WEBVTT 00:00:00.050 --> 00:00:02.330 The following content is provided under a Creative 00:00:02.330 --> 00:00:03.610 Commons license. 00:00:03.610 --> 00:00:05.980 Your support will help MIT OpenCourseWare 00:00:05.980 --> 00:00:09.495 continue to offer high quality educational resources for free. 00:00:09.495 --> 00:00:12.540 To make a donation or to view additional materials 00:00:12.540 --> 00:00:15.610 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:15.610 --> 00:00:21.170 at ocw.mit.edu. 00:00:21.170 --> 00:00:23.070 PROFESSOR: And this last little bit 00:00:23.070 --> 00:00:24.902 is something which is not yet on the Web. 00:00:24.902 --> 00:00:26.860 But, anyway, when I was walking out of the room 00:00:26.860 --> 00:00:29.740 last time, I noticed that I'd written down 00:00:29.740 --> 00:00:32.070 the wrong formula for c_1 - c_2. 00:00:32.070 --> 00:00:35.380 There's a misprint, there's a minus sign that's wrong. 00:00:35.380 --> 00:00:39.180 I claimed last time that c_1 - c_2 was +1/2. 00:00:39.180 --> 00:00:40.580 But, actually, it's -1/2. 00:00:40.580 --> 00:00:42.080 If you go through the calculation 00:00:42.080 --> 00:00:45.780 that we did with the antiderivative of sin x cos x, 00:00:45.780 --> 00:00:48.500 we get these two possible answers. 00:00:48.500 --> 00:00:52.900 And if they're to be equal, then if we just subtract them 00:00:52.900 --> 00:00:56.300 we get c_1 - c_2 + 1/2 = 0. 00:00:56.300 --> 00:01:01.740 So c_1 - c_2 = 1/2. 00:01:01.740 --> 00:01:03.930 So, those are all of the corrections. 00:01:03.930 --> 00:01:06.530 Again, everything here will be on the Web. 00:01:06.530 --> 00:01:14.660 But just wanted to make it all clear to you. 00:01:14.660 --> 00:01:15.410 So here we are. 00:01:15.410 --> 00:01:19.900 This is our last day of the second unit, Applications 00:01:19.900 --> 00:01:22.010 of Differentiation. 00:01:22.010 --> 00:01:30.210 And I have one of the most fun topics to introduce to you. 00:01:30.210 --> 00:01:32.530 Which is differential equations. 00:01:32.530 --> 00:01:35.410 Now, we have a whole course on differential equations, 00:01:35.410 --> 00:01:38.140 which is called 18.03. 00:01:38.140 --> 00:01:43.840 And so we're only going to do just a little bit. 00:01:43.840 --> 00:01:52.600 But I'm going to teach you one technique. 00:01:52.600 --> 00:01:58.250 Which fits in precisely with what we've been doing already. 00:01:58.250 --> 00:02:05.280 Which is differentials. 00:02:05.280 --> 00:02:08.480 The first and simplest kind of differential equation 00:02:08.480 --> 00:02:12.770 is the rate of change of x with respect to y 00:02:12.770 --> 00:02:16.770 is equal to some function f(x). 00:02:16.770 --> 00:02:19.020 Now, that's a perfectly good differential equation. 00:02:19.020 --> 00:02:21.420 And we already discussed last time 00:02:21.420 --> 00:02:26.204 that the solution, that is, the function y, 00:02:26.204 --> 00:02:27.620 is going to be the antiderivative, 00:02:27.620 --> 00:02:33.100 or the integral, of x. 00:02:33.100 --> 00:02:35.880 Now, for the purposes of today, we're 00:02:35.880 --> 00:02:40.597 going to consider this problem to be solved. 00:02:40.597 --> 00:02:41.930 That is, you can always do this. 00:02:41.930 --> 00:02:44.200 You can always take antiderivatives. 00:02:44.200 --> 00:02:51.560 And for our purposes now, that is for now, 00:02:51.560 --> 00:03:08.380 we only have one technique to find antiderivatives. 00:03:08.380 --> 00:03:15.170 And that's called substitution. 00:03:15.170 --> 00:03:18.750 It has a very small variant, which 00:03:18.750 --> 00:03:27.940 we called advanced guessing. 00:03:27.940 --> 00:03:29.820 And that works just as well. 00:03:29.820 --> 00:03:32.490 And that's basically all that you'll ever need to do. 00:03:32.490 --> 00:03:36.457 As a practical matter, these are the ones you'll face for now. 00:03:36.457 --> 00:03:38.540 Ones that you can actually see what the answer is, 00:03:38.540 --> 00:03:42.640 or you'll have to make a substitution. 00:03:42.640 --> 00:03:48.120 Now, the first tricky example, or the first maybe interesting 00:03:48.120 --> 00:03:50.190 example of a differential equation, 00:03:50.190 --> 00:04:00.750 which I'll call Example 2, is going to be the following. 00:04:00.750 --> 00:04:07.510 d/dx + x acting on y is equal to 0. 00:04:07.510 --> 00:04:10.580 So that's our first differential equation that 00:04:10.580 --> 00:04:12.780 were going to try to solve. 00:04:12.780 --> 00:04:18.630 Apart from this standard antiderivative approach. 00:04:18.630 --> 00:04:24.350 This operation here has a name. 00:04:24.350 --> 00:04:27.490 This actually has a name, it's called the annihilation 00:04:27.490 --> 00:04:33.460 operator. 00:04:33.460 --> 00:04:43.780 And it's called that in quantum mechanics. 00:04:43.780 --> 00:04:45.950 And there's a corresponding creation operator 00:04:45.950 --> 00:04:50.780 where you change the sign from plus to minus. 00:04:50.780 --> 00:04:53.660 And this is one of the simplest differential equations. 00:04:53.660 --> 00:04:55.550 The reason why it's studied in quantum 00:04:55.550 --> 00:04:58.360 mechanics all it that it has very simple solutions 00:04:58.360 --> 00:05:00.500 that you can just write out. 00:05:00.500 --> 00:05:02.920 So we're going to solve this equation. 00:05:02.920 --> 00:05:05.680 It's the one that governs the ground state 00:05:05.680 --> 00:05:08.670 of the harmonic oscillator. 00:05:08.670 --> 00:05:10.790 So it has a lot of fancy words associated with it, 00:05:10.790 --> 00:05:12.706 but it's a fairly simple differential equation 00:05:12.706 --> 00:05:14.430 and it works perfectly by the method 00:05:14.430 --> 00:05:17.270 that we're going to propose. 00:05:17.270 --> 00:05:20.820 So the first step in this solution 00:05:20.820 --> 00:05:26.800 is just to rewrite the equation by putting 00:05:26.800 --> 00:05:29.090 one of the terms on the right-hand side. 00:05:29.090 --> 00:05:32.300 So this is dy/dx = -xy. 00:05:35.700 --> 00:05:37.700 Now, here is where you see the difference 00:05:37.700 --> 00:05:41.210 between this type of equation and the previous type. 00:05:41.210 --> 00:05:43.130 In the previous equation, we just 00:05:43.130 --> 00:05:45.660 had a function of x on the right-hand side. 00:05:45.660 --> 00:05:50.260 But here, the rate of change depends on both x and y. 00:05:50.260 --> 00:05:51.830 So it's not clear at all that we can 00:05:51.830 --> 00:05:55.110 solve this kind of equation. 00:05:55.110 --> 00:05:57.370 But there is a remarkable trick which 00:05:57.370 --> 00:05:59.420 works very well in this case. 00:05:59.420 --> 00:06:02.710 Which is to use multiplication. 00:06:02.710 --> 00:06:06.340 To use this idea of differential that we talked about last time. 00:06:06.340 --> 00:06:14.700 Namely, we divide by y and multiply by dx. 00:06:14.700 --> 00:06:17.660 So now we've separated the equation. 00:06:17.660 --> 00:06:20.750 We've separated out the differentials. 00:06:20.750 --> 00:06:22.960 And what's going to be important for us 00:06:22.960 --> 00:06:27.410 is that the left-hand side is expressed solely in terms of y 00:06:27.410 --> 00:06:30.170 and the right-hand side is expressed solely in terms of x. 00:06:30.170 --> 00:06:33.400 And we'll go through this in careful detail. 00:06:33.400 --> 00:06:36.410 So now, the idea is if you've set up the equation in terms 00:06:36.410 --> 00:06:39.680 of differentials as opposed to ratios of differentials, 00:06:39.680 --> 00:06:44.680 or rates of change, now I can use Leibniz's notation 00:06:44.680 --> 00:06:47.600 and integrate these differentials. 00:06:47.600 --> 00:06:55.420 Take their antiderivatives. 00:06:55.420 --> 00:07:02.070 And we know what each of these is. 00:07:02.070 --> 00:07:17.920 Namely, the left-hand side is just-- Ah, well, that's tough. 00:07:17.920 --> 00:07:24.250 OK. 00:07:24.250 --> 00:07:29.410 I had an au pair who actually did a lot of Tae Kwan Do. 00:07:29.410 --> 00:07:32.870 She could definitely defeat any of you in any encounter, 00:07:32.870 --> 00:07:34.940 I promise. 00:07:34.940 --> 00:07:35.440 OK. 00:07:35.440 --> 00:07:37.870 Anyway. 00:07:37.870 --> 00:07:38.970 So, let's go back. 00:07:38.970 --> 00:07:41.970 We want to take the antiderivative of this. 00:07:41.970 --> 00:07:48.130 So remember, this is the function 00:07:48.130 --> 00:07:50.140 whose derivative is 1/y. 00:07:50.140 --> 00:07:52.320 And now there's a slight novelty here. 00:07:52.320 --> 00:07:54.860 Here we're differentiating the variable as x, 00:07:54.860 --> 00:07:58.220 and here we're differentiating the variable as y. 00:07:58.220 --> 00:08:02.860 So the antiderivative here is ln y. 00:08:02.860 --> 00:08:07.970 And the antiderivative on the other side is -x^2 / 2. 00:08:07.970 --> 00:08:10.400 And they differ by a constant. 00:08:10.400 --> 00:08:17.620 So we have this relationship here. 00:08:17.620 --> 00:08:19.660 Now, that's almost the end of the story. 00:08:19.660 --> 00:08:23.100 We have to exponentiate to express y in terms of x. 00:08:23.100 --> 00:08:26.970 So, e^(ln y) = e^(-x^2 / 2) + c. 00:08:29.880 --> 00:08:36.010 And now I can rewrite that as y is equal to-- I'll write as A 00:08:36.010 --> 00:08:40.120 e^(-x^2 / 2), where A = e^c. 00:08:43.870 --> 00:08:47.940 And incidentally, we're just taking the case y positive 00:08:47.940 --> 00:08:48.440 here. 00:08:48.440 --> 00:08:50.920 We'll talk about what happens when 00:08:50.920 --> 00:08:55.590 y is negative in a few minutes. 00:08:55.590 --> 00:08:57.270 So here's the answer to the question, 00:08:57.270 --> 00:09:02.040 almost, except for this fact that I picked out y positive. 00:09:02.040 --> 00:09:09.890 Really, the solution is y is equal to any multiple 00:09:09.890 --> 00:09:11.780 of e^(-x^2 / 2). 00:09:11.780 --> 00:09:19.310 Any constant a; a positive, negative, or 0. 00:09:19.310 --> 00:09:22.240 Any constant will do. 00:09:22.240 --> 00:09:24.640 And we should double-check that to make sure. 00:09:24.640 --> 00:09:32.930 If you take d/dx of y right, that's going to be a d/dx 00:09:32.930 --> 00:09:33.950 e^(-x^2 / 2). 00:09:36.570 --> 00:09:38.220 And now by the chain rule, you can 00:09:38.220 --> 00:09:42.080 see that this is a times the factor of -x, that's 00:09:42.080 --> 00:09:45.080 the derivative of the exponent, with respect to x, 00:09:45.080 --> 00:09:48.190 times the exponential. 00:09:48.190 --> 00:09:50.770 And now you just rearrange that. 00:09:50.770 --> 00:09:54.430 That's -xy. 00:09:54.430 --> 00:09:55.860 So it does check. 00:09:55.860 --> 00:09:57.360 These are solutions to the equation. 00:09:57.360 --> 00:09:58.560 The a didn't matter. 00:09:58.560 --> 00:10:05.370 It didn't matter whether it was positive or negative. 00:10:05.370 --> 00:10:08.750 This function is known as the normal distribution, 00:10:08.750 --> 00:10:11.310 so it fits beautifully with a lot of probability 00:10:11.310 --> 00:10:15.710 and probabilistic interpretation of quantum mechanics. 00:10:15.710 --> 00:10:22.630 This is sort of where the particle is. 00:10:22.630 --> 00:10:25.380 So next, what I'd like to do is just 00:10:25.380 --> 00:10:30.920 go through the method in general and point out when it works. 00:10:30.920 --> 00:10:32.960 And then I'll make a few comments just 00:10:32.960 --> 00:10:37.650 to make sure that you understand the technicalities of dealing 00:10:37.650 --> 00:10:39.590 with constants and so forth. 00:10:39.590 --> 00:10:42.385 So, first of all, the general method of separation 00:10:42.385 --> 00:10:53.830 of variables. 00:10:53.830 --> 00:10:55.360 And here's when it works. 00:10:55.360 --> 00:10:58.470 It works when you're faced with a differential 00:10:58.470 --> 00:11:03.410 equation of the form f(x) g(y). 00:11:03.410 --> 00:11:05.920 That's the situation that we had. 00:11:05.920 --> 00:11:08.150 And I'll just illustrate that. 00:11:08.150 --> 00:11:09.820 Just to remind you here. 00:11:09.820 --> 00:11:11.650 Here's our equation. 00:11:11.650 --> 00:11:13.120 It's in that form. 00:11:13.120 --> 00:11:22.190 And the function f(x) is -x, and the function g(y) is just y. 00:11:22.190 --> 00:11:26.670 And now, the way the method works is, this separation step. 00:11:26.670 --> 00:11:30.370 From here to here, this is the key step. 00:11:30.370 --> 00:11:35.070 This is the only conceptually remarkable step, 00:11:35.070 --> 00:11:37.360 which all has to do with the fact 00:11:37.360 --> 00:11:40.160 that Leibniz fixed his notations up so that this 00:11:40.160 --> 00:11:42.190 works perfectly. 00:11:42.190 --> 00:11:48.320 And so that involves taking the y, so dividing by g(y), 00:11:48.320 --> 00:11:53.170 and multiplying by dx, it's comfortable 00:11:53.170 --> 00:11:55.540 because it feels like ordinary arithmetic, 00:11:55.540 --> 00:11:59.430 even though these are differentials. 00:11:59.430 --> 00:12:02.710 And then, we just antidifferentiate. 00:12:02.710 --> 00:12:10.560 So we have a function, H, which is the integral of dy / g(y), 00:12:10.560 --> 00:12:12.440 and we have another function which 00:12:12.440 --> 00:12:15.910 is F. Note they are functions of completely different variables 00:12:15.910 --> 00:12:16.740 here. 00:12:16.740 --> 00:12:20.850 Integral of f(x) dx. 00:12:20.850 --> 00:12:23.864 Now, in our example we did that. 00:12:23.864 --> 00:12:25.530 We carried out this antidifferentiation, 00:12:25.530 --> 00:12:29.680 and this function turned out to be ln y, 00:12:29.680 --> 00:12:39.630 and this function turned out to be -x^2 / 2. 00:12:39.630 --> 00:12:42.350 And then we write the relationship. 00:12:42.350 --> 00:12:45.790 Which is that if these are both antiderivatives 00:12:45.790 --> 00:12:48.890 of the same thing, then they have to differ by a constant. 00:12:48.890 --> 00:12:55.870 Or, in other words, H(y) has to equal to F(x) + c. 00:12:55.870 --> 00:13:10.130 Where c is constant. 00:13:10.130 --> 00:13:15.150 Now, notice that this kind of equation 00:13:15.150 --> 00:13:20.450 is what we call an implicit equation. 00:13:20.450 --> 00:13:23.930 It's not quite a formula for y, directly. 00:13:23.930 --> 00:13:26.330 It defines y implicitly. 00:13:26.330 --> 00:13:29.630 That's that top line up here. 00:13:29.630 --> 00:13:33.020 That's the implicit equation. 00:13:33.020 --> 00:13:35.110 In order to make it an explicit equation, which 00:13:35.110 --> 00:13:38.780 is what is underneath, what I have to do is take the inverse. 00:13:38.780 --> 00:13:41.260 So I write it as y = H^(-1)(F(x) + c). 00:13:45.080 --> 00:13:48.140 Now, in real life the calculus part is often pretty easy. 00:13:48.140 --> 00:13:52.620 And it can be quite messy to do the inverse operation. 00:13:52.620 --> 00:13:55.850 So sometimes we just leave it alone in the implicit form. 00:13:55.850 --> 00:13:58.250 But it's also satisfying, sometimes, 00:13:58.250 --> 00:14:09.290 to write it in the final form here. 00:14:09.290 --> 00:14:14.160 Now I've got to give you a few little pieces of commentary 00:14:14.160 --> 00:14:16.640 before-- For those of you walked in a little bit late, 00:14:16.640 --> 00:14:25.660 this will all be on the Web. 00:14:25.660 --> 00:14:31.180 So just a few pieces of commentary. 00:14:31.180 --> 00:14:36.230 So if you like, some remarks. 00:14:36.230 --> 00:14:51.140 The first remark is that I could have written natural log 00:14:51.140 --> 00:14:55.850 of absolute y is equal to -x^2 / 2 + c. 00:14:58.390 --> 00:15:01.871 We learned last time that the antiderivative works also 00:15:01.871 --> 00:15:02.870 for the negative values. 00:15:02.870 --> 00:15:08.490 So this would work for y not equal to 0. 00:15:08.490 --> 00:15:10.900 Both for positive and negative values. 00:15:10.900 --> 00:15:13.990 And you can see that that would have captured most 00:15:13.990 --> 00:15:15.580 of the rest of the solution. 00:15:15.580 --> 00:15:21.870 Namely, |y| would be equal to A e^(-x^2 / 2), 00:15:21.870 --> 00:15:24.800 by the same reasoning as before. 00:15:24.800 --> 00:15:29.052 And then that would mean that y was plus or minus A e^(-x^2 / 00:15:29.052 --> 00:15:34.800 2), which is really just what we got. 00:15:34.800 --> 00:15:38.120 Because, in fact, I didn't bother with this. 00:15:38.120 --> 00:15:40.087 Because actually in most-- and the reason why 00:15:40.087 --> 00:15:42.420 I'm going through this, by the way, carefully this time, 00:15:42.420 --> 00:15:44.878 is that you're going to be faced with this very frequently. 00:15:44.878 --> 00:15:47.290 The exponential function comes up all the time. 00:15:47.290 --> 00:15:49.710 And so, therefore, you want to be completely comfortable 00:15:49.710 --> 00:15:52.300 dealing with it. 00:15:52.300 --> 00:15:54.510 So this time I had the positive A, 00:15:54.510 --> 00:15:56.540 while the negative A fits in either this way, 00:15:56.540 --> 00:15:57.740 or I can throw it in. 00:15:57.740 --> 00:15:59.970 Because I know that that's going to work that way. 00:15:59.970 --> 00:16:03.370 But of course, I double-checked to be confident. 00:16:03.370 --> 00:16:07.380 Now, this still leaves out one value. 00:16:07.380 --> 00:16:12.295 So, this still leaves out-- So, if you like, 00:16:12.295 --> 00:16:14.420 what I have here now is a is equal to plus or minus 00:16:14.420 --> 00:16:18.540 capital A. The capital A one being the positive one. 00:16:18.540 --> 00:16:20.920 But this still leaves out one case. 00:16:20.920 --> 00:16:23.480 Which is y = 0. 00:16:23.480 --> 00:16:27.180 Which is an extremely boring solution, but nevertheless 00:16:27.180 --> 00:16:28.760 a solution to this problem. 00:16:28.760 --> 00:16:32.330 If you plug in 0 here for y, you get 0. 00:16:32.330 --> 00:16:34.600 If you plug in 0 here for y, you get 00:16:34.600 --> 00:16:36.520 that these two sides are equal. 00:16:36.520 --> 00:16:38.080 0 = 0. 00:16:38.080 --> 00:16:40.630 Not a very interesting answer to the question. 00:16:40.630 --> 00:16:42.290 But it's still an answer. 00:16:42.290 --> 00:16:43.930 And so y = 0 is left out.. 00:16:43.930 --> 00:16:52.520 Well, that's not so surprising that we missed that solution. 00:16:52.520 --> 00:16:56.360 Because in the process of carrying out these operations, 00:16:56.360 --> 00:16:58.250 I divided by y. 00:16:58.250 --> 00:17:02.290 I did that right here. 00:17:02.290 --> 00:17:03.600 So, that's what happens. 00:17:03.600 --> 00:17:05.941 If you're going to do various non-linear operations, 00:17:05.941 --> 00:17:08.190 in particular, if you're going to divide by something, 00:17:08.190 --> 00:17:10.564 if it happens to be 0 you're going to miss that solution. 00:17:10.564 --> 00:17:13.860 You might have problems with that solution. 00:17:13.860 --> 00:17:16.870 But we have to live with that because we want to get ahead. 00:17:16.870 --> 00:17:20.520 And we want to get the formulas for various solutions. 00:17:20.520 --> 00:17:22.840 So that's the first remark that I wanted to make. 00:17:22.840 --> 00:17:30.340 And now, the second one is almost related 00:17:30.340 --> 00:17:33.380 to what I was just discussing right here. 00:17:33.380 --> 00:17:37.240 That I'm erasing. 00:17:37.240 --> 00:17:39.420 And that's the following. 00:17:39.420 --> 00:17:52.100 I could have also written ln y + c_1 = -x^2 / 2 + c_2. 00:17:52.100 --> 00:17:54.300 Where c_1 and c_2 are different constants. 00:17:54.300 --> 00:17:57.210 When I'm faced with this antidifferentiation, 00:17:57.210 --> 00:17:59.460 I just taught you last time, that you want 00:17:59.460 --> 00:18:02.230 to have an arbitrary constant. 00:18:02.230 --> 00:18:06.760 Here and there, in both slots. 00:18:06.760 --> 00:18:09.480 So I perfectly well could have written this down. 00:18:09.480 --> 00:18:17.990 But notice that I can rewrite this as ln y = -x^2 / 2 + c_2 - 00:18:17.990 --> 00:18:20.190 c_1. 00:18:20.190 --> 00:18:22.250 I can subtract. 00:18:22.250 --> 00:18:25.385 And then, if I just combine these two guys together 00:18:25.385 --> 00:18:29.060 and name them c, I have a different constant. 00:18:29.060 --> 00:18:32.010 In other words, it's superfluous and redundant 00:18:32.010 --> 00:18:35.060 to have two arbitrary constants here, 00:18:35.060 --> 00:18:38.260 because they can always be combined into one. 00:18:38.260 --> 00:18:47.010 So two constants are superfluous. 00:18:47.010 --> 00:18:54.430 Can always be combined. 00:18:54.430 --> 00:18:56.490 So we just never do it this first way. 00:18:56.490 --> 00:19:05.260 It's just extra writing, it's a waste of time. 00:19:05.260 --> 00:19:08.120 There's one other subtle remark, which you won't actually 00:19:08.120 --> 00:19:10.020 appreciate until you've done several problems 00:19:10.020 --> 00:19:11.280 in this direction. 00:19:11.280 --> 00:19:14.490 Which is that the constant appears 00:19:14.490 --> 00:19:18.900 additive here, in this first solution to the problem. 00:19:18.900 --> 00:19:22.590 But when I do this nonlinear operation of exponentiation, 00:19:22.590 --> 00:19:26.390 it now becomes multiplicative constant. 00:19:26.390 --> 00:19:31.006 And so, in general, there's a free constant somewhere 00:19:31.006 --> 00:19:31.630 in the problem. 00:19:31.630 --> 00:19:35.490 But it's not always an additive constant. 00:19:35.490 --> 00:19:38.330 It's only an additive constant right at the first step 00:19:38.330 --> 00:19:39.890 when you take the antiderivative. 00:19:39.890 --> 00:19:42.265 And then after that, when you do all your other nonlinear 00:19:42.265 --> 00:19:45.440 operations, it can turn into anything at all. 00:19:45.440 --> 00:19:47.980 So you should always expect it to be something slightly more 00:19:47.980 --> 00:19:49.563 interesting than an additive constant. 00:19:49.563 --> 00:19:59.060 Although occasionally it stays an additive constant. 00:19:59.060 --> 00:20:01.180 The last little bit of commentary 00:20:01.180 --> 00:20:06.010 that I want to make just goes back to the original problem 00:20:06.010 --> 00:20:06.810 here. 00:20:06.810 --> 00:20:09.680 Which is right here. 00:20:09.680 --> 00:20:11.190 The example 1. 00:20:11.190 --> 00:20:14.490 And I want to solve it, even though this is simpleminded. 00:20:14.490 --> 00:20:21.490 But Example 1 via separation. 00:20:21.490 --> 00:20:25.290 So that you see our variables. 00:20:25.290 --> 00:20:28.440 So that you see what it does. 00:20:28.440 --> 00:20:34.230 The situation is this. 00:20:34.230 --> 00:20:35.890 And the separation just means you 00:20:35.890 --> 00:20:38.570 put the dx on the other side. 00:20:38.570 --> 00:20:44.030 So this is dy = f(x) dx. 00:20:44.030 --> 00:20:54.680 And then we integrate. 00:20:54.680 --> 00:20:58.170 And the antiderivative of dy is just y. 00:20:58.170 --> 00:21:03.490 So this is the solution to the problem. 00:21:03.490 --> 00:21:05.170 And it's just what we wrote before; 00:21:05.170 --> 00:21:07.480 it's just a funny notation. 00:21:07.480 --> 00:21:19.480 And it comes to the same thing as the antiderivative. 00:21:19.480 --> 00:21:23.240 OK, so now we're going to go on to a trickier problem. 00:21:23.240 --> 00:21:24.090 A trickier example. 00:21:24.090 --> 00:21:26.420 We need one or two more just to get some practice 00:21:26.420 --> 00:21:29.330 with this method. 00:21:29.330 --> 00:21:31.730 Everybody happy so far? 00:21:31.730 --> 00:21:32.240 Question. 00:21:32.240 --> 00:21:53.472 STUDENT: [INAUDIBLE] 00:21:53.472 --> 00:21:55.180 PROFESSOR: So, the question is, how do we 00:21:55.180 --> 00:21:58.150 deal with this ambiguity. 00:21:58.150 --> 00:22:03.020 I'm summarizing very, very, briefly what I heard. 00:22:03.020 --> 00:22:06.530 Well, you know, sometimes a > 0, sometimes a < 0, 00:22:06.530 --> 00:22:07.550 sometimes it's not. 00:22:07.550 --> 00:22:12.810 So there's a name for this guy. 00:22:12.810 --> 00:22:20.136 Which is that this is what's called the general solution. 00:22:20.136 --> 00:22:22.010 In other words, the whole family of solutions 00:22:22.010 --> 00:22:24.460 is the answer to the question. 00:22:24.460 --> 00:22:28.020 Now, it could be that you're given extra information. 00:22:28.020 --> 00:22:31.760 If you're given extra information, that might be, 00:22:31.760 --> 00:22:33.527 and this is very typical in such problems, 00:22:33.527 --> 00:22:35.610 you have the rate of change of the function, which 00:22:35.610 --> 00:22:36.510 is what we've given. 00:22:36.510 --> 00:22:39.780 But you might also have the place where it starts. 00:22:39.780 --> 00:22:44.579 Which would be, say, it starts at 3. 00:22:44.579 --> 00:22:46.620 Now, if you have that extra piece of information, 00:22:46.620 --> 00:22:50.670 then you can nail down exactly which function it is. 00:22:50.670 --> 00:22:52.420 If you do that, if you plug in 3, 00:22:52.420 --> 00:22:57.860 you see that a times e^(-0^2 / 2) is equal to 3. 00:22:57.860 --> 00:23:00.300 So a = 3. 00:23:00.300 --> 00:23:02.720 And the answer is y = 3e^(-x^2 / 2). 00:23:06.140 --> 00:23:08.940 And similarly, if it's negative, if it starts out negative, 00:23:08.940 --> 00:23:10.100 it'll stay negative. 00:23:10.100 --> 00:23:11.100 For instance. 00:23:11.100 --> 00:23:14.870 If it starts out 0, it'll stay 0, this particular function 00:23:14.870 --> 00:23:16.100 here. 00:23:16.100 --> 00:23:18.110 So the answer to your question is how 00:23:18.110 --> 00:23:19.700 you deal with the ambiguity. 00:23:19.700 --> 00:23:23.620 The answer is that you simply say what the solution is. 00:23:23.620 --> 00:23:25.200 And the solution is not one function, 00:23:25.200 --> 00:23:26.324 it's a family of functions. 00:23:26.324 --> 00:23:30.340 It's a list and you have to have what's known as a parameter. 00:23:30.340 --> 00:23:32.440 And that parameter gets nailed down 00:23:32.440 --> 00:23:35.240 if you tell me more information about the function. 00:23:35.240 --> 00:23:37.654 Not the rate of change, but something about the values 00:23:37.654 --> 00:23:38.320 of the function. 00:23:46.620 --> 00:23:53.660 STUDENT: [INAUDIBLE] 00:23:53.660 --> 00:23:55.720 PROFESSOR: The general solution is this solution. 00:23:55.720 --> 00:23:56.553 STUDENT: [INAUDIBLE] 00:23:56.553 --> 00:23:58.176 PROFESSOR: And I'm showing you here 00:23:58.176 --> 00:24:00.300 that you could get to most of the general solution. 00:24:00.300 --> 00:24:04.570 There's one thing that's left out, namely the case a = 0. 00:24:04.570 --> 00:24:08.120 So, in other words, I would not go through this method. 00:24:08.120 --> 00:24:10.690 I would only use this, which is simpler. 00:24:10.690 --> 00:24:13.590 But then I have to understand that I haven't gotten 00:24:13.590 --> 00:24:15.410 all of the solutions this way. 00:24:15.410 --> 00:24:19.325 I'm going to need to throw in all the rest of the solutions. 00:24:19.325 --> 00:24:20.950 So in the back of your head, you always 00:24:20.950 --> 00:24:23.779 have to have something like this in mind. 00:24:23.779 --> 00:24:25.570 So that you can generate all the solutions. 00:24:25.570 --> 00:24:28.510 This is very suggestive, right? 00:24:28.510 --> 00:24:31.840 The restriction, it turns that the restriction A > 0 is 00:24:31.840 --> 00:24:40.660 superfluous, is unnecessary. 00:24:40.660 --> 00:24:46.180 But that, we only get by further thought and by checking. 00:24:46.180 --> 00:24:46.890 Another question? 00:24:46.890 --> 00:24:47.389 Over here. 00:24:47.389 --> 00:24:52.210 STUDENT: [INAUDIBLE] 00:24:52.210 --> 00:24:54.630 PROFESSOR: The aim of differential equations 00:24:54.630 --> 00:24:55.600 is to solve them. 00:24:55.600 --> 00:24:59.372 Just as with algebraic equations. 00:24:59.372 --> 00:25:01.330 Usually, differential equations are telling you 00:25:01.330 --> 00:25:04.300 something about the balance between an acceleration 00:25:04.300 --> 00:25:05.980 and a velocity. 00:25:05.980 --> 00:25:09.840 If you have a falling object, it might have a resistance. 00:25:09.840 --> 00:25:11.210 It's telling you something. 00:25:11.210 --> 00:25:13.700 So, actually, sometimes in applied problems, 00:25:13.700 --> 00:25:16.450 formulating what differential equation describe 00:25:16.450 --> 00:25:18.310 this situation is very important. 00:25:18.310 --> 00:25:21.910 In order to see that that's the right thing, 00:25:21.910 --> 00:25:24.330 you have to have solved it to see that it fits 00:25:24.330 --> 00:25:25.780 the data that you're getting. 00:25:25.780 --> 00:25:28.570 STUDENT: [INAUDIBLE] 00:25:28.570 --> 00:25:31.720 PROFESSOR: The question is, can you solve for x instead of y. 00:25:31.720 --> 00:25:36.250 The answer is, sure. 00:25:36.250 --> 00:25:38.356 That's the same thing as-- so that 00:25:38.356 --> 00:25:40.230 would be the inverse function of the function 00:25:40.230 --> 00:25:42.520 that we're officially looking for. 00:25:42.520 --> 00:25:43.960 But yeah, it's legal. 00:25:43.960 --> 00:25:46.150 In other words, oftentimes we're stuck 00:25:46.150 --> 00:25:48.835 with just the implicit, some implicit formula 00:25:48.835 --> 00:25:51.540 and sometimes we're stuck with a formula x is a function of y 00:25:51.540 --> 00:25:54.730 versus y is a function of x. 00:25:54.730 --> 00:25:57.850 The way in which the function is specified 00:25:57.850 --> 00:26:00.780 is something that can be complicated. 00:26:00.780 --> 00:26:02.810 As you'll see in the next example, 00:26:02.810 --> 00:26:04.760 it's not necessarily the best thing 00:26:04.760 --> 00:26:07.530 to think about a function-- y as a function of x. 00:26:07.530 --> 00:26:12.170 Well, in the fourth example. 00:26:12.170 --> 00:26:27.000 Alright, we're going to go on and do our next example here. 00:26:27.000 --> 00:26:32.440 So the third example is going to be taken 00:26:32.440 --> 00:26:36.090 as a kind of geometry problem. 00:26:36.090 --> 00:26:38.990 I'll draw a picture of it. 00:26:38.990 --> 00:26:44.180 Suppose you have a curve with the following property. 00:26:44.180 --> 00:26:50.730 If you take a point on the curve, and you take the ray, 00:26:50.730 --> 00:26:56.224 you take the ray from the origin to the curve, well, that's not 00:26:56.224 --> 00:26:57.390 going to be one that I want. 00:26:57.390 --> 00:27:00.350 I think I'm going to want something which is steeper. 00:27:00.350 --> 00:27:02.130 Because what I'm going to insist is 00:27:02.130 --> 00:27:09.050 that the tangent line be twice as steep as the ray 00:27:09.050 --> 00:27:10.490 from the origin. 00:27:10.490 --> 00:27:19.600 So, in other words, slope of tangent line 00:27:19.600 --> 00:27:31.540 equals twice slope of ray from origin. 00:27:31.540 --> 00:27:34.110 So the slope of this orange line is twice 00:27:34.110 --> 00:27:39.410 the slope of the pink line. 00:27:39.410 --> 00:27:41.240 Now, these kinds of geometric problems 00:27:41.240 --> 00:27:48.700 can be written very succinctly with differential equations. 00:27:48.700 --> 00:27:51.530 Namely, it's just the following. dy / dx, 00:27:51.530 --> 00:27:55.340 that's the slope of the tangent line, is equal to, 00:27:55.340 --> 00:27:58.030 well remember what the slope of this ray is, 00:27:58.030 --> 00:28:00.700 if this point-- I need a notation. 00:28:00.700 --> 00:28:04.520 At this point is (x, y) which is a point on the curve. 00:28:04.520 --> 00:28:07.860 So the slope of this pink line is what? 00:28:07.860 --> 00:28:09.650 STUDENT: [INAUDIBLE] 00:28:09.650 --> 00:28:12.610 PROFESSOR: y/x. 00:28:12.610 --> 00:28:20.810 So if it's twice it, there's the equation. 00:28:20.810 --> 00:28:28.040 OK, now, we only have one method for solving these equations. 00:28:28.040 --> 00:28:29.890 So let's use it. 00:28:29.890 --> 00:28:31.620 It says to separate variables. 00:28:31.620 --> 00:28:41.000 So I write dy / y here, is equal to 2 dx / x. 00:28:41.000 --> 00:28:42.530 That's the basic separation. 00:28:42.530 --> 00:28:47.990 That's the procedure that we're always going to use. 00:28:47.990 --> 00:28:54.640 And now if I integrate that, I find 00:28:54.640 --> 00:29:03.250 that on the right-hand side I have the logarithm of y. 00:29:03.250 --> 00:29:05.380 And on the left-hand-- Sorry, on the left-hand side 00:29:05.380 --> 00:29:06.590 I have the logarithm of y. 00:29:06.590 --> 00:29:10.500 On the right-hand side, I have twice the logarithm 00:29:10.500 --> 00:29:20.150 of x, plus a constant. 00:29:20.150 --> 00:29:27.330 So let's see what happens to this example. 00:29:27.330 --> 00:29:29.846 This is an implicit equation, and of course we 00:29:29.846 --> 00:29:31.970 have the problems of the plus or minus signs, which 00:29:31.970 --> 00:29:38.070 I'm not going to worry about until later. 00:29:38.070 --> 00:29:40.320 So let's exponentiate and see what happens. 00:29:40.320 --> 00:29:43.600 We get e^(ln y) = e^(2 ln x + c). 00:29:47.340 --> 00:29:51.940 So, again, this is y on the left-hand side. 00:29:51.940 --> 00:29:54.010 And on the right-hand side, if you think about it 00:29:54.010 --> 00:29:55.770 for a second, it's (e^(ln x))^2. 00:29:59.050 --> 00:30:00.370 Which is x^2. 00:30:00.370 --> 00:30:02.680 So this is x^2, and then there's an e^c. 00:30:02.680 --> 00:30:06.390 So that's another one of these A factors here. 00:30:06.390 --> 00:30:13.240 A = e^c. 00:30:13.240 --> 00:30:20.160 So the answer is, well, I'll draw the picture. 00:30:20.160 --> 00:30:22.530 And I'm going to cheat as I did before. 00:30:22.530 --> 00:30:24.550 We skipped the case y negative. 00:30:24.550 --> 00:30:30.236 We really only did the case y positive, so far. 00:30:30.236 --> 00:30:31.860 But if you think about it for a second, 00:30:31.860 --> 00:30:33.490 and we'll check it in a second, you're 00:30:33.490 --> 00:30:36.390 going to get all of these parabolas here. 00:30:36.390 --> 00:30:40.970 So the solution is this family of functions. 00:30:40.970 --> 00:30:44.330 And they can be bending down. 00:30:44.330 --> 00:30:45.660 As well as up. 00:30:45.660 --> 00:30:48.140 So these are the solutions to this equation. 00:30:48.140 --> 00:30:50.410 Every single one of these curves has the property 00:30:50.410 --> 00:30:52.750 that if you pick a point on it, the tangent line 00:30:52.750 --> 00:30:58.050 has twice the slope of the ray to the origin. 00:30:58.050 --> 00:31:01.840 And the formula, if you like, of the general solution is y = 00:31:01.840 --> 00:31:08.960 ax^2, a is any constant. 00:31:08.960 --> 00:31:09.460 Question? 00:31:09.460 --> 00:31:21.844 STUDENT: [INAUDIBLE] 00:31:21.844 --> 00:31:22.510 PROFESSOR: Yeah. 00:31:22.510 --> 00:31:28.960 So again - so first of all, so there 00:31:28.960 --> 00:31:30.110 are two approaches to this. 00:31:30.110 --> 00:31:32.900 One is to check it, and make sure that it's right. 00:31:32.900 --> 00:31:35.140 When a formula works for some family of values, 00:31:35.140 --> 00:31:36.740 sometimes it works for others. 00:31:36.740 --> 00:31:39.650 But another one is to realize that these things will usually 00:31:39.650 --> 00:31:40.970 work out this way. 00:31:40.970 --> 00:31:45.459 Because in this argument here, I allow the absolute value. 00:31:45.459 --> 00:31:47.750 And that would have been a perfectly legal thing for me 00:31:47.750 --> 00:31:48.250 to do. 00:31:48.250 --> 00:31:51.220 I could have put in absolute values here. 00:31:51.220 --> 00:31:55.690 In which case, I would've gotten that the absolute value of this 00:31:55.690 --> 00:31:56.890 was equal to that. 00:31:56.890 --> 00:32:02.370 And now you see I've covered the plus and minus cases. 00:32:02.370 --> 00:32:03.880 So it's that same idea. 00:32:03.880 --> 00:32:11.180 This implies that y is equal to either Ax^2 or -Ax^2, 00:32:11.180 --> 00:32:14.100 depending on which sign you pick. 00:32:14.100 --> 00:32:21.210 So that allows me for the curves above and curves below. 00:32:21.210 --> 00:32:25.470 Because it's really true that the antiderivative here is this 00:32:25.470 --> 00:32:26.240 function. 00:32:26.240 --> 00:32:28.820 It's defined for y negative. 00:32:28.820 --> 00:32:33.840 So let's just double-check. 00:32:33.840 --> 00:32:39.460 In this case, what's happening, we have y = ax^2 and we want 00:32:39.460 --> 00:32:44.410 to compute dy/dx to make sure that it satisfies the equation 00:32:44.410 --> 00:32:46.040 that I started out with. 00:32:46.040 --> 00:32:50.890 And what I see here is that this is 2ax. 00:32:50.890 --> 00:32:53.370 And now I'm going to write this in a suggestive way. 00:32:53.370 --> 00:33:00.330 I'm going to write it as 2ax^2 / x. 00:33:00.330 --> 00:33:06.610 And, sure enough, this is 2y / x. 00:33:06.610 --> 00:33:08.810 It does not matter whether a-- it 00:33:08.810 --> 00:33:17.370 works for a positive, a negative, a equals 0. 00:33:17.370 --> 00:33:24.180 It's OK. 00:33:24.180 --> 00:33:29.770 Again, we didn't pick up by this method the a = 0 case. 00:33:29.770 --> 00:33:35.350 And that's not surprising because we divided by y. 00:33:35.350 --> 00:33:39.660 There's another thing to watch out about, about this example. 00:33:39.660 --> 00:33:41.990 So there's another warning. 00:33:41.990 --> 00:33:44.910 Which I have to give you. 00:33:44.910 --> 00:33:47.130 And this is a subtlety which you definitely 00:33:47.130 --> 00:33:50.090 won't get to in any detail until you 00:33:50.090 --> 00:33:54.070 get to a higher level ordinary differential equations course, 00:33:54.070 --> 00:33:56.980 but I do want to warn you about it right now. 00:33:56.980 --> 00:34:05.310 Which is that if you look at the equation, 00:34:05.310 --> 00:34:14.100 you need to watch out that it's undefined at x = 0. 00:34:14.100 --> 00:34:15.700 It's undefined at x = 0. 00:34:15.700 --> 00:34:20.350 We also divided by x, and x is also a problem. 00:34:20.350 --> 00:34:24.690 Now, that actually has an important consequence. 00:34:24.690 --> 00:34:27.820 Which is that, strangely, knowing the value here 00:34:27.820 --> 00:34:31.040 and knowing the rate of change doesn't specify this function. 00:34:31.040 --> 00:34:33.180 This is bad. 00:34:33.180 --> 00:34:36.160 And it violates one of our pieces of intuition. 00:34:36.160 --> 00:34:38.720 And what's going wrong is that the rate of change 00:34:38.720 --> 00:34:40.600 was not specified. 00:34:40.600 --> 00:34:43.560 It's undefined at x = 0. 00:34:43.560 --> 00:34:45.260 So there's a problem here, and in fact 00:34:45.260 --> 00:34:48.220 if you think carefully about what this function is doing, 00:34:48.220 --> 00:34:53.510 it could come in on one branch and leave on a completely 00:34:53.510 --> 00:34:56.050 different branch. 00:34:56.050 --> 00:35:01.541 It doesn't really have to obey any rule across x = 0. 00:35:01.541 --> 00:35:03.540 So you should really be thinking of these things 00:35:03.540 --> 00:35:05.860 as rays emanating from the origin. 00:35:05.860 --> 00:35:10.140 The origin was a special point in the whole geometric problem. 00:35:10.140 --> 00:35:15.080 Rather than as being complete parabolas. 00:35:15.080 --> 00:35:16.460 But that's a very subtle point. 00:35:16.460 --> 00:35:23.270 I don't expect you to be able to say anything about it. 00:35:23.270 --> 00:35:25.920 But I just want to warn you that it really is true 00:35:25.920 --> 00:35:30.640 that when x = 0 there's a problem for this differential 00:35:30.640 --> 00:35:33.810 equation. 00:35:33.810 --> 00:35:46.570 So now, let me say our next problem. 00:35:46.570 --> 00:35:47.690 Next example. 00:35:47.690 --> 00:35:52.370 Just another geometry question. 00:35:52.370 --> 00:36:01.430 So here's Example 4. 00:36:01.430 --> 00:36:04.430 I'm just going to use the example that we've already got. 00:36:04.430 --> 00:36:09.090 Because there's only so much time left here. 00:36:09.090 --> 00:36:23.480 The fourth example is to take the curves perpendicular 00:36:23.480 --> 00:36:31.610 to the parabolas. 00:36:31.610 --> 00:36:33.332 This is another geometry problem. 00:36:33.332 --> 00:36:35.040 And by specifying that the the curves are 00:36:35.040 --> 00:36:37.020 perpendicular to these parabolas, 00:36:37.020 --> 00:36:44.500 I'm telling you what their slope is. 00:36:44.500 --> 00:36:47.000 So let's think about that. 00:36:47.000 --> 00:36:48.900 What's the new equation? 00:36:48.900 --> 00:36:56.270 The new diff. eq. is the following. 00:36:56.270 --> 00:37:01.550 It's that the slope is equal to the negative reciprocal 00:37:01.550 --> 00:37:05.610 of the slope of the tangent line. 00:37:05.610 --> 00:37:14.850 Of tangent to the parabola. 00:37:14.850 --> 00:37:16.800 So that's the equation. 00:37:16.800 --> 00:37:19.270 That's actually fairly easy to write down, 00:37:19.270 --> 00:37:26.570 because it's -1 divided by 2 y/x. 00:37:26.570 --> 00:37:32.281 That's the slope of the parabola. 00:37:32.281 --> 00:37:32.780 2y/x. 00:37:36.860 --> 00:37:38.300 So let's rewrite that. 00:37:38.300 --> 00:37:52.160 Now, this is-- the x goes in the numerator, so it's -x/(2y). 00:37:52.160 --> 00:37:57.990 And now I want to solve this one. 00:37:57.990 --> 00:38:01.780 Well, again, there's only one technique. 00:38:01.780 --> 00:38:10.240 Which is we're going to separate variables. 00:38:10.240 --> 00:38:12.010 And we separate the differentials here, 00:38:12.010 --> 00:38:18.201 so we get 2y dy = -x dx. 00:38:18.201 --> 00:38:20.200 That's just looking at the equation that I have, 00:38:20.200 --> 00:38:22.640 which is right over here. 00:38:22.640 --> 00:38:30.760 dy/dx = -x/(2y), and cross-multiplying to get this. 00:38:30.760 --> 00:38:33.100 And now I can take the antiderivative. 00:38:33.100 --> 00:38:35.850 This is y^2. 00:38:35.850 --> 00:38:40.670 And the antiderivative over here is -x^2 / 2 + c. 00:38:44.410 --> 00:38:57.410 And so, the solutions are x^2 / 2 + y^2 is equal to some c, 00:38:57.410 --> 00:39:02.600 some constant c. 00:39:02.600 --> 00:39:06.460 Now, this time, things don't work the same. 00:39:06.460 --> 00:39:09.600 And you can't expect them always to work the same. 00:39:09.600 --> 00:39:11.550 I claimed that this must be true. 00:39:11.550 --> 00:39:16.910 But unfortunately I cannot insist that every c will work. 00:39:16.910 --> 00:39:21.040 As you can see here, only the positive numbers c 00:39:21.040 --> 00:39:24.980 can work here. 00:39:24.980 --> 00:39:28.720 So the picture is that something slightly different 00:39:28.720 --> 00:39:29.530 happened here. 00:39:29.530 --> 00:39:31.310 And you have to live with this. 00:39:31.310 --> 00:39:33.570 Is that sometimes not all the constants will work. 00:39:33.570 --> 00:39:36.270 Because there's more to the problem than just 00:39:36.270 --> 00:39:37.880 the antidifferentiation. 00:39:37.880 --> 00:39:40.870 And here there are fewer answers rather than more answers. 00:39:40.870 --> 00:39:43.110 In the other case we had to add in some answers, 00:39:43.110 --> 00:39:45.030 here we have to take them away. 00:39:45.030 --> 00:39:46.890 Some of them don't make any sense. 00:39:46.890 --> 00:39:48.644 And the only ones we can get are the ones 00:39:48.644 --> 00:39:50.060 which are of this form, where this 00:39:50.060 --> 00:39:53.804 is, say, some radius squared. 00:39:53.804 --> 00:39:55.470 Well maybe I shouldn't call it a radius. 00:39:55.470 --> 00:39:58.800 I'll just call it a parameter, a^2. 00:39:58.800 --> 00:40:05.940 And these are of course ellipses. 00:40:05.940 --> 00:40:09.080 And you can see that the ellipses, 00:40:09.080 --> 00:40:13.990 the length here is the square root of 2a 00:40:13.990 --> 00:40:18.160 and the semi-axis, vertical semi-axis, is a. 00:40:18.160 --> 00:40:20.110 So this is the kind of ellipse that we've got. 00:40:20.110 --> 00:40:24.140 And I draw it on the previous diagram, 00:40:24.140 --> 00:40:28.100 I think it's somewhat suggestive here. 00:40:28.100 --> 00:40:30.280 There, ellipses are kind of eggs. 00:40:30.280 --> 00:40:32.820 They're a little bit longer than they are high. 00:40:32.820 --> 00:40:37.160 And they go like this. 00:40:37.160 --> 00:40:40.960 And if I drew them pretty much right, 00:40:40.960 --> 00:40:43.230 they should be making right angles. 00:40:43.230 --> 00:40:49.710 At all of these places. 00:40:49.710 --> 00:40:53.530 OK, last little bit here. 00:40:53.530 --> 00:40:57.930 Again, you've got to be very careful with these solutions. 00:40:57.930 --> 00:41:06.760 And so there's a warning here too. 00:41:06.760 --> 00:41:08.500 So let's take a look at the-- This 00:41:08.500 --> 00:41:10.924 is the implicit solution to the equation. 00:41:10.924 --> 00:41:13.090 And this is the one that tells us what the shape is. 00:41:13.090 --> 00:41:16.350 But we can also have the explicit solution. 00:41:16.350 --> 00:41:18.170 And if I solve for the explicit solution, 00:41:18.170 --> 00:41:25.400 it's y is equal to either plus square root of (a^2 - x^2 / 2), 00:41:25.400 --> 00:41:32.080 or y is equal to minus the square root of (a^2 - x^2 / 2). 00:41:32.080 --> 00:41:39.770 These are the explicit solutions. 00:41:39.770 --> 00:41:41.350 And now, we notice something that we 00:41:41.350 --> 00:41:43.850 should have noticed before. 00:41:43.850 --> 00:41:50.490 Which is that an ellipse is not a function. 00:41:50.490 --> 00:41:54.070 It's only the top half, if you like, 00:41:54.070 --> 00:41:56.610 that's giving you a solution to this equation. 00:41:56.610 --> 00:41:58.740 Or maybe the bottom half that's giving it 00:41:58.740 --> 00:42:00.450 the solution to the equation. 00:42:00.450 --> 00:42:07.480 So the one over here, this one is the top halves. 00:42:07.480 --> 00:42:15.270 And this one over here is the bottom halves. 00:42:15.270 --> 00:42:18.570 And there's something else that's interesting. 00:42:18.570 --> 00:42:28.560 Which is that we have a problem at y = 0. y = 0 00:42:28.560 --> 00:42:33.550 is where x = square root of 2a. 00:42:33.550 --> 00:42:35.960 That's when we get to this end here. 00:42:35.960 --> 00:42:38.800 And what happens is the solution comes around and it stops. 00:42:38.800 --> 00:42:44.820 It has a vertical slope. 00:42:44.820 --> 00:42:48.910 Vertical slope. 00:42:48.910 --> 00:42:56.420 And the solution stops. 00:42:56.420 --> 00:43:00.020 But really, that's not so unreasonable. 00:43:00.020 --> 00:43:01.890 After all, look at the formula. 00:43:01.890 --> 00:43:03.660 There was a y in the denominator here. 00:43:03.660 --> 00:43:08.500 When y = 0, the slope should be infinite. 00:43:08.500 --> 00:43:12.530 And so this equation is just giving us 00:43:12.530 --> 00:43:14.490 what it geometrically and intuitively 00:43:14.490 --> 00:43:15.800 should be giving us. 00:43:15.800 --> 00:43:22.790 At that stage. 00:43:22.790 --> 00:43:26.221 So that is the introduction to ordinary differential 00:43:26.221 --> 00:43:26.720 equations. 00:43:26.720 --> 00:43:28.740 Again, there's only one technique 00:43:28.740 --> 00:43:32.580 which is-- We're not done yet, we have a whole four minutes 00:43:32.580 --> 00:43:34.030 left and we're going to review. 00:43:34.030 --> 00:43:39.220 Now, so fortunately, this review is very short. 00:43:39.220 --> 00:43:42.170 Fortunately for you, I handed out to you 00:43:42.170 --> 00:43:44.920 exactly what you're going to be covering on the test. 00:43:44.920 --> 00:43:48.440 And it's what's printed here but there's a whole two 00:43:48.440 --> 00:43:51.000 pages of discussion. 00:43:51.000 --> 00:43:59.500 And I want to give you very, very clear-cut instructions 00:43:59.500 --> 00:44:00.000 here. 00:44:00.000 --> 00:44:04.780 This is usually the hardest test of this course. 00:44:04.780 --> 00:44:07.170 People usually do terribly on it. 00:44:07.170 --> 00:44:11.840 And I'm going to try to stop that 00:44:11.840 --> 00:44:14.850 by making it a little bit easier. 00:44:14.850 --> 00:44:17.620 And now here's what we're going to do. 00:44:17.620 --> 00:44:21.150 I'm telling you exactly what type of problems 00:44:21.150 --> 00:44:22.447 are going to be on the test. 00:44:22.447 --> 00:44:23.280 These are these six. 00:44:23.280 --> 00:44:25.940 It's also written on your sheet, your handout. 00:44:25.940 --> 00:44:29.012 It's also just what was asked on last year's test. 00:44:29.012 --> 00:44:31.220 You should go and you should look at last year's test 00:44:31.220 --> 00:44:33.420 and see what types of problems they are. 00:44:33.420 --> 00:44:36.210 I really, really, am going to ask the same questions, 00:44:36.210 --> 00:44:38.900 or the same type of questions. 00:44:38.900 --> 00:44:41.860 Not the same questions. 00:44:41.860 --> 00:44:44.960 So that's what's going to happen on the test. 00:44:44.960 --> 00:44:48.680 And let me just tell you, say one thing, which 00:44:48.680 --> 00:44:51.210 is the main theme of the class. 00:44:51.210 --> 00:44:52.290 And I will open up. 00:44:52.290 --> 00:44:54.330 We'll have time for one question after that. 00:44:54.330 --> 00:44:58.540 The main theme of this unit is simply the following. 00:44:58.540 --> 00:45:05.460 That information about derivative and sometimes 00:45:05.460 --> 00:45:11.010 maybe the second derivative, tells us 00:45:11.010 --> 00:45:17.315 information about f itself. 00:45:17.315 --> 00:45:19.940 And that's just what were doing here with ordinary differential 00:45:19.940 --> 00:45:20.490 equations. 00:45:20.490 --> 00:45:22.865 And that was what we were doing way at the beginning when 00:45:22.865 --> 00:45:23.825 we did approximations.