:43.010 ","End":"04:46.550","Text":"and you tell it what you want to estimate,"},{"Start":"04:46.550 ","End":"04:48.410","Text":"and it gives you the answer,"},{"Start":"04:48.410 ","End":"04:51.950","Text":"and this is what I got computer aided,"},{"Start":"04:51.950 ","End":"04:55.065","Text":"so no further details."},{"Start":"04:55.065 ","End":"04:59.895","Text":"Move on to Part C. Once again,"},{"Start":"04:59.895 ","End":"05:03.210","Text":"I used a computer aided program,"},{"Start":"05:03.210 ","End":"05:08.135","Text":"and this time with the step with the 0.01,"},{"Start":"05:08.135 ","End":"05:14.010","Text":"it gave me the estimate for y(1) as this."},{"Start":"05:14.440 ","End":"05:18.020","Text":"Now in Part D,"},{"Start":"05:18.020 ","End":"05:26.345","Text":"this time we have something that claims to be a solution to the initial value problem."},{"Start":"05:26.345 ","End":"05:29.240","Text":"We can\u0027t always find solutions,"},{"Start":"05:29.240 ","End":"05:32.630","Text":"and where it came from it doesn\u0027t really matter."},{"Start":"05:32.630 ","End":"05:34.040","Text":"If you want to know,"},{"Start":"05:34.040 ","End":"05:41.300","Text":"I just solve this equation using first-order linear theory of that."},{"Start":"05:41.300 ","End":"05:43.250","Text":"But it doesn\u0027t matter."},{"Start":"05:43.250 ","End":"05:45.865","Text":"If I present your solution,"},{"Start":"05:45.865 ","End":"05:48.215","Text":"you can check that it really is."},{"Start":"05:48.215 ","End":"05:53.375","Text":"It has to satisfy the differential equation and the initial condition."},{"Start":"05:53.375 ","End":"05:57.335","Text":"I\u0027ll check the initial condition first."},{"Start":"05:57.335 ","End":"06:00.230","Text":"Y(0) means I put x=0,"},{"Start":"06:00.230 ","End":"06:03.750","Text":"so I get e^0 minus 0 minus 1,"},{"Start":"06:03.750 ","End":"06:05.800","Text":"which is 1 minus 0 minus 1,"},{"Start":"06:05.800 ","End":"06:07.065","Text":"which is 0,"},{"Start":"06:07.065 ","End":"06:09.495","Text":"and this is what we wanted."},{"Start":"06:09.495 ","End":"06:12.135","Text":"We\u0027re okay there."},{"Start":"06:12.135 ","End":"06:16.560","Text":"The differential equation part,"},{"Start":"06:16.560 ","End":"06:20.790","Text":"let me start from the left-hand side and I\u0027ll reach the right-hand side,"},{"Start":"06:21.500 ","End":"06:26.210","Text":"y\u0027 means that we want to differentiate this."},{"Start":"06:26.210 ","End":"06:30.215","Text":"Y\u0027 is e^x, derivative is e^x."},{"Start":"06:30.215 ","End":"06:33.290","Text":"Minus x derivative is minus 1,"},{"Start":"06:33.290 ","End":"06:35.885","Text":"and the minus 1 gives nothing."},{"Start":"06:35.885 ","End":"06:42.695","Text":"E^x minus 1 looks very much like this."},{"Start":"06:42.695 ","End":"06:46.804","Text":"If I just subtract x and add x,"},{"Start":"06:46.804 ","End":"06:51.650","Text":"then I\u0027ll get exactly x plus this,"},{"Start":"06:51.650 ","End":"06:58.610","Text":"which is y. Y\u0027 is equal to x plus y as required here,"},{"Start":"06:58.610 ","End":"07:03.165","Text":"so we\u0027re done with this part."},{"Start":"07:03.165 ","End":"07:05.675","Text":"This really is a solution."},{"Start":"07:05.675 ","End":"07:08.425","Text":"Let me put a picture in here."},{"Start":"07:08.425 ","End":"07:13.500","Text":"I borrowed this sketch from a previous exercise, it helps."},{"Start":"07:13.500 ","End":"07:16.250","Text":"We had to draw 3 solutions there."},{"Start":"07:16.250 ","End":"07:18.070","Text":"Ignore the top and the bottom one,"},{"Start":"07:18.070 ","End":"07:20.765","Text":"this is the one that we\u0027re interested in."},{"Start":"07:20.765 ","End":"07:25.240","Text":"This one passes through the point 0, 0."},{"Start":"07:25.240 ","End":"07:28.850","Text":"That was one of the conditions for that question there."},{"Start":"07:28.850 ","End":"07:32.900","Text":"We see that y(0) is 0."},{"Start":"07:32.900 ","End":"07:35.255","Text":"We want to know what is y(1)."},{"Start":"07:35.255 ","End":"07:37.730","Text":"Now, 1 is here,"},{"Start":"07:37.730 ","End":"07:42.120","Text":"so we have to go up until we hit the curve,"},{"Start":"07:42.430 ","End":"07:46.340","Text":"then we go across and we read the value here."},{"Start":"07:46.340 ","End":"07:49.280","Text":"I see it\u0027s between 0.5 and 1."},{"Start":"07:49.280 ","End":"07:52.670","Text":"Looks like 0.7 something, hard to say,"},{"Start":"07:52.670 ","End":"07:56.855","Text":"so let\u0027s see if we get 0.7 something."},{"Start":"07:56.855 ","End":"08:00.035","Text":"What we get, y(1),"},{"Start":"08:00.035 ","End":"08:03.330","Text":"we get by plugging x=1 in here,"},{"Start":"08:03.330 ","End":"08:06.780","Text":"so we get e^1 minus 1 minus 1."},{"Start":"08:06.780 ","End":"08:10.630","Text":"Basically this is just e minus 2."},{"Start":"08:11.060 ","End":"08:15.090","Text":"If you look up e on the calculator,"},{"Start":"08:15.090 ","End":"08:18.640","Text":"I happened to memorize it this many places,"},{"Start":"08:18.640 ","End":"08:24.960","Text":"that e is 2.7182818281828, repeats anyway."},{"Start":"08:24.960 ","End":"08:27.795","Text":"I just subtract the 2 from it and this is what we get."},{"Start":"08:27.795 ","End":"08:29.565","Text":"That looks about right."},{"Start":"08:29.565 ","End":"08:32.930","Text":"Now we have 4 different values,"},{"Start":"08:32.930 ","End":"08:37.055","Text":"and I\u0027ll arrange these values in"},{"Start":"08:37.055 ","End":"08:42.740","Text":"a table after I\u0027ve truncated them or rounded them to 3 decimal places."},{"Start":"08:42.740 ","End":"08:46.225","Text":"Here they are collected."},{"Start":"08:46.225 ","End":"08:48.435","Text":"The 3 steps we had,"},{"Start":"08:48.435 ","End":"08:50.070","Text":"0.25, 0.1,"},{"Start":"08:50.070 ","End":"08:53.690","Text":"0.01 and these are the 3 values I rounded,"},{"Start":"08:53.690 ","End":"08:56.630","Text":"and the final one is the exact value,"},{"Start":"08:56.630 ","End":"08:59.250","Text":"also 3 decimal places."},{"Start":"09:00.340 ","End":"09:06.080","Text":"Notice that in this case, at any rate,"},{"Start":"09:06.080 ","End":"09:08.510","Text":"when we decrease h,"},{"Start":"09:08.510 ","End":"09:09.890","Text":"it gets smaller and smaller,"},{"Start":"09:09.890 ","End":"09:12.920","Text":"we get closer and closer to the exact value."},{"Start":"09:12.920 ","End":"09:15.080","Text":"This may not always happen,"},{"Start":"09:15.080 ","End":"09:18.184","Text":"but in general, this is how it works."},{"Start":"09:18.184 ","End":"09:21.310","Text":"This is what I wrote down."},{"Start":"09:21.590 ","End":"09:25.740","Text":"For this exercise, we are done."}],"ID":26148},{"Watched":false,"Name":"Example","Duration":"7m 46s","ChapterTopicVideoID":25330,"CourseChapterTopicPlaylistID":153003,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.540","Text":"Now let\u0027s look at an example of this technique."},{"Start":"00:03.540 ","End":"00:06.150","Text":"We\u0027re given the initial value problem,"},{"Start":"00:06.150 ","End":"00:11.955","Text":"y\u0027 equals y and y(0)=1 and there are 3 parts."},{"Start":"00:11.955 ","End":"00:17.145","Text":"Find the first 3 Picard approximations to the solution for this."},{"Start":"00:17.145 ","End":"00:20.360","Text":"In other words, we want y_n for n=1,"},{"Start":"00:20.360 ","End":"00:23.695","Text":"2, and 3; y_1(x), y_2(x), y_3(x)."},{"Start":"00:23.695 ","End":"00:26.150","Text":"Then we want to find a general expression,"},{"Start":"00:26.150 ","End":"00:33.110","Text":"y_n(x) in terms of n and prove it using induction on n. The last part"},{"Start":"00:33.110 ","End":"00:36.140","Text":"is to first solve the initial value problem"},{"Start":"00:36.140 ","End":"00:40.739","Text":"directly using our techniques for differential equations,"},{"Start":"00:40.739 ","End":"00:45.260","Text":"and then to show that the nth Picard approximation that we"},{"Start":"00:45.260 ","End":"00:50.390","Text":"found in b converges to the solution when n goes to infinity."},{"Start":"00:50.390 ","End":"00:56.720","Text":"Now, the f(x,y) which is the function on the right here is just y,"},{"Start":"00:56.720 ","End":"01:01.435","Text":"and x_0 is 0 and y_0 is 1."},{"Start":"01:01.435 ","End":"01:03.710","Text":"The problem is determined by f,"},{"Start":"01:03.710 ","End":"01:06.185","Text":"by x_0 and by y_0."},{"Start":"01:06.185 ","End":"01:09.140","Text":"Step 1 from the 3 step process."},{"Start":"01:09.140 ","End":"01:12.358","Text":"Y_0 is a constant function,"},{"Start":"01:12.358 ","End":"01:17.090","Text":"and we take it just to be the initial value of y which is 1."},{"Start":"01:17.090 ","End":"01:18.890","Text":"That\u0027s our first approximation,"},{"Start":"01:18.890 ","End":"01:20.165","Text":"the constant function 1."},{"Start":"01:20.165 ","End":"01:27.995","Text":"Step 2 is to use the recursive definition to try and find general y_n."},{"Start":"01:27.995 ","End":"01:33.630","Text":"Now let\u0027s simplify this because we know what f is,"},{"Start":"01:33.630 ","End":"01:35.010","Text":"we know what x_0 is,"},{"Start":"01:35.010 ","End":"01:36.705","Text":"and we know what y_0 is."},{"Start":"01:36.705 ","End":"01:39.345","Text":"Y_0 is 1,"},{"Start":"01:39.345 ","End":"01:42.375","Text":"x_0 here is 0,"},{"Start":"01:42.375 ","End":"01:44.595","Text":"and f(x,y) is y,"},{"Start":"01:44.595 ","End":"01:48.360","Text":"so f of this and this is this part here."},{"Start":"01:48.360 ","End":"01:52.890","Text":"We can just replace all this by just y_n(t)."},{"Start":"01:53.860 ","End":"01:58.670","Text":"Now let\u0027s start substituting values of n. Well,"},{"Start":"01:58.670 ","End":"02:01.310","Text":"we know y_0(x)=1,"},{"Start":"02:01.310 ","End":"02:03.830","Text":"and now we can apply this to get y_1."},{"Start":"02:03.830 ","End":"02:09.025","Text":"It\u0027s 1 plus the integral from 0-x(y_0t)."},{"Start":"02:09.025 ","End":"02:12.495","Text":"Y_0 is the constant 1,"},{"Start":"02:12.495 ","End":"02:14.060","Text":"so we have this integral."},{"Start":"02:14.060 ","End":"02:17.675","Text":"The integral of 1 is t in terms of dt,"},{"Start":"02:17.675 ","End":"02:23.255","Text":"but we take it between 0 and x so it\u0027s 1 plus x minus 0."},{"Start":"02:23.255 ","End":"02:29.180","Text":"Now we just have to plug this back again into this formula here."},{"Start":"02:29.180 ","End":"02:33.340","Text":"We get y_2 is 1 plus the integral of y_1."},{"Start":"02:33.340 ","End":"02:36.885","Text":"Next time we\u0027ll have 3 here and here 2."},{"Start":"02:36.885 ","End":"02:39.300","Text":"We know what y_1 is it\u0027s 1 plus x,"},{"Start":"02:39.300 ","End":"02:44.328","Text":"so y_1(t) is 1 plus t. Take this integral."},{"Start":"02:44.328 ","End":"02:49.020","Text":"This is t plus t^2 over 2 from"},{"Start":"02:49.020 ","End":"02:54.585","Text":"0 to x which comes out to be 1 from here."},{"Start":"02:54.585 ","End":"02:56.720","Text":"Here this gives us x,"},{"Start":"02:56.720 ","End":"02:59.315","Text":"and this gives us x^2 over 2."},{"Start":"02:59.315 ","End":"03:02.085","Text":"Go to the next step."},{"Start":"03:02.085 ","End":"03:07.470","Text":"Y_3 is 1 plus integral from naught to x(y_2)."},{"Start":"03:07.470 ","End":"03:10.380","Text":"We know y_2 in terms of x,"},{"Start":"03:10.380 ","End":"03:15.980","Text":"just replace x with t. Take the integral, we get this."},{"Start":"03:15.980 ","End":"03:17.240","Text":"0 to x,"},{"Start":"03:17.240 ","End":"03:18.620","Text":"0 gives us nothing."},{"Start":"03:18.620 ","End":"03:23.205","Text":"X gives us x plus x^2 over 2 plus x^3 over 6."},{"Start":"03:23.205 ","End":"03:25.620","Text":"We found y_1, y_2,"},{"Start":"03:25.620 ","End":"03:30.070","Text":"and y_3, so that concludes Part a."},{"Start":"03:30.070 ","End":"03:35.255","Text":"Now Part b to remind you is to find a general expression for y_n."},{"Start":"03:35.255 ","End":"03:37.130","Text":"We found y_1, y_2,"},{"Start":"03:37.130 ","End":"03:41.480","Text":"and y_3, and now we have to make a guess what y_n is."},{"Start":"03:41.480 ","End":"03:43.720","Text":"I think the pattern is clear."},{"Start":"03:43.720 ","End":"03:47.845","Text":"The claim is that y_n is 1 plus x plus x^2 over 2."},{"Start":"03:47.845 ","End":"03:49.460","Text":"Obviously 2 is 2 factorial,"},{"Start":"03:49.460 ","End":"03:51.155","Text":"6 is 3 factorial."},{"Start":"03:51.155 ","End":"03:53.960","Text":"The guesses is that it goes up to the n"},{"Start":"03:53.960 ","End":"03:58.320","Text":"factorial because in the case of 2 we stopped at x^2 over 2 factorial."},{"Start":"03:58.320 ","End":"04:00.735","Text":"With 3, we stopped at x=3."},{"Start":"04:00.735 ","End":"04:03.884","Text":"With n we\u0027ll stop at x^n over n factorial."},{"Start":"04:03.884 ","End":"04:05.235","Text":"That\u0027s what it looks like,"},{"Start":"04:05.235 ","End":"04:14.180","Text":"and we\u0027re going to prove it by induction on n. Now let\u0027s prove the induction step."},{"Start":"04:14.180 ","End":"04:20.255","Text":"We had a recursion formula above adapted for our case and it came out like this."},{"Start":"04:20.255 ","End":"04:23.590","Text":"This is the recursive step from n to n plus 1."},{"Start":"04:23.590 ","End":"04:27.710","Text":"We\u0027re going to use the induction hypothesis that we know what y_n is."},{"Start":"04:27.710 ","End":"04:31.630","Text":"It\u0027s given by this, only here we have t instead of x."},{"Start":"04:31.630 ","End":"04:37.490","Text":"We can use the induction hypothesis to say that this is equal to this."},{"Start":"04:37.490 ","End":"04:39.864","Text":"So 1 is just 1,"},{"Start":"04:39.864 ","End":"04:42.545","Text":"and the integral of this comes out to be this."},{"Start":"04:42.545 ","End":"04:47.170","Text":"The last term will be x^n plus 1 over n plus 1,"},{"Start":"04:47.170 ","End":"04:49.055","Text":"and there\u0027s an n factorial here."},{"Start":"04:49.055 ","End":"04:51.900","Text":"Now there\u0027s a recursive definition to factorial,"},{"Start":"04:51.900 ","End":"04:54.900","Text":"but this is n plus 1 factorial,"},{"Start":"04:54.900 ","End":"04:56.750","Text":"and also I dropped this term."},{"Start":"04:56.750 ","End":"04:58.610","Text":"It swallowed up in the dot, dot,"},{"Start":"04:58.610 ","End":"05:04.300","Text":"dot just so it looks more like what we had here and it\u0027s the same but with n plus 1"},{"Start":"05:04.300 ","End":"05:11.315","Text":"instead of n. That proves that y_n is indeed given by this formula,"},{"Start":"05:11.315 ","End":"05:13.760","Text":"and that\u0027s part b."},{"Start":"05:13.760 ","End":"05:15.950","Text":"Let\u0027s go on to part c,"},{"Start":"05:15.950 ","End":"05:18.095","Text":"and the remainder; this is what it said."},{"Start":"05:18.095 ","End":"05:21.290","Text":"We have to solve the initial value problem directly and"},{"Start":"05:21.290 ","End":"05:25.127","Text":"then show that the nth approximation;"},{"Start":"05:25.127 ","End":"05:28.595","Text":"Picard approximation converges to it."},{"Start":"05:28.595 ","End":"05:30.853","Text":"The direct solution."},{"Start":"05:30.853 ","End":"05:35.585","Text":"The IVP is y\u0027 equals y, y(0)=1."},{"Start":"05:35.585 ","End":"05:37.580","Text":"We\u0027re going to use separation of variables."},{"Start":"05:37.580 ","End":"05:40.550","Text":"Write y\u0027 as dy by dx,"},{"Start":"05:40.550 ","End":"05:45.530","Text":"and then bring the ys to the left and the xs to the right,"},{"Start":"05:45.530 ","End":"05:49.535","Text":"and then this integral is natural log(y)."},{"Start":"05:49.535 ","End":"05:52.010","Text":"Sometimes we say absolute value of y,"},{"Start":"05:52.010 ","End":"05:57.340","Text":"but we know y is positive at least the part that goes through 0,"},{"Start":"05:57.340 ","End":"05:58.490","Text":"1 y is 1,"},{"Start":"05:58.490 ","End":"06:00.110","Text":"so we\u0027ll just take the positive part."},{"Start":"06:00.110 ","End":"06:06.015","Text":"Integral of 1 is x plus C. We know that when x is 0, y=1."},{"Start":"06:06.015 ","End":"06:11.930","Text":"If we plug that in here we get natural log of 1 is 0 plus C. 0 is 0 plus C,"},{"Start":"06:11.930 ","End":"06:13.520","Text":"so C is 0."},{"Start":"06:13.520 ","End":"06:18.380","Text":"If C is 0, then we just get natural log of y equals"},{"Start":"06:18.380 ","End":"06:23.209","Text":"x because the C is 0 and then we exponentiate both sides."},{"Start":"06:23.209 ","End":"06:25.985","Text":"We\u0027ve got y which is a function of x,"},{"Start":"06:25.985 ","End":"06:28.550","Text":"is equal to e^x."},{"Start":"06:28.550 ","End":"06:31.450","Text":"That\u0027s the solution to the IVP."},{"Start":"06:31.450 ","End":"06:33.235","Text":"A remark."},{"Start":"06:33.235 ","End":"06:39.995","Text":"Sometimes we check if y=0 is a solution to a differential equation because it often is."},{"Start":"06:39.995 ","End":"06:44.540","Text":"In this case, y identically equal=0 is a solution."},{"Start":"06:44.540 ","End":"06:47.600","Text":"It does satisfy y\u0027=y,"},{"Start":"06:47.600 ","End":"06:50.178","Text":"but it doesn\u0027t satisfy the initial condition,"},{"Start":"06:50.178 ","End":"06:52.565","Text":"because in that case y of 0 is 0."},{"Start":"06:52.565 ","End":"06:55.520","Text":"That\u0027s just a remark that the obvious thing is"},{"Start":"06:55.520 ","End":"06:58.460","Text":"not really a solution because it\u0027s an IVP;"},{"Start":"06:58.460 ","End":"07:01.740","Text":"initial value problem, not just an ODE."},{"Start":"07:01.870 ","End":"07:07.410","Text":"Now let\u0027s do the other 1/2 and see if we get e^x the other way."},{"Start":"07:07.410 ","End":"07:09.410","Text":"Y_n(x) is this expression,"},{"Start":"07:09.410 ","End":"07:11.645","Text":"and now we let n go to infinity."},{"Start":"07:11.645 ","End":"07:17.435","Text":"Partial sums of an infinite sequence converge to the infinite series."},{"Start":"07:17.435 ","End":"07:24.065","Text":"Anyway, this converges too and we know this is e^x for all x."},{"Start":"07:24.065 ","End":"07:26.719","Text":"This is equal to y(x),"},{"Start":"07:26.719 ","End":"07:30.290","Text":"so we got the same thing both ways directly and as"},{"Start":"07:30.290 ","End":"07:36.035","Text":"the limit of the Picard approximations."},{"Start":"07:36.035 ","End":"07:39.140","Text":"The nth Picard approximation tends to"},{"Start":"07:39.140 ","End":"07:42.725","Text":"the solution as n goes to infinity as was required,"},{"Start":"07:42.725 ","End":"07:47.190","Text":"and that concludes this example."}],"ID":26147},{"Watched":false,"Name":"Exercise 2","Duration":"4m 58s","ChapterTopicVideoID":25332,"CourseChapterTopicPlaylistID":153003,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.175","Text":"In this exercise, we\u0027re given an initial value problem as written here."},{"Start":"00:05.175 ","End":"00:09.405","Text":"We have to find the first 3 Picard approximations y_1,"},{"Start":"00:09.405 ","End":"00:11.400","Text":"y_2, and y_3."},{"Start":"00:11.400 ","End":"00:17.145","Text":"Then to write a general expression for the nth approximation."},{"Start":"00:17.145 ","End":"00:22.170","Text":"The last part is to find the limit of y_n as n goes to"},{"Start":"00:22.170 ","End":"00:27.675","Text":"infinity and show that this really is a solution to the initial value problem."},{"Start":"00:27.675 ","End":"00:30.780","Text":"Start with part a. Now, f(x,"},{"Start":"00:30.780 ","End":"00:34.200","Text":"y) is 2x 1 plus y from here."},{"Start":"00:34.200 ","End":"00:36.180","Text":"We have x naught and y naught,"},{"Start":"00:36.180 ","End":"00:39.450","Text":"both 0 from here and from here."},{"Start":"00:39.450 ","End":"00:42.005","Text":"The first step is to take"},{"Start":"00:42.005 ","End":"00:47.570","Text":"our 0 approximation is to let y naught of x be the constant y naught,"},{"Start":"00:47.570 ","End":"00:50.840","Text":"which is 0 here."},{"Start":"00:50.840 ","End":"00:53.795","Text":"The next step is to use the formula."},{"Start":"00:53.795 ","End":"00:57.755","Text":"That\u0027s the standard formula for Picard approximation."},{"Start":"00:57.755 ","End":"01:03.065","Text":"Each time we put y_n here and we get the following y_n."},{"Start":"01:03.065 ","End":"01:08.210","Text":"Let\u0027s simplify this for our case because we know x naught,"},{"Start":"01:08.210 ","End":"01:11.090","Text":"y naught, and f from here."},{"Start":"01:11.090 ","End":"01:13.295","Text":"x naught is this,"},{"Start":"01:13.295 ","End":"01:21.764","Text":"y naught is this 0 plus which we don\u0027t need and f(t) and y_n(t) from the formula,"},{"Start":"01:21.764 ","End":"01:25.580","Text":"we replace x by t and replace y by y_n."},{"Start":"01:25.580 ","End":"01:29.015","Text":"This is our recursion formula for our case."},{"Start":"01:29.015 ","End":"01:31.655","Text":"Now let\u0027s start applying it."},{"Start":"01:31.655 ","End":"01:34.670","Text":"From y naught, we\u0027re going to get y_1."},{"Start":"01:34.670 ","End":"01:36.590","Text":"y_1 from this formula,"},{"Start":"01:36.590 ","End":"01:39.770","Text":"is this, just substitute y naught here,"},{"Start":"01:39.770 ","End":"01:46.765","Text":"and y naught is 0 so 2t 1 plus 0 is just 2t."},{"Start":"01:46.765 ","End":"01:53.210","Text":"The integral of 2t is t^2 from 0 to x gives us x^2."},{"Start":"01:53.210 ","End":"01:55.295","Text":"Now we have y_1(x)."},{"Start":"01:55.295 ","End":"01:58.100","Text":"From y_1, we can get y_2,"},{"Start":"01:58.100 ","End":"02:00.245","Text":"again, substitute in this formula."},{"Start":"02:00.245 ","End":"02:02.240","Text":"We know what y_1 is."},{"Start":"02:02.240 ","End":"02:05.750","Text":"It\u0027s x^2, so y_1(t) is t^2."},{"Start":"02:05.750 ","End":"02:08.269","Text":"There are several ways of computing this integral."},{"Start":"02:08.269 ","End":"02:10.970","Text":"One way would be to substitute s=t^2."},{"Start":"02:10.970 ","End":"02:12.605","Text":"Anyway, whatever it is,"},{"Start":"02:12.605 ","End":"02:18.245","Text":"this integral comes out to be x^2 plus x^4 over 2. That\u0027s y_2."},{"Start":"02:18.245 ","End":"02:21.620","Text":"Then from y_2 we can get y_3 again,"},{"Start":"02:21.620 ","End":"02:23.300","Text":"with the same formula."},{"Start":"02:23.300 ","End":"02:29.450","Text":"Remember that y_2(x) was x^2 plus x^2 over 2 so"},{"Start":"02:29.450 ","End":"02:33.080","Text":"y_2 of t is same thing just with t. You"},{"Start":"02:33.080 ","End":"02:37.730","Text":"can write the 2 as 2 factorial and you\u0027ll see why."},{"Start":"02:37.730 ","End":"02:40.070","Text":"When we integrate this,"},{"Start":"02:40.070 ","End":"02:44.600","Text":"this part gives us t^6 over 6,"},{"Start":"02:44.600 ","End":"02:46.505","Text":"which is 3 factorial."},{"Start":"02:46.505 ","End":"02:49.175","Text":"This is what we get for y_3."},{"Start":"02:49.175 ","End":"02:52.715","Text":"In part b, when we have to find a general expression,"},{"Start":"02:52.715 ","End":"02:55.865","Text":"it\u0027s fairly clear that if this is y_3 and if you look back,"},{"Start":"02:55.865 ","End":"02:59.195","Text":"y and y_2 follow this pattern each time we add another term,"},{"Start":"02:59.195 ","End":"03:02.210","Text":"we can guess that y_n(x) is this,"},{"Start":"03:02.210 ","End":"03:03.880","Text":"but we keep going."},{"Start":"03:03.880 ","End":"03:07.130","Text":"In the numerator, we have twice the n,"},{"Start":"03:07.130 ","End":"03:09.680","Text":"because y_3 we go up to 6,"},{"Start":"03:09.680 ","End":"03:11.105","Text":"it\u0027s 2, 4, 6."},{"Start":"03:11.105 ","End":"03:15.650","Text":"It\u0027s going to be 2, 4, 6 up to 2n and in the denominator n factorial."},{"Start":"03:15.650 ","End":"03:17.630","Text":"We could prove this by induction,"},{"Start":"03:17.630 ","End":"03:22.685","Text":"but no need because when we take the limit will see that it really get a solution."},{"Start":"03:22.685 ","End":"03:25.220","Text":"We want the limit of this expression."},{"Start":"03:25.220 ","End":"03:28.880","Text":"It\u0027s fairly clear that we just keep adding terms."},{"Start":"03:28.880 ","End":"03:35.545","Text":"The limit of the partial sum goes to the infinite sum instead of from 1,"},{"Start":"03:35.545 ","End":"03:37.880","Text":"2n(x)^ 2n over n factorial."},{"Start":"03:37.880 ","End":"03:40.025","Text":"We\u0027ll just take the sum to infinity."},{"Start":"03:40.025 ","End":"03:43.055","Text":"Just stressing this by adding a plus dot, dot, dot,"},{"Start":"03:43.055 ","End":"03:44.630","Text":"meaning instead of the finite sum,"},{"Start":"03:44.630 ","End":"03:48.455","Text":"the infinite sum assuming it really does converge."},{"Start":"03:48.455 ","End":"03:51.115","Text":"But if we show that this is a solution,"},{"Start":"03:51.115 ","End":"03:54.785","Text":"then that\u0027s good enough because that\u0027s what we really want."},{"Start":"03:54.785 ","End":"03:56.840","Text":"What is this series?"},{"Start":"03:56.840 ","End":"04:02.900","Text":"Well, if you remember that e^x is this series and you replace x with x^2,"},{"Start":"04:02.900 ","End":"04:04.610","Text":"and we almost get this."},{"Start":"04:04.610 ","End":"04:07.009","Text":"We get the same thing as here,"},{"Start":"04:07.009 ","End":"04:09.245","Text":"except with a 1 plus here."},{"Start":"04:09.245 ","End":"04:12.440","Text":"What we have to do is subtract 1 from each of the x^2"},{"Start":"04:12.440 ","End":"04:16.325","Text":"and we get exactly what this limit is."},{"Start":"04:16.325 ","End":"04:21.470","Text":"The final step is to check that it really is a solution to the initial value problem so"},{"Start":"04:21.470 ","End":"04:28.175","Text":"y is this, y\u0027 is 2xe^x^2."},{"Start":"04:28.175 ","End":"04:29.720","Text":"The minus 1 gives us nothing."},{"Start":"04:29.720 ","End":"04:32.450","Text":"Here we get the inner derivative, which is 2x."},{"Start":"04:32.450 ","End":"04:36.350","Text":"This is 2x and instead of e^x^2,"},{"Start":"04:36.350 ","End":"04:38.010","Text":"we can fiddle with it."},{"Start":"04:38.010 ","End":"04:42.635","Text":"We get an e^x^2 minus 1 just by adding a 1 plus here."},{"Start":"04:42.635 ","End":"04:48.230","Text":"This is exactly y so it\u0027s 2x(1 plus y)."},{"Start":"04:48.230 ","End":"04:50.720","Text":"The initial condition is satisfied."},{"Start":"04:50.720 ","End":"04:53.225","Text":"Of course, it\u0027s just mentally check if x is 0,"},{"Start":"04:53.225 ","End":"04:58.890","Text":"e^0 is 1 minus 1 is 0. That\u0027s it. We\u0027re done."}],"ID":26149}],"Thumbnail":null,"ID":153003}]

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