1 00:00:00,050 --> 00:00:02,330 The following content is provided under a Creative 2 00:00:02,330 --> 00:00:03,610 Commons license. 3 00:00:03,610 --> 00:00:05,980 Your support will help MIT OpenCourseWare 4 00:00:05,980 --> 00:00:09,495 continue to offer high quality educational resources for free. 5 00:00:09,495 --> 00:00:12,540 To make a donation or to view additional materials 6 00:00:12,540 --> 00:00:15,610 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:15,610 --> 00:00:21,170 at ocw.mit.edu. 8 00:00:21,170 --> 00:00:23,070 PROFESSOR: And this last little bit 9 00:00:23,070 --> 00:00:24,902 is something which is not yet on the Web. 10 00:00:24,902 --> 00:00:26,860 But, anyway, when I was walking out of the room 11 00:00:26,860 --> 00:00:29,740 last time, I noticed that I'd written down 12 00:00:29,740 --> 00:00:32,070 the wrong formula for c_1 - c_2. 13 00:00:32,070 --> 00:00:35,380 There's a misprint, there's a minus sign that's wrong. 14 00:00:35,380 --> 00:00:39,180 I claimed last time that c_1 - c_2 was +1/2. 15 00:00:39,180 --> 00:00:40,580 But, actually, it's -1/2. 16 00:00:40,580 --> 00:00:42,080 If you go through the calculation 17 00:00:42,080 --> 00:00:45,780 that we did with the antiderivative of sin x cos x, 18 00:00:45,780 --> 00:00:48,500 we get these two possible answers. 19 00:00:48,500 --> 00:00:52,900 And if they're to be equal, then if we just subtract them 20 00:00:52,900 --> 00:00:56,300 we get c_1 - c_2 + 1/2 = 0. 21 00:00:56,300 --> 00:01:01,740 So c_1 - c_2 = 1/2. 22 00:01:01,740 --> 00:01:03,930 So, those are all of the corrections. 23 00:01:03,930 --> 00:01:06,530 Again, everything here will be on the Web. 24 00:01:06,530 --> 00:01:14,660 But just wanted to make it all clear to you. 25 00:01:14,660 --> 00:01:15,410 So here we are. 26 00:01:15,410 --> 00:01:19,900 This is our last day of the second unit, Applications 27 00:01:19,900 --> 00:01:22,010 of Differentiation. 28 00:01:22,010 --> 00:01:30,210 And I have one of the most fun topics to introduce to you. 29 00:01:30,210 --> 00:01:32,530 Which is differential equations. 30 00:01:32,530 --> 00:01:35,410 Now, we have a whole course on differential equations, 31 00:01:35,410 --> 00:01:38,140 which is called 18.03. 32 00:01:38,140 --> 00:01:43,840 And so we're only going to do just a little bit. 33 00:01:43,840 --> 00:01:52,600 But I'm going to teach you one technique. 34 00:01:52,600 --> 00:01:58,250 Which fits in precisely with what we've been doing already. 35 00:01:58,250 --> 00:02:05,280 Which is differentials. 36 00:02:05,280 --> 00:02:08,480 The first and simplest kind of differential equation 37 00:02:08,480 --> 00:02:12,770 is the rate of change of x with respect to y 38 00:02:12,770 --> 00:02:16,770 is equal to some function f(x). 39 00:02:16,770 --> 00:02:19,020 Now, that's a perfectly good differential equation. 40 00:02:19,020 --> 00:02:21,420 And we already discussed last time 41 00:02:21,420 --> 00:02:26,204 that the solution, that is, the function y, 42 00:02:26,204 --> 00:02:27,620 is going to be the antiderivative, 43 00:02:27,620 --> 00:02:33,100 or the integral, of x. 44 00:02:33,100 --> 00:02:35,880 Now, for the purposes of today, we're 45 00:02:35,880 --> 00:02:40,597 going to consider this problem to be solved. 46 00:02:40,597 --> 00:02:41,930 That is, you can always do this. 47 00:02:41,930 --> 00:02:44,200 You can always take antiderivatives. 48 00:02:44,200 --> 00:02:51,560 And for our purposes now, that is for now, 49 00:02:51,560 --> 00:03:08,380 we only have one technique to find antiderivatives. 50 00:03:08,380 --> 00:03:15,170 And that's called substitution. 51 00:03:15,170 --> 00:03:18,750 It has a very small variant, which 52 00:03:18,750 --> 00:03:27,940 we called advanced guessing. 53 00:03:27,940 --> 00:03:29,820 And that works just as well. 54 00:03:29,820 --> 00:03:32,490 And that's basically all that you'll ever need to do. 55 00:03:32,490 --> 00:03:36,457 As a practical matter, these are the ones you'll face for now. 56 00:03:36,457 --> 00:03:38,540 Ones that you can actually see what the answer is, 57 00:03:38,540 --> 00:03:42,640 or you'll have to make a substitution. 58 00:03:42,640 --> 00:03:48,120 Now, the first tricky example, or the first maybe interesting 59 00:03:48,120 --> 00:03:50,190 example of a differential equation, 60 00:03:50,190 --> 00:04:00,750 which I'll call Example 2, is going to be the following. 61 00:04:00,750 --> 00:04:07,510 d/dx + x acting on y is equal to 0. 62 00:04:07,510 --> 00:04:10,580 So that's our first differential equation that 63 00:04:10,580 --> 00:04:12,780 were going to try to solve. 64 00:04:12,780 --> 00:04:18,630 Apart from this standard antiderivative approach. 65 00:04:18,630 --> 00:04:24,350 This operation here has a name. 66 00:04:24,350 --> 00:04:27,490 This actually has a name, it's called the annihilation 67 00:04:27,490 --> 00:04:33,460 operator. 68 00:04:33,460 --> 00:04:43,780 And it's called that in quantum mechanics. 69 00:04:43,780 --> 00:04:45,950 And there's a corresponding creation operator 70 00:04:45,950 --> 00:04:50,780 where you change the sign from plus to minus. 71 00:04:50,780 --> 00:04:53,660 And this is one of the simplest differential equations. 72 00:04:53,660 --> 00:04:55,550 The reason why it's studied in quantum 73 00:04:55,550 --> 00:04:58,360 mechanics all it that it has very simple solutions 74 00:04:58,360 --> 00:05:00,500 that you can just write out. 75 00:05:00,500 --> 00:05:02,920 So we're going to solve this equation. 76 00:05:02,920 --> 00:05:05,680 It's the one that governs the ground state 77 00:05:05,680 --> 00:05:08,670 of the harmonic oscillator. 78 00:05:08,670 --> 00:05:10,790 So it has a lot of fancy words associated with it, 79 00:05:10,790 --> 00:05:12,706 but it's a fairly simple differential equation 80 00:05:12,706 --> 00:05:14,430 and it works perfectly by the method 81 00:05:14,430 --> 00:05:17,270 that we're going to propose. 82 00:05:17,270 --> 00:05:20,820 So the first step in this solution 83 00:05:20,820 --> 00:05:26,800 is just to rewrite the equation by putting 84 00:05:26,800 --> 00:05:29,090 one of the terms on the right-hand side. 85 00:05:29,090 --> 00:05:32,300 So this is dy/dx = -xy. 86 00:05:35,700 --> 00:05:37,700 Now, here is where you see the difference 87 00:05:37,700 --> 00:05:41,210 between this type of equation and the previous type. 88 00:05:41,210 --> 00:05:43,130 In the previous equation, we just 89 00:05:43,130 --> 00:05:45,660 had a function of x on the right-hand side. 90 00:05:45,660 --> 00:05:50,260 But here, the rate of change depends on both x and y. 91 00:05:50,260 --> 00:05:51,830 So it's not clear at all that we can 92 00:05:51,830 --> 00:05:55,110 solve this kind of equation. 93 00:05:55,110 --> 00:05:57,370 But there is a remarkable trick which 94 00:05:57,370 --> 00:05:59,420 works very well in this case. 95 00:05:59,420 --> 00:06:02,710 Which is to use multiplication. 96 00:06:02,710 --> 00:06:06,340 To use this idea of differential that we talked about last time. 97 00:06:06,340 --> 00:06:14,700 Namely, we divide by y and multiply by dx. 98 00:06:14,700 --> 00:06:17,660 So now we've separated the equation. 99 00:06:17,660 --> 00:06:20,750 We've separated out the differentials. 100 00:06:20,750 --> 00:06:22,960 And what's going to be important for us 101 00:06:22,960 --> 00:06:27,410 is that the left-hand side is expressed solely in terms of y 102 00:06:27,410 --> 00:06:30,170 and the right-hand side is expressed solely in terms of x. 103 00:06:30,170 --> 00:06:33,400 And we'll go through this in careful detail. 104 00:06:33,400 --> 00:06:36,410 So now, the idea is if you've set up the equation in terms 105 00:06:36,410 --> 00:06:39,680 of differentials as opposed to ratios of differentials, 106 00:06:39,680 --> 00:06:44,680 or rates of change, now I can use Leibniz's notation 107 00:06:44,680 --> 00:06:47,600 and integrate these differentials. 108 00:06:47,600 --> 00:06:55,420 Take their antiderivatives. 109 00:06:55,420 --> 00:07:02,070 And we know what each of these is. 110 00:07:02,070 --> 00:07:17,920 Namely, the left-hand side is just-- Ah, well, that's tough. 111 00:07:17,920 --> 00:07:24,250 OK. 112 00:07:24,250 --> 00:07:29,410 I had an au pair who actually did a lot of Tae Kwan Do. 113 00:07:29,410 --> 00:07:32,870 She could definitely defeat any of you in any encounter, 114 00:07:32,870 --> 00:07:34,940 I promise. 115 00:07:34,940 --> 00:07:35,440 OK. 116 00:07:35,440 --> 00:07:37,870 Anyway. 117 00:07:37,870 --> 00:07:38,970 So, let's go back. 118 00:07:38,970 --> 00:07:41,970 We want to take the antiderivative of this. 119 00:07:41,970 --> 00:07:48,130 So remember, this is the function 120 00:07:48,130 --> 00:07:50,140 whose derivative is 1/y. 121 00:07:50,140 --> 00:07:52,320 And now there's a slight novelty here. 122 00:07:52,320 --> 00:07:54,860 Here we're differentiating the variable as x, 123 00:07:54,860 --> 00:07:58,220 and here we're differentiating the variable as y. 124 00:07:58,220 --> 00:08:02,860 So the antiderivative here is ln y. 125 00:08:02,860 --> 00:08:07,970 And the antiderivative on the other side is -x^2 / 2. 126 00:08:07,970 --> 00:08:10,400 And they differ by a constant. 127 00:08:10,400 --> 00:08:17,620 So we have this relationship here. 128 00:08:17,620 --> 00:08:19,660 Now, that's almost the end of the story. 129 00:08:19,660 --> 00:08:23,100 We have to exponentiate to express y in terms of x. 130 00:08:23,100 --> 00:08:26,970 So, e^(ln y) = e^(-x^2 / 2) + c. 131 00:08:29,880 --> 00:08:36,010 And now I can rewrite that as y is equal to-- I'll write as A 132 00:08:36,010 --> 00:08:40,120 e^(-x^2 / 2), where A = e^c. 133 00:08:43,870 --> 00:08:47,940 And incidentally, we're just taking the case y positive 134 00:08:47,940 --> 00:08:48,440 here. 135 00:08:48,440 --> 00:08:50,920 We'll talk about what happens when 136 00:08:50,920 --> 00:08:55,590 y is negative in a few minutes. 137 00:08:55,590 --> 00:08:57,270 So here's the answer to the question, 138 00:08:57,270 --> 00:09:02,040 almost, except for this fact that I picked out y positive. 139 00:09:02,040 --> 00:09:09,890 Really, the solution is y is equal to any multiple 140 00:09:09,890 --> 00:09:11,780 of e^(-x^2 / 2). 141 00:09:11,780 --> 00:09:19,310 Any constant a; a positive, negative, or 0. 142 00:09:19,310 --> 00:09:22,240 Any constant will do. 143 00:09:22,240 --> 00:09:24,640 And we should double-check that to make sure. 144 00:09:24,640 --> 00:09:32,930 If you take d/dx of y right, that's going to be a d/dx 145 00:09:32,930 --> 00:09:33,950 e^(-x^2 / 2). 146 00:09:36,570 --> 00:09:38,220 And now by the chain rule, you can 147 00:09:38,220 --> 00:09:42,080 see that this is a times the factor of -x, that's 148 00:09:42,080 --> 00:09:45,080 the derivative of the exponent, with respect to x, 149 00:09:45,080 --> 00:09:48,190 times the exponential. 150 00:09:48,190 --> 00:09:50,770 And now you just rearrange that. 151 00:09:50,770 --> 00:09:54,430 That's -xy. 152 00:09:54,430 --> 00:09:55,860 So it does check. 153 00:09:55,860 --> 00:09:57,360 These are solutions to the equation. 154 00:09:57,360 --> 00:09:58,560 The a didn't matter. 155 00:09:58,560 --> 00:10:05,370 It didn't matter whether it was positive or negative. 156 00:10:05,370 --> 00:10:08,750 This function is known as the normal distribution, 157 00:10:08,750 --> 00:10:11,310 so it fits beautifully with a lot of probability 158 00:10:11,310 --> 00:10:15,710 and probabilistic interpretation of quantum mechanics. 159 00:10:15,710 --> 00:10:22,630 This is sort of where the particle is. 160 00:10:22,630 --> 00:10:25,380 So next, what I'd like to do is just 161 00:10:25,380 --> 00:10:30,920 go through the method in general and point out when it works. 162 00:10:30,920 --> 00:10:32,960 And then I'll make a few comments just 163 00:10:32,960 --> 00:10:37,650 to make sure that you understand the technicalities of dealing 164 00:10:37,650 --> 00:10:39,590 with constants and so forth. 165 00:10:39,590 --> 00:10:42,385 So, first of all, the general method of separation 166 00:10:42,385 --> 00:10:53,830 of variables. 167 00:10:53,830 --> 00:10:55,360 And here's when it works. 168 00:10:55,360 --> 00:10:58,470 It works when you're faced with a differential 169 00:10:58,470 --> 00:11:03,410 equation of the form f(x) g(y). 170 00:11:03,410 --> 00:11:05,920 That's the situation that we had. 171 00:11:05,920 --> 00:11:08,150 And I'll just illustrate that. 172 00:11:08,150 --> 00:11:09,820 Just to remind you here. 173 00:11:09,820 --> 00:11:11,650 Here's our equation. 174 00:11:11,650 --> 00:11:13,120 It's in that form. 175 00:11:13,120 --> 00:11:22,190 And the function f(x) is -x, and the function g(y) is just y. 176 00:11:22,190 --> 00:11:26,670 And now, the way the method works is, this separation step. 177 00:11:26,670 --> 00:11:30,370 From here to here, this is the key step. 178 00:11:30,370 --> 00:11:35,070 This is the only conceptually remarkable step, 179 00:11:35,070 --> 00:11:37,360 which all has to do with the fact 180 00:11:37,360 --> 00:11:40,160 that Leibniz fixed his notations up so that this 181 00:11:40,160 --> 00:11:42,190 works perfectly. 182 00:11:42,190 --> 00:11:48,320 And so that involves taking the y, so dividing by g(y), 183 00:11:48,320 --> 00:11:53,170 and multiplying by dx, it's comfortable 184 00:11:53,170 --> 00:11:55,540 because it feels like ordinary arithmetic, 185 00:11:55,540 --> 00:11:59,430 even though these are differentials. 186 00:11:59,430 --> 00:12:02,710 And then, we just antidifferentiate. 187 00:12:02,710 --> 00:12:10,560 So we have a function, H, which is the integral of dy / g(y), 188 00:12:10,560 --> 00:12:12,440 and we have another function which 189 00:12:12,440 --> 00:12:15,910 is F. Note they are functions of completely different variables 190 00:12:15,910 --> 00:12:16,740 here. 191 00:12:16,740 --> 00:12:20,850 Integral of f(x) dx. 192 00:12:20,850 --> 00:12:23,864 Now, in our example we did that. 193 00:12:23,864 --> 00:12:25,530 We carried out this antidifferentiation, 194 00:12:25,530 --> 00:12:29,680 and this function turned out to be ln y, 195 00:12:29,680 --> 00:12:39,630 and this function turned out to be -x^2 / 2. 196 00:12:39,630 --> 00:12:42,350 And then we write the relationship. 197 00:12:42,350 --> 00:12:45,790 Which is that if these are both antiderivatives 198 00:12:45,790 --> 00:12:48,890 of the same thing, then they have to differ by a constant. 199 00:12:48,890 --> 00:12:55,870 Or, in other words, H(y) has to equal to F(x) + c. 200 00:12:55,870 --> 00:13:10,130 Where c is constant. 201 00:13:10,130 --> 00:13:15,150 Now, notice that this kind of equation 202 00:13:15,150 --> 00:13:20,450 is what we call an implicit equation. 203 00:13:20,450 --> 00:13:23,930 It's not quite a formula for y, directly. 204 00:13:23,930 --> 00:13:26,330 It defines y implicitly. 205 00:13:26,330 --> 00:13:29,630 That's that top line up here. 206 00:13:29,630 --> 00:13:33,020 That's the implicit equation. 207 00:13:33,020 --> 00:13:35,110 In order to make it an explicit equation, which 208 00:13:35,110 --> 00:13:38,780 is what is underneath, what I have to do is take the inverse. 209 00:13:38,780 --> 00:13:41,260 So I write it as y = H^(-1)(F(x) + c). 210 00:13:45,080 --> 00:13:48,140 Now, in real life the calculus part is often pretty easy. 211 00:13:48,140 --> 00:13:52,620 And it can be quite messy to do the inverse operation. 212 00:13:52,620 --> 00:13:55,850 So sometimes we just leave it alone in the implicit form. 213 00:13:55,850 --> 00:13:58,250 But it's also satisfying, sometimes, 214 00:13:58,250 --> 00:14:09,290 to write it in the final form here. 215 00:14:09,290 --> 00:14:14,160 Now I've got to give you a few little pieces of commentary 216 00:14:14,160 --> 00:14:16,640 before-- For those of you walked in a little bit late, 217 00:14:16,640 --> 00:14:25,660 this will all be on the Web. 218 00:14:25,660 --> 00:14:31,180 So just a few pieces of commentary. 219 00:14:31,180 --> 00:14:36,230 So if you like, some remarks. 220 00:14:36,230 --> 00:14:51,140 The first remark is that I could have written natural log 221 00:14:51,140 --> 00:14:55,850 of absolute y is equal to -x^2 / 2 + c. 222 00:14:58,390 --> 00:15:01,871 We learned last time that the antiderivative works also 223 00:15:01,871 --> 00:15:02,870 for the negative values. 224 00:15:02,870 --> 00:15:08,490 So this would work for y not equal to 0. 225 00:15:08,490 --> 00:15:10,900 Both for positive and negative values. 226 00:15:10,900 --> 00:15:13,990 And you can see that that would have captured most 227 00:15:13,990 --> 00:15:15,580 of the rest of the solution. 228 00:15:15,580 --> 00:15:21,870 Namely, |y| would be equal to A e^(-x^2 / 2), 229 00:15:21,870 --> 00:15:24,800 by the same reasoning as before. 230 00:15:24,800 --> 00:15:29,052 And then that would mean that y was plus or minus A e^(-x^2 / 231 00:15:29,052 --> 00:15:34,800 2), which is really just what we got. 232 00:15:34,800 --> 00:15:38,120 Because, in fact, I didn't bother with this. 233 00:15:38,120 --> 00:15:40,087 Because actually in most-- and the reason why 234 00:15:40,087 --> 00:15:42,420 I'm going through this, by the way, carefully this time, 235 00:15:42,420 --> 00:15:44,878 is that you're going to be faced with this very frequently. 236 00:15:44,878 --> 00:15:47,290 The exponential function comes up all the time. 237 00:15:47,290 --> 00:15:49,710 And so, therefore, you want to be completely comfortable 238 00:15:49,710 --> 00:15:52,300 dealing with it. 239 00:15:52,300 --> 00:15:54,510 So this time I had the positive A, 240 00:15:54,510 --> 00:15:56,540 while the negative A fits in either this way, 241 00:15:56,540 --> 00:15:57,740 or I can throw it in. 242 00:15:57,740 --> 00:15:59,970 Because I know that that's going to work that way. 243 00:15:59,970 --> 00:16:03,370 But of course, I double-checked to be confident. 244 00:16:03,370 --> 00:16:07,380 Now, this still leaves out one value. 245 00:16:07,380 --> 00:16:12,295 So, this still leaves out-- So, if you like, 246 00:16:12,295 --> 00:16:14,420 what I have here now is a is equal to plus or minus 247 00:16:14,420 --> 00:16:18,540 capital A. The capital A one being the positive one. 248 00:16:18,540 --> 00:16:20,920 But this still leaves out one case. 249 00:16:20,920 --> 00:16:23,480 Which is y = 0. 250 00:16:23,480 --> 00:16:27,180 Which is an extremely boring solution, but nevertheless 251 00:16:27,180 --> 00:16:28,760 a solution to this problem. 252 00:16:28,760 --> 00:16:32,330 If you plug in 0 here for y, you get 0. 253 00:16:32,330 --> 00:16:34,600 If you plug in 0 here for y, you get 254 00:16:34,600 --> 00:16:36,520 that these two sides are equal. 255 00:16:36,520 --> 00:16:38,080 0 = 0. 256 00:16:38,080 --> 00:16:40,630 Not a very interesting answer to the question. 257 00:16:40,630 --> 00:16:42,290 But it's still an answer. 258 00:16:42,290 --> 00:16:43,930 And so y = 0 is left out.. 259 00:16:43,930 --> 00:16:52,520 Well, that's not so surprising that we missed that solution. 260 00:16:52,520 --> 00:16:56,360 Because in the process of carrying out these operations, 261 00:16:56,360 --> 00:16:58,250 I divided by y. 262 00:16:58,250 --> 00:17:02,290 I did that right here. 263 00:17:02,290 --> 00:17:03,600 So, that's what happens. 264 00:17:03,600 --> 00:17:05,941 If you're going to do various non-linear operations, 265 00:17:05,941 --> 00:17:08,190 in particular, if you're going to divide by something, 266 00:17:08,190 --> 00:17:10,564 if it happens to be 0 you're going to miss that solution. 267 00:17:10,564 --> 00:17:13,860 You might have problems with that solution. 268 00:17:13,860 --> 00:17:16,870 But we have to live with that because we want to get ahead. 269 00:17:16,870 --> 00:17:20,520 And we want to get the formulas for various solutions. 270 00:17:20,520 --> 00:17:22,840 So that's the first remark that I wanted to make. 271 00:17:22,840 --> 00:17:30,340 And now, the second one is almost related 272 00:17:30,340 --> 00:17:33,380 to what I was just discussing right here. 273 00:17:33,380 --> 00:17:37,240 That I'm erasing. 274 00:17:37,240 --> 00:17:39,420 And that's the following. 275 00:17:39,420 --> 00:17:52,100 I could have also written ln y + c_1 = -x^2 / 2 + c_2. 276 00:17:52,100 --> 00:17:54,300 Where c_1 and c_2 are different constants. 277 00:17:54,300 --> 00:17:57,210 When I'm faced with this antidifferentiation, 278 00:17:57,210 --> 00:17:59,460 I just taught you last time, that you want 279 00:17:59,460 --> 00:18:02,230 to have an arbitrary constant. 280 00:18:02,230 --> 00:18:06,760 Here and there, in both slots. 281 00:18:06,760 --> 00:18:09,480 So I perfectly well could have written this down. 282 00:18:09,480 --> 00:18:17,990 But notice that I can rewrite this as ln y = -x^2 / 2 + c_2 - 283 00:18:17,990 --> 00:18:20,190 c_1. 284 00:18:20,190 --> 00:18:22,250 I can subtract. 285 00:18:22,250 --> 00:18:25,385 And then, if I just combine these two guys together 286 00:18:25,385 --> 00:18:29,060 and name them c, I have a different constant. 287 00:18:29,060 --> 00:18:32,010 In other words, it's superfluous and redundant 288 00:18:32,010 --> 00:18:35,060 to have two arbitrary constants here, 289 00:18:35,060 --> 00:18:38,260 because they can always be combined into one. 290 00:18:38,260 --> 00:18:47,010 So two constants are superfluous. 291 00:18:47,010 --> 00:18:54,430 Can always be combined. 292 00:18:54,430 --> 00:18:56,490 So we just never do it this first way. 293 00:18:56,490 --> 00:19:05,260 It's just extra writing, it's a waste of time. 294 00:19:05,260 --> 00:19:08,120 There's one other subtle remark, which you won't actually 295 00:19:08,120 --> 00:19:10,020 appreciate until you've done several problems 296 00:19:10,020 --> 00:19:11,280 in this direction. 297 00:19:11,280 --> 00:19:14,490 Which is that the constant appears 298 00:19:14,490 --> 00:19:18,900 additive here, in this first solution to the problem. 299 00:19:18,900 --> 00:19:22,590 But when I do this nonlinear operation of exponentiation, 300 00:19:22,590 --> 00:19:26,390 it now becomes multiplicative constant. 301 00:19:26,390 --> 00:19:31,006 And so, in general, there's a free constant somewhere 302 00:19:31,006 --> 00:19:31,630 in the problem. 303 00:19:31,630 --> 00:19:35,490 But it's not always an additive constant. 304 00:19:35,490 --> 00:19:38,330 It's only an additive constant right at the first step 305 00:19:38,330 --> 00:19:39,890 when you take the antiderivative. 306 00:19:39,890 --> 00:19:42,265 And then after that, when you do all your other nonlinear 307 00:19:42,265 --> 00:19:45,440 operations, it can turn into anything at all. 308 00:19:45,440 --> 00:19:47,980 So you should always expect it to be something slightly more 309 00:19:47,980 --> 00:19:49,563 interesting than an additive constant. 310 00:19:49,563 --> 00:19:59,060 Although occasionally it stays an additive constant. 311 00:19:59,060 --> 00:20:01,180 The last little bit of commentary 312 00:20:01,180 --> 00:20:06,010 that I want to make just goes back to the original problem 313 00:20:06,010 --> 00:20:06,810 here. 314 00:20:06,810 --> 00:20:09,680 Which is right here. 315 00:20:09,680 --> 00:20:11,190 The example 1. 316 00:20:11,190 --> 00:20:14,490 And I want to solve it, even though this is simpleminded. 317 00:20:14,490 --> 00:20:21,490 But Example 1 via separation. 318 00:20:21,490 --> 00:20:25,290 So that you see our variables. 319 00:20:25,290 --> 00:20:28,440 So that you see what it does. 320 00:20:28,440 --> 00:20:34,230 The situation is this. 321 00:20:34,230 --> 00:20:35,890 And the separation just means you 322 00:20:35,890 --> 00:20:38,570 put the dx on the other side. 323 00:20:38,570 --> 00:20:44,030 So this is dy = f(x) dx. 324 00:20:44,030 --> 00:20:54,680 And then we integrate. 325 00:20:54,680 --> 00:20:58,170 And the antiderivative of dy is just y. 326 00:20:58,170 --> 00:21:03,490 So this is the solution to the problem. 327 00:21:03,490 --> 00:21:05,170 And it's just what we wrote before; 328 00:21:05,170 --> 00:21:07,480 it's just a funny notation. 329 00:21:07,480 --> 00:21:19,480 And it comes to the same thing as the antiderivative. 330 00:21:19,480 --> 00:21:23,240 OK, so now we're going to go on to a trickier problem. 331 00:21:23,240 --> 00:21:24,090 A trickier example. 332 00:21:24,090 --> 00:21:26,420 We need one or two more just to get some practice 333 00:21:26,420 --> 00:21:29,330 with this method. 334 00:21:29,330 --> 00:21:31,730 Everybody happy so far? 335 00:21:31,730 --> 00:21:32,240 Question. 336 00:21:32,240 --> 00:21:53,472 STUDENT: [INAUDIBLE] 337 00:21:53,472 --> 00:21:55,180 PROFESSOR: So, the question is, how do we 338 00:21:55,180 --> 00:21:58,150 deal with this ambiguity. 339 00:21:58,150 --> 00:22:03,020 I'm summarizing very, very, briefly what I heard. 340 00:22:03,020 --> 00:22:06,530 Well, you know, sometimes a > 0, sometimes a < 0, 341 00:22:06,530 --> 00:22:07,550 sometimes it's not. 342 00:22:07,550 --> 00:22:12,810 So there's a name for this guy. 343 00:22:12,810 --> 00:22:20,136 Which is that this is what's called the general solution. 344 00:22:20,136 --> 00:22:22,010 In other words, the whole family of solutions 345 00:22:22,010 --> 00:22:24,460 is the answer to the question. 346 00:22:24,460 --> 00:22:28,020 Now, it could be that you're given extra information. 347 00:22:28,020 --> 00:22:31,760 If you're given extra information, that might be, 348 00:22:31,760 --> 00:22:33,527 and this is very typical in such problems, 349 00:22:33,527 --> 00:22:35,610 you have the rate of change of the function, which 350 00:22:35,610 --> 00:22:36,510 is what we've given. 351 00:22:36,510 --> 00:22:39,780 But you might also have the place where it starts. 352 00:22:39,780 --> 00:22:44,579 Which would be, say, it starts at 3. 353 00:22:44,579 --> 00:22:46,620 Now, if you have that extra piece of information, 354 00:22:46,620 --> 00:22:50,670 then you can nail down exactly which function it is. 355 00:22:50,670 --> 00:22:52,420 If you do that, if you plug in 3, 356 00:22:52,420 --> 00:22:57,860 you see that a times e^(-0^2 / 2) is equal to 3. 357 00:22:57,860 --> 00:23:00,300 So a = 3. 358 00:23:00,300 --> 00:23:02,720 And the answer is y = 3e^(-x^2 / 2). 359 00:23:06,140 --> 00:23:08,940 And similarly, if it's negative, if it starts out negative, 360 00:23:08,940 --> 00:23:10,100 it'll stay negative. 361 00:23:10,100 --> 00:23:11,100 For instance. 362 00:23:11,100 --> 00:23:14,870 If it starts out 0, it'll stay 0, this particular function 363 00:23:14,870 --> 00:23:16,100 here. 364 00:23:16,100 --> 00:23:18,110 So the answer to your question is how 365 00:23:18,110 --> 00:23:19,700 you deal with the ambiguity. 366 00:23:19,700 --> 00:23:23,620 The answer is that you simply say what the solution is. 367 00:23:23,620 --> 00:23:25,200 And the solution is not one function, 368 00:23:25,200 --> 00:23:26,324 it's a family of functions. 369 00:23:26,324 --> 00:23:30,340 It's a list and you have to have what's known as a parameter. 370 00:23:30,340 --> 00:23:32,440 And that parameter gets nailed down 371 00:23:32,440 --> 00:23:35,240 if you tell me more information about the function. 372 00:23:35,240 --> 00:23:37,654 Not the rate of change, but something about the values 373 00:23:37,654 --> 00:23:38,320 of the function. 374 00:23:46,620 --> 00:23:53,660 STUDENT: [INAUDIBLE] 375 00:23:53,660 --> 00:23:55,720 PROFESSOR: The general solution is this solution. 376 00:23:55,720 --> 00:23:56,553 STUDENT: [INAUDIBLE] 377 00:23:56,553 --> 00:23:58,176 PROFESSOR: And I'm showing you here 378 00:23:58,176 --> 00:24:00,300 that you could get to most of the general solution. 379 00:24:00,300 --> 00:24:04,570 There's one thing that's left out, namely the case a = 0. 380 00:24:04,570 --> 00:24:08,120 So, in other words, I would not go through this method. 381 00:24:08,120 --> 00:24:10,690 I would only use this, which is simpler. 382 00:24:10,690 --> 00:24:13,590 But then I have to understand that I haven't gotten 383 00:24:13,590 --> 00:24:15,410 all of the solutions this way. 384 00:24:15,410 --> 00:24:19,325 I'm going to need to throw in all the rest of the solutions. 385 00:24:19,325 --> 00:24:20,950 So in the back of your head, you always 386 00:24:20,950 --> 00:24:23,779 have to have something like this in mind. 387 00:24:23,779 --> 00:24:25,570 So that you can generate all the solutions. 388 00:24:25,570 --> 00:24:28,510 This is very suggestive, right? 389 00:24:28,510 --> 00:24:31,840 The restriction, it turns that the restriction A > 0 is 390 00:24:31,840 --> 00:24:40,660 superfluous, is unnecessary. 391 00:24:40,660 --> 00:24:46,180 But that, we only get by further thought and by checking. 392 00:24:46,180 --> 00:24:46,890 Another question? 393 00:24:46,890 --> 00:24:47,389 Over here. 394 00:24:47,389 --> 00:24:52,210 STUDENT: [INAUDIBLE] 395 00:24:52,210 --> 00:24:54,630 PROFESSOR: The aim of differential equations 396 00:24:54,630 --> 00:24:55,600 is to solve them. 397 00:24:55,600 --> 00:24:59,372 Just as with algebraic equations. 398 00:24:59,372 --> 00:25:01,330 Usually, differential equations are telling you 399 00:25:01,330 --> 00:25:04,300 something about the balance between an acceleration 400 00:25:04,300 --> 00:25:05,980 and a velocity. 401 00:25:05,980 --> 00:25:09,840 If you have a falling object, it might have a resistance. 402 00:25:09,840 --> 00:25:11,210 It's telling you something. 403 00:25:11,210 --> 00:25:13,700 So, actually, sometimes in applied problems, 404 00:25:13,700 --> 00:25:16,450 formulating what differential equation describe 405 00:25:16,450 --> 00:25:18,310 this situation is very important. 406 00:25:18,310 --> 00:25:21,910 In order to see that that's the right thing, 407 00:25:21,910 --> 00:25:24,330 you have to have solved it to see that it fits 408 00:25:24,330 --> 00:25:25,780 the data that you're getting. 409 00:25:25,780 --> 00:25:28,570 STUDENT: [INAUDIBLE] 410 00:25:28,570 --> 00:25:31,720 PROFESSOR: The question is, can you solve for x instead of y. 411 00:25:31,720 --> 00:25:36,250 The answer is, sure. 412 00:25:36,250 --> 00:25:38,356 That's the same thing as-- so that 413 00:25:38,356 --> 00:25:40,230 would be the inverse function of the function 414 00:25:40,230 --> 00:25:42,520 that we're officially looking for. 415 00:25:42,520 --> 00:25:43,960 But yeah, it's legal. 416 00:25:43,960 --> 00:25:46,150 In other words, oftentimes we're stuck 417 00:25:46,150 --> 00:25:48,835 with just the implicit, some implicit formula 418 00:25:48,835 --> 00:25:51,540 and sometimes we're stuck with a formula x is a function of y 419 00:25:51,540 --> 00:25:54,730 versus y is a function of x. 420 00:25:54,730 --> 00:25:57,850 The way in which the function is specified 421 00:25:57,850 --> 00:26:00,780 is something that can be complicated. 422 00:26:00,780 --> 00:26:02,810 As you'll see in the next example, 423 00:26:02,810 --> 00:26:04,760 it's not necessarily the best thing 424 00:26:04,760 --> 00:26:07,530 to think about a function-- y as a function of x. 425 00:26:07,530 --> 00:26:12,170 Well, in the fourth example. 426 00:26:12,170 --> 00:26:27,000 Alright, we're going to go on and do our next example here. 427 00:26:27,000 --> 00:26:32,440 So the third example is going to be taken 428 00:26:32,440 --> 00:26:36,090 as a kind of geometry problem. 429 00:26:36,090 --> 00:26:38,990 I'll draw a picture of it. 430 00:26:38,990 --> 00:26:44,180 Suppose you have a curve with the following property. 431 00:26:44,180 --> 00:26:50,730 If you take a point on the curve, and you take the ray, 432 00:26:50,730 --> 00:26:56,224 you take the ray from the origin to the curve, well, that's not 433 00:26:56,224 --> 00:26:57,390 going to be one that I want. 434 00:26:57,390 --> 00:27:00,350 I think I'm going to want something which is steeper. 435 00:27:00,350 --> 00:27:02,130 Because what I'm going to insist is 436 00:27:02,130 --> 00:27:09,050 that the tangent line be twice as steep as the ray 437 00:27:09,050 --> 00:27:10,490 from the origin. 438 00:27:10,490 --> 00:27:19,600 So, in other words, slope of tangent line 439 00:27:19,600 --> 00:27:31,540 equals twice slope of ray from origin. 440 00:27:31,540 --> 00:27:34,110 So the slope of this orange line is twice 441 00:27:34,110 --> 00:27:39,410 the slope of the pink line. 442 00:27:39,410 --> 00:27:41,240 Now, these kinds of geometric problems 443 00:27:41,240 --> 00:27:48,700 can be written very succinctly with differential equations. 444 00:27:48,700 --> 00:27:51,530 Namely, it's just the following. dy / dx, 445 00:27:51,530 --> 00:27:55,340 that's the slope of the tangent line, is equal to, 446 00:27:55,340 --> 00:27:58,030 well remember what the slope of this ray is, 447 00:27:58,030 --> 00:28:00,700 if this point-- I need a notation. 448 00:28:00,700 --> 00:28:04,520 At this point is (x, y) which is a point on the curve. 449 00:28:04,520 --> 00:28:07,860 So the slope of this pink line is what? 450 00:28:07,860 --> 00:28:09,650 STUDENT: [INAUDIBLE] 451 00:28:09,650 --> 00:28:12,610 PROFESSOR: y/x. 452 00:28:12,610 --> 00:28:20,810 So if it's twice it, there's the equation. 453 00:28:20,810 --> 00:28:28,040 OK, now, we only have one method for solving these equations. 454 00:28:28,040 --> 00:28:29,890 So let's use it. 455 00:28:29,890 --> 00:28:31,620 It says to separate variables. 456 00:28:31,620 --> 00:28:41,000 So I write dy / y here, is equal to 2 dx / x. 457 00:28:41,000 --> 00:28:42,530 That's the basic separation. 458 00:28:42,530 --> 00:28:47,990 That's the procedure that we're always going to use. 459 00:28:47,990 --> 00:28:54,640 And now if I integrate that, I find 460 00:28:54,640 --> 00:29:03,250 that on the right-hand side I have the logarithm of y. 461 00:29:03,250 --> 00:29:05,380 And on the left-hand-- Sorry, on the left-hand side 462 00:29:05,380 --> 00:29:06,590 I have the logarithm of y. 463 00:29:06,590 --> 00:29:10,500 On the right-hand side, I have twice the logarithm 464 00:29:10,500 --> 00:29:20,150 of x, plus a constant. 465 00:29:20,150 --> 00:29:27,330 So let's see what happens to this example. 466 00:29:27,330 --> 00:29:29,846 This is an implicit equation, and of course we 467 00:29:29,846 --> 00:29:31,970 have the problems of the plus or minus signs, which 468 00:29:31,970 --> 00:29:38,070 I'm not going to worry about until later. 469 00:29:38,070 --> 00:29:40,320 So let's exponentiate and see what happens. 470 00:29:40,320 --> 00:29:43,600 We get e^(ln y) = e^(2 ln x + c). 471 00:29:47,340 --> 00:29:51,940 So, again, this is y on the left-hand side. 472 00:29:51,940 --> 00:29:54,010 And on the right-hand side, if you think about it 473 00:29:54,010 --> 00:29:55,770 for a second, it's (e^(ln x))^2. 474 00:29:59,050 --> 00:30:00,370 Which is x^2. 475 00:30:00,370 --> 00:30:02,680 So this is x^2, and then there's an e^c. 476 00:30:02,680 --> 00:30:06,390 So that's another one of these A factors here. 477 00:30:06,390 --> 00:30:13,240 A = e^c. 478 00:30:13,240 --> 00:30:20,160 So the answer is, well, I'll draw the picture. 479 00:30:20,160 --> 00:30:22,530 And I'm going to cheat as I did before. 480 00:30:22,530 --> 00:30:24,550 We skipped the case y negative. 481 00:30:24,550 --> 00:30:30,236 We really only did the case y positive, so far. 482 00:30:30,236 --> 00:30:31,860 But if you think about it for a second, 483 00:30:31,860 --> 00:30:33,490 and we'll check it in a second, you're 484 00:30:33,490 --> 00:30:36,390 going to get all of these parabolas here. 485 00:30:36,390 --> 00:30:40,970 So the solution is this family of functions. 486 00:30:40,970 --> 00:30:44,330 And they can be bending down. 487 00:30:44,330 --> 00:30:45,660 As well as up. 488 00:30:45,660 --> 00:30:48,140 So these are the solutions to this equation. 489 00:30:48,140 --> 00:30:50,410 Every single one of these curves has the property 490 00:30:50,410 --> 00:30:52,750 that if you pick a point on it, the tangent line 491 00:30:52,750 --> 00:30:58,050 has twice the slope of the ray to the origin. 492 00:30:58,050 --> 00:31:01,840 And the formula, if you like, of the general solution is y = 493 00:31:01,840 --> 00:31:08,960 ax^2, a is any constant. 494 00:31:08,960 --> 00:31:09,460 Question? 495 00:31:09,460 --> 00:31:21,844 STUDENT: [INAUDIBLE] 496 00:31:21,844 --> 00:31:22,510 PROFESSOR: Yeah. 497 00:31:22,510 --> 00:31:28,960 So again - so first of all, so there 498 00:31:28,960 --> 00:31:30,110 are two approaches to this. 499 00:31:30,110 --> 00:31:32,900 One is to check it, and make sure that it's right. 500 00:31:32,900 --> 00:31:35,140 When a formula works for some family of values, 501 00:31:35,140 --> 00:31:36,740 sometimes it works for others. 502 00:31:36,740 --> 00:31:39,650 But another one is to realize that these things will usually 503 00:31:39,650 --> 00:31:40,970 work out this way. 504 00:31:40,970 --> 00:31:45,459 Because in this argument here, I allow the absolute value. 505 00:31:45,459 --> 00:31:47,750 And that would have been a perfectly legal thing for me 506 00:31:47,750 --> 00:31:48,250 to do. 507 00:31:48,250 --> 00:31:51,220 I could have put in absolute values here. 508 00:31:51,220 --> 00:31:55,690 In which case, I would've gotten that the absolute value of this 509 00:31:55,690 --> 00:31:56,890 was equal to that. 510 00:31:56,890 --> 00:32:02,370 And now you see I've covered the plus and minus cases. 511 00:32:02,370 --> 00:32:03,880 So it's that same idea. 512 00:32:03,880 --> 00:32:11,180 This implies that y is equal to either Ax^2 or -Ax^2, 513 00:32:11,180 --> 00:32:14,100 depending on which sign you pick. 514 00:32:14,100 --> 00:32:21,210 So that allows me for the curves above and curves below. 515 00:32:21,210 --> 00:32:25,470 Because it's really true that the antiderivative here is this 516 00:32:25,470 --> 00:32:26,240 function. 517 00:32:26,240 --> 00:32:28,820 It's defined for y negative. 518 00:32:28,820 --> 00:32:33,840 So let's just double-check. 519 00:32:33,840 --> 00:32:39,460 In this case, what's happening, we have y = ax^2 and we want 520 00:32:39,460 --> 00:32:44,410 to compute dy/dx to make sure that it satisfies the equation 521 00:32:44,410 --> 00:32:46,040 that I started out with. 522 00:32:46,040 --> 00:32:50,890 And what I see here is that this is 2ax. 523 00:32:50,890 --> 00:32:53,370 And now I'm going to write this in a suggestive way. 524 00:32:53,370 --> 00:33:00,330 I'm going to write it as 2ax^2 / x. 525 00:33:00,330 --> 00:33:06,610 And, sure enough, this is 2y / x. 526 00:33:06,610 --> 00:33:08,810 It does not matter whether a-- it 527 00:33:08,810 --> 00:33:17,370 works for a positive, a negative, a equals 0. 528 00:33:17,370 --> 00:33:24,180 It's OK. 529 00:33:24,180 --> 00:33:29,770 Again, we didn't pick up by this method the a = 0 case. 530 00:33:29,770 --> 00:33:35,350 And that's not surprising because we divided by y. 531 00:33:35,350 --> 00:33:39,660 There's another thing to watch out about, about this example. 532 00:33:39,660 --> 00:33:41,990 So there's another warning. 533 00:33:41,990 --> 00:33:44,910 Which I have to give you. 534 00:33:44,910 --> 00:33:47,130 And this is a subtlety which you definitely 535 00:33:47,130 --> 00:33:50,090 won't get to in any detail until you 536 00:33:50,090 --> 00:33:54,070 get to a higher level ordinary differential equations course, 537 00:33:54,070 --> 00:33:56,980 but I do want to warn you about it right now. 538 00:33:56,980 --> 00:34:05,310 Which is that if you look at the equation, 539 00:34:05,310 --> 00:34:14,100 you need to watch out that it's undefined at x = 0. 540 00:34:14,100 --> 00:34:15,700 It's undefined at x = 0. 541 00:34:15,700 --> 00:34:20,350 We also divided by x, and x is also a problem. 542 00:34:20,350 --> 00:34:24,690 Now, that actually has an important consequence. 543 00:34:24,690 --> 00:34:27,820 Which is that, strangely, knowing the value here 544 00:34:27,820 --> 00:34:31,040 and knowing the rate of change doesn't specify this function. 545 00:34:31,040 --> 00:34:33,180 This is bad. 546 00:34:33,180 --> 00:34:36,160 And it violates one of our pieces of intuition. 547 00:34:36,160 --> 00:34:38,720 And what's going wrong is that the rate of change 548 00:34:38,720 --> 00:34:40,600 was not specified. 549 00:34:40,600 --> 00:34:43,560 It's undefined at x = 0. 550 00:34:43,560 --> 00:34:45,260 So there's a problem here, and in fact 551 00:34:45,260 --> 00:34:48,220 if you think carefully about what this function is doing, 552 00:34:48,220 --> 00:34:53,510 it could come in on one branch and leave on a completely 553 00:34:53,510 --> 00:34:56,050 different branch. 554 00:34:56,050 --> 00:35:01,541 It doesn't really have to obey any rule across x = 0. 555 00:35:01,541 --> 00:35:03,540 So you should really be thinking of these things 556 00:35:03,540 --> 00:35:05,860 as rays emanating from the origin. 557 00:35:05,860 --> 00:35:10,140 The origin was a special point in the whole geometric problem. 558 00:35:10,140 --> 00:35:15,080 Rather than as being complete parabolas. 559 00:35:15,080 --> 00:35:16,460 But that's a very subtle point. 560 00:35:16,460 --> 00:35:23,270 I don't expect you to be able to say anything about it. 561 00:35:23,270 --> 00:35:25,920 But I just want to warn you that it really is true 562 00:35:25,920 --> 00:35:30,640 that when x = 0 there's a problem for this differential 563 00:35:30,640 --> 00:35:33,810 equation. 564 00:35:33,810 --> 00:35:46,570 So now, let me say our next problem. 565 00:35:46,570 --> 00:35:47,690 Next example. 566 00:35:47,690 --> 00:35:52,370 Just another geometry question. 567 00:35:52,370 --> 00:36:01,430 So here's Example 4. 568 00:36:01,430 --> 00:36:04,430 I'm just going to use the example that we've already got. 569 00:36:04,430 --> 00:36:09,090 Because there's only so much time left here. 570 00:36:09,090 --> 00:36:23,480 The fourth example is to take the curves perpendicular 571 00:36:23,480 --> 00:36:31,610 to the parabolas. 572 00:36:31,610 --> 00:36:33,332 This is another geometry problem. 573 00:36:33,332 --> 00:36:35,040 And by specifying that the the curves are 574 00:36:35,040 --> 00:36:37,020 perpendicular to these parabolas, 575 00:36:37,020 --> 00:36:44,500 I'm telling you what their slope is. 576 00:36:44,500 --> 00:36:47,000 So let's think about that. 577 00:36:47,000 --> 00:36:48,900 What's the new equation? 578 00:36:48,900 --> 00:36:56,270 The new diff. eq. is the following. 579 00:36:56,270 --> 00:37:01,550 It's that the slope is equal to the negative reciprocal 580 00:37:01,550 --> 00:37:05,610 of the slope of the tangent line. 581 00:37:05,610 --> 00:37:14,850 Of tangent to the parabola. 582 00:37:14,850 --> 00:37:16,800 So that's the equation. 583 00:37:16,800 --> 00:37:19,270 That's actually fairly easy to write down, 584 00:37:19,270 --> 00:37:26,570 because it's -1 divided by 2 y/x. 585 00:37:26,570 --> 00:37:32,281 That's the slope of the parabola. 586 00:37:32,281 --> 00:37:32,780 2y/x. 587 00:37:36,860 --> 00:37:38,300 So let's rewrite that. 588 00:37:38,300 --> 00:37:52,160 Now, this is-- the x goes in the numerator, so it's -x/(2y). 589 00:37:52,160 --> 00:37:57,990 And now I want to solve this one. 590 00:37:57,990 --> 00:38:01,780 Well, again, there's only one technique. 591 00:38:01,780 --> 00:38:10,240 Which is we're going to separate variables. 592 00:38:10,240 --> 00:38:12,010 And we separate the differentials here, 593 00:38:12,010 --> 00:38:18,201 so we get 2y dy = -x dx. 594 00:38:18,201 --> 00:38:20,200 That's just looking at the equation that I have, 595 00:38:20,200 --> 00:38:22,640 which is right over here. 596 00:38:22,640 --> 00:38:30,760 dy/dx = -x/(2y), and cross-multiplying to get this. 597 00:38:30,760 --> 00:38:33,100 And now I can take the antiderivative. 598 00:38:33,100 --> 00:38:35,850 This is y^2. 599 00:38:35,850 --> 00:38:40,670 And the antiderivative over here is -x^2 / 2 + c. 600 00:38:44,410 --> 00:38:57,410 And so, the solutions are x^2 / 2 + y^2 is equal to some c, 601 00:38:57,410 --> 00:39:02,600 some constant c. 602 00:39:02,600 --> 00:39:06,460 Now, this time, things don't work the same. 603 00:39:06,460 --> 00:39:09,600 And you can't expect them always to work the same. 604 00:39:09,600 --> 00:39:11,550 I claimed that this must be true. 605 00:39:11,550 --> 00:39:16,910 But unfortunately I cannot insist that every c will work. 606 00:39:16,910 --> 00:39:21,040 As you can see here, only the positive numbers c 607 00:39:21,040 --> 00:39:24,980 can work here. 608 00:39:24,980 --> 00:39:28,720 So the picture is that something slightly different 609 00:39:28,720 --> 00:39:29,530 happened here. 610 00:39:29,530 --> 00:39:31,310 And you have to live with this. 611 00:39:31,310 --> 00:39:33,570 Is that sometimes not all the constants will work. 612 00:39:33,570 --> 00:39:36,270 Because there's more to the problem than just 613 00:39:36,270 --> 00:39:37,880 the antidifferentiation. 614 00:39:37,880 --> 00:39:40,870 And here there are fewer answers rather than more answers. 615 00:39:40,870 --> 00:39:43,110 In the other case we had to add in some answers, 616 00:39:43,110 --> 00:39:45,030 here we have to take them away. 617 00:39:45,030 --> 00:39:46,890 Some of them don't make any sense. 618 00:39:46,890 --> 00:39:48,644 And the only ones we can get are the ones 619 00:39:48,644 --> 00:39:50,060 which are of this form, where this 620 00:39:50,060 --> 00:39:53,804 is, say, some radius squared. 621 00:39:53,804 --> 00:39:55,470 Well maybe I shouldn't call it a radius. 622 00:39:55,470 --> 00:39:58,800 I'll just call it a parameter, a^2. 623 00:39:58,800 --> 00:40:05,940 And these are of course ellipses. 624 00:40:05,940 --> 00:40:09,080 And you can see that the ellipses, 625 00:40:09,080 --> 00:40:13,990 the length here is the square root of 2a 626 00:40:13,990 --> 00:40:18,160 and the semi-axis, vertical semi-axis, is a. 627 00:40:18,160 --> 00:40:20,110 So this is the kind of ellipse that we've got. 628 00:40:20,110 --> 00:40:24,140 And I draw it on the previous diagram, 629 00:40:24,140 --> 00:40:28,100 I think it's somewhat suggestive here. 630 00:40:28,100 --> 00:40:30,280 There, ellipses are kind of eggs. 631 00:40:30,280 --> 00:40:32,820 They're a little bit longer than they are high. 632 00:40:32,820 --> 00:40:37,160 And they go like this. 633 00:40:37,160 --> 00:40:40,960 And if I drew them pretty much right, 634 00:40:40,960 --> 00:40:43,230 they should be making right angles. 635 00:40:43,230 --> 00:40:49,710 At all of these places. 636 00:40:49,710 --> 00:40:53,530 OK, last little bit here. 637 00:40:53,530 --> 00:40:57,930 Again, you've got to be very careful with these solutions. 638 00:40:57,930 --> 00:41:06,760 And so there's a warning here too. 639 00:41:06,760 --> 00:41:08,500 So let's take a look at the-- This 640 00:41:08,500 --> 00:41:10,924 is the implicit solution to the equation. 641 00:41:10,924 --> 00:41:13,090 And this is the one that tells us what the shape is. 642 00:41:13,090 --> 00:41:16,350 But we can also have the explicit solution. 643 00:41:16,350 --> 00:41:18,170 And if I solve for the explicit solution, 644 00:41:18,170 --> 00:41:25,400 it's y is equal to either plus square root of (a^2 - x^2 / 2), 645 00:41:25,400 --> 00:41:32,080 or y is equal to minus the square root of (a^2 - x^2 / 2). 646 00:41:32,080 --> 00:41:39,770 These are the explicit solutions. 647 00:41:39,770 --> 00:41:41,350 And now, we notice something that we 648 00:41:41,350 --> 00:41:43,850 should have noticed before. 649 00:41:43,850 --> 00:41:50,490 Which is that an ellipse is not a function. 650 00:41:50,490 --> 00:41:54,070 It's only the top half, if you like, 651 00:41:54,070 --> 00:41:56,610 that's giving you a solution to this equation. 652 00:41:56,610 --> 00:41:58,740 Or maybe the bottom half that's giving it 653 00:41:58,740 --> 00:42:00,450 the solution to the equation. 654 00:42:00,450 --> 00:42:07,480 So the one over here, this one is the top halves. 655 00:42:07,480 --> 00:42:15,270 And this one over here is the bottom halves. 656 00:42:15,270 --> 00:42:18,570 And there's something else that's interesting. 657 00:42:18,570 --> 00:42:28,560 Which is that we have a problem at y = 0. y = 0 658 00:42:28,560 --> 00:42:33,550 is where x = square root of 2a. 659 00:42:33,550 --> 00:42:35,960 That's when we get to this end here. 660 00:42:35,960 --> 00:42:38,800 And what happens is the solution comes around and it stops. 661 00:42:38,800 --> 00:42:44,820 It has a vertical slope. 662 00:42:44,820 --> 00:42:48,910 Vertical slope. 663 00:42:48,910 --> 00:42:56,420 And the solution stops. 664 00:42:56,420 --> 00:43:00,020 But really, that's not so unreasonable. 665 00:43:00,020 --> 00:43:01,890 After all, look at the formula. 666 00:43:01,890 --> 00:43:03,660 There was a y in the denominator here. 667 00:43:03,660 --> 00:43:08,500 When y = 0, the slope should be infinite. 668 00:43:08,500 --> 00:43:12,530 And so this equation is just giving us 669 00:43:12,530 --> 00:43:14,490 what it geometrically and intuitively 670 00:43:14,490 --> 00:43:15,800 should be giving us. 671 00:43:15,800 --> 00:43:22,790 At that stage. 672 00:43:22,790 --> 00:43:26,221 So that is the introduction to ordinary differential 673 00:43:26,221 --> 00:43:26,720 equations. 674 00:43:26,720 --> 00:43:28,740 Again, there's only one technique 675 00:43:28,740 --> 00:43:32,580 which is-- We're not done yet, we have a whole four minutes 676 00:43:32,580 --> 00:43:34,030 left and we're going to review. 677 00:43:34,030 --> 00:43:39,220 Now, so fortunately, this review is very short. 678 00:43:39,220 --> 00:43:42,170 Fortunately for you, I handed out to you 679 00:43:42,170 --> 00:43:44,920 exactly what you're going to be covering on the test. 680 00:43:44,920 --> 00:43:48,440 And it's what's printed here but there's a whole two 681 00:43:48,440 --> 00:43:51,000 pages of discussion. 682 00:43:51,000 --> 00:43:59,500 And I want to give you very, very clear-cut instructions 683 00:43:59,500 --> 00:44:00,000 here. 684 00:44:00,000 --> 00:44:04,780 This is usually the hardest test of this course. 685 00:44:04,780 --> 00:44:07,170 People usually do terribly on it. 686 00:44:07,170 --> 00:44:11,840 And I'm going to try to stop that 687 00:44:11,840 --> 00:44:14,850 by making it a little bit easier. 688 00:44:14,850 --> 00:44:17,620 And now here's what we're going to do. 689 00:44:17,620 --> 00:44:21,150 I'm telling you exactly what type of problems 690 00:44:21,150 --> 00:44:22,447 are going to be on the test. 691 00:44:22,447 --> 00:44:23,280 These are these six. 692 00:44:23,280 --> 00:44:25,940 It's also written on your sheet, your handout. 693 00:44:25,940 --> 00:44:29,012 It's also just what was asked on last year's test. 694 00:44:29,012 --> 00:44:31,220 You should go and you should look at last year's test 695 00:44:31,220 --> 00:44:33,420 and see what types of problems they are. 696 00:44:33,420 --> 00:44:36,210 I really, really, am going to ask the same questions, 697 00:44:36,210 --> 00:44:38,900 or the same type of questions. 698 00:44:38,900 --> 00:44:41,860 Not the same questions. 699 00:44:41,860 --> 00:44:44,960 So that's what's going to happen on the test. 700 00:44:44,960 --> 00:44:48,680 And let me just tell you, say one thing, which 701 00:44:48,680 --> 00:44:51,210 is the main theme of the class. 702 00:44:51,210 --> 00:44:52,290 And I will open up. 703 00:44:52,290 --> 00:44:54,330 We'll have time for one question after that. 704 00:44:54,330 --> 00:44:58,540 The main theme of this unit is simply the following. 705 00:44:58,540 --> 00:45:05,460 That information about derivative and sometimes 706 00:45:05,460 --> 00:45:11,010 maybe the second derivative, tells us 707 00:45:11,010 --> 00:45:17,315 information about f itself. 708 00:45:17,315 --> 00:45:19,940 And that's just what were doing here with ordinary differential 709 00:45:19,940 --> 00:45:20,490 equations. 710 00:45:20,490 --> 00:45:22,865 And that was what we were doing way at the beginning when 711 00:45:22,865 --> 00:45:23,825 we did approximations.