1 00:00:04,000 --> 00:00:10,000 The topic for today is how to change variables. 2 00:00:08,000 --> 00:00:14,000 So, we're talking about substitutions and differential 3 00:00:13,000 --> 00:00:19,000 equations, or changing variables. 4 00:00:16,000 --> 00:00:22,000 That might seem like a sort of fussy thing to talk about in the 5 00:00:22,000 --> 00:00:28,000 third or fourth lecture, but the reason is that so far, 6 00:00:27,000 --> 00:00:33,000 you know how to solve two kinds of differential equations, 7 00:00:32,000 --> 00:00:38,000 two kinds of first-order differential equations, 8 00:00:36,000 --> 00:00:42,000 one where you can separate variables, and the linear 9 00:00:41,000 --> 00:00:47,000 equation that we talked about last time. 10 00:00:47,000 --> 00:00:53,000 Now, the sad fact is that in some sense, those are the only 11 00:00:51,000 --> 00:00:57,000 two general methods there are, that those are the only two 12 00:00:56,000 --> 00:01:02,000 kinds of equations that can always be solved. 13 00:01:00,000 --> 00:01:06,000 Well, what about all the others? 14 00:01:02,000 --> 00:01:08,000 The answer is that to a great extent, all the other equations 15 00:01:06,000 --> 00:01:12,000 that can be solved, the solution can be done by 16 00:01:09,000 --> 00:01:15,000 changing the variables in the equation to reduce it to one of 17 00:01:14,000 --> 00:01:20,000 the cases that we can already do. 18 00:01:16,000 --> 00:01:22,000 Now, I'm going to give you two examples of that, 19 00:01:19,000 --> 00:01:25,000 two significant examples of that today. 20 00:01:22,000 --> 00:01:28,000 But, ultimately, as you will see, 21 00:01:24,000 --> 00:01:30,000 the way the equations are solved is by changing them into 22 00:01:28,000 --> 00:01:34,000 a linear equation, or an equation where the 23 00:01:31,000 --> 00:01:37,000 variables are separable. However, that's for a few 24 00:01:36,000 --> 00:01:42,000 minutes. The first change of variables 25 00:01:39,000 --> 00:01:45,000 that I want to talk about is an almost trivial one. 26 00:01:43,000 --> 00:01:49,000 But it's the most common kind there is, and you've already had 27 00:01:47,000 --> 00:01:53,000 it in physics class. But I think it's so important 28 00:01:51,000 --> 00:01:57,000 in the science and engineering subjects that it's a good idea, 29 00:01:55,000 --> 00:02:01,000 even in 18.03, to call attention to it 30 00:01:58,000 --> 00:02:04,000 explicitly. So, in that sense, 31 00:02:01,000 --> 00:02:07,000 the most common change of variables is the one simple one 32 00:02:06,000 --> 00:02:12,000 called scaling. So, again, the kind of equation 33 00:02:11,000 --> 00:02:17,000 I'm talking about is a general first-order equation. 34 00:02:15,000 --> 00:02:21,000 And, scaling simply means to change the coordinates, 35 00:02:20,000 --> 00:02:26,000 in effect, or axes, to change the coordinates on 36 00:02:24,000 --> 00:02:30,000 the axes to scale the axes, to either stretch them or 37 00:02:29,000 --> 00:02:35,000 contract them. So, what does the change of 38 00:02:34,000 --> 00:02:40,000 variable actually look like? Well, it means you introduce 39 00:02:39,000 --> 00:02:45,000 new variables, where x1 is equal to x times 40 00:02:42,000 --> 00:02:48,000 something or times a constant. I'll write it as divided by a 41 00:02:48,000 --> 00:02:54,000 constant, since that tends to be a little bit more the way people 42 00:02:53,000 --> 00:02:59,000 think of it. And y, the same. 43 00:02:56,000 --> 00:03:02,000 So, the new variable y1 is related to the old one by an 44 00:03:00,000 --> 00:03:06,000 equation of that form. So, a, b are constants. 45 00:03:06,000 --> 00:03:12,000 So, those are the equations. Why does one do this? 46 00:03:12,000 --> 00:03:18,000 Well, for a lot of reasons. But, maybe we can list them. 47 00:03:19,000 --> 00:03:25,000 You, for example, could be changing units. 48 00:03:24,000 --> 00:03:30,000 That's a common reason in physics. 49 00:03:28,000 --> 00:03:34,000 Changing the units that he used, you would have to make a 50 00:03:35,000 --> 00:03:41,000 change of coordinates of this form. 51 00:03:41,000 --> 00:03:47,000 Perhaps the even more important reason is to, 52 00:03:45,000 --> 00:03:51,000 sometimes it's used to make the variables dimensionless. 53 00:03:50,000 --> 00:03:56,000 In other words, so that the variables become 54 00:03:55,000 --> 00:04:01,000 pure numbers, with no units attached to them. 55 00:03:59,000 --> 00:04:05,000 Since you are well aware of the tortures involved in dealing 56 00:04:05,000 --> 00:04:11,000 with units in physics, the point of making variables, 57 00:04:10,000 --> 00:04:16,000 I'm sorry, dimensionless, I don't have to sell that. 58 00:04:17,000 --> 00:04:23,000 Dimensionless, i.e. 59 00:04:18,000 --> 00:04:24,000 no units, without any units attached. 60 00:04:21,000 --> 00:04:27,000 It just represents the number three, not three seconds, 61 00:04:26,000 --> 00:04:32,000 or three grahams, or anything like that. 62 00:04:30,000 --> 00:04:36,000 And, the third reason is to reduce or simplify the 63 00:04:34,000 --> 00:04:40,000 constants: reduce the number or simplify the constants in the 64 00:04:40,000 --> 00:04:46,000 equation. Reduce their number is self 65 00:04:44,000 --> 00:04:50,000 explanatory. Simplify means make them less, 66 00:04:48,000 --> 00:04:54,000 either dimensionless also, or if you can't do that, 67 00:04:53,000 --> 00:04:59,000 at least less dependent upon the critical units than the old 68 00:04:59,000 --> 00:05:05,000 ones were. Let me give you a very simple 69 00:05:04,000 --> 00:05:10,000 example which will illustrate most of these things. 70 00:05:08,000 --> 00:05:14,000 It's the equation. It's a version of the cooling 71 00:05:12,000 --> 00:05:18,000 law, which applies at very high temperatures, 72 00:05:16,000 --> 00:05:22,000 and it runs. So, it's like Newton's cooling 73 00:05:20,000 --> 00:05:26,000 laws, except it's the internal and external temperatures vary, 74 00:05:25,000 --> 00:05:31,000 what's important is not the first power as in Newton's Law, 75 00:05:30,000 --> 00:05:36,000 but the fourth power. So, it's a constant. 76 00:05:34,000 --> 00:05:40,000 And, the difference is, now, it's the external 77 00:05:38,000 --> 00:05:44,000 temperature, which, just so there won't be so many 78 00:05:42,000 --> 00:05:48,000 capital T's in the equation, I'm going to call M, 79 00:05:46,000 --> 00:05:52,000 to the forth power minus T to the forth power. 80 00:05:50,000 --> 00:05:56,000 So, T is the internal 81 00:05:53,000 --> 00:05:59,000 temperature, the thing we are interested in. 82 00:05:57,000 --> 00:06:03,000 And, M is the external constant, which I'll assume, 83 00:06:01,000 --> 00:06:07,000 now, is a constant external temperature. 84 00:06:06,000 --> 00:06:12,000 So, this is valid if big temperature differences, 85 00:06:10,000 --> 00:06:16,000 Newton's Law, breaks down and one needs a 86 00:06:14,000 --> 00:06:20,000 different one. Now, you are free to solve that 87 00:06:19,000 --> 00:06:25,000 equation just as it stands, if you can. 88 00:06:23,000 --> 00:06:29,000 There are difficulties connected with it because you're 89 00:06:28,000 --> 00:06:34,000 dealing with the fourth powers, of course. 90 00:06:34,000 --> 00:06:40,000 But, before you do that, one should scale. 91 00:06:37,000 --> 00:06:43,000 How should I scale? Well, I'm going to scale by 92 00:06:42,000 --> 00:06:48,000 relating T to M. So, that is very likely to use 93 00:06:46,000 --> 00:06:52,000 is T1 equals T divided by M. 94 00:06:50,000 --> 00:06:56,000 This is now dimensionless because M, of course, 95 00:06:55,000 --> 00:07:01,000 has the units of temperature, degrees Celsius, 96 00:06:59,000 --> 00:07:05,000 degrees absolute, whatever it is, 97 00:07:02,000 --> 00:07:08,000 as does T. And therefore, 98 00:07:06,000 --> 00:07:12,000 by taking the ratio of the two, there are no units attached to 99 00:07:11,000 --> 00:07:17,000 it. So, this is dimensionless. 100 00:07:14,000 --> 00:07:20,000 Now, how actually do I change the variable in the equation? 101 00:07:19,000 --> 00:07:25,000 Well, watch this. It's an utterly trivial idea, 102 00:07:23,000 --> 00:07:29,000 and utterly important. Don't slog around doing it this 103 00:07:27,000 --> 00:07:33,000 way, trying to stuff it in, and divide first. 104 00:07:31,000 --> 00:07:37,000 Instead, do the inverse. In other words, 105 00:07:36,000 --> 00:07:42,000 write it instead as T equals MT1, the reason being that it's 106 00:07:41,000 --> 00:07:47,000 T that's facing you in that equation, and therefore T you 107 00:07:46,000 --> 00:07:52,000 want to substitute for. So, let's do it. 108 00:07:49,000 --> 00:07:55,000 The new equation will be what? Well, dT-- Since this is a 109 00:07:54,000 --> 00:08:00,000 constant, the left-hand side becomes dT1 / dt times M equals 110 00:07:59,000 --> 00:08:05,000 k times M to the forth minus M to the forth T1 to the forth, 111 00:08:04,000 --> 00:08:10,000 so I'm going to factor 112 00:08:09,000 --> 00:08:15,000 out that M to the forth, and make it one minus T1 to the 113 00:08:14,000 --> 00:08:20,000 forth, okay? 114 00:08:18,000 --> 00:08:24,000 Now, I could divide through by M and get rid of one of those, 115 00:08:23,000 --> 00:08:29,000 and so, the new equation, now, is dT1 / dt, 116 00:08:27,000 --> 00:08:33,000 d time, is equal to-- Now, I have k M cubed out 117 00:08:32,000 --> 00:08:38,000 front here. I'm going to just give that a 118 00:08:36,000 --> 00:08:42,000 new name, k1. Essentially, 119 00:08:39,000 --> 00:08:45,000 it's the same equation. It's no harder to solve and no 120 00:08:44,000 --> 00:08:50,000 easier to solve than the original one. 121 00:08:47,000 --> 00:08:53,000 But it's been simplified. For one, I think it looks 122 00:08:52,000 --> 00:08:58,000 better. So, to compare the two, 123 00:08:55,000 --> 00:09:01,000 I'll put this one up in green, and this one in green, 124 00:09:00,000 --> 00:09:06,000 too, just to convince you it's the same, but indicate that it's 125 00:09:06,000 --> 00:09:12,000 the same equation. Notice, so, T1 has been 126 00:09:11,000 --> 00:09:17,000 rendered, is now dimensionless. So, I don't have to even ask 127 00:09:16,000 --> 00:09:22,000 when I solve this equation, oh, please tell me what the 128 00:09:20,000 --> 00:09:26,000 units of temperature are. How you are measuring 129 00:09:24,000 --> 00:09:30,000 temperature makes no difference to this equation. 130 00:09:27,000 --> 00:09:33,000 k1 still has units. What units does it have? 131 00:09:32,000 --> 00:09:38,000 It's been simplified because it now has the units of, 132 00:09:37,000 --> 00:09:43,000 since this is dimensionless and this is dimensionless, 133 00:09:42,000 --> 00:09:48,000 it has the units of inverse time. 134 00:09:45,000 --> 00:09:51,000 So, k1, whereas it had units involving both degrees and 135 00:09:51,000 --> 00:09:57,000 seconds before, now it has inverse time as its 136 00:09:55,000 --> 00:10:01,000 units. And, moreover, 137 00:09:57,000 --> 00:10:03,000 there's one less constant. So, one less constant in the 138 00:10:02,000 --> 00:10:08,000 equation. It just looks better. 139 00:10:07,000 --> 00:10:13,000 This business, I think you know that k1, 140 00:10:11,000 --> 00:10:17,000 the process of forming k1 out of k M cubed is 141 00:10:18,000 --> 00:10:24,000 called lumping constants. I think they use standard 142 00:10:23,000 --> 00:10:29,000 terminology in physics and engineering courses. 143 00:10:30,000 --> 00:10:36,000 Try to get all the constants together like this. 144 00:10:33,000 --> 00:10:39,000 And then you lump them there. They are lumped for you, 145 00:10:36,000 --> 00:10:42,000 and then you just give the lump a new name. 146 00:10:39,000 --> 00:10:45,000 So, that's an example of scaling. 147 00:10:41,000 --> 00:10:47,000 Watch out for when you can use that. 148 00:10:43,000 --> 00:10:49,000 For example, it would have probably been a 149 00:10:46,000 --> 00:10:52,000 good thing to use in the first problem set when you were 150 00:10:50,000 --> 00:10:56,000 handling this problem of drug elimination and hormone 151 00:10:53,000 --> 00:10:59,000 elimination production inside of the thing. 152 00:10:56,000 --> 00:11:02,000 You could lump constants, and as was done to some extent 153 00:11:00,000 --> 00:11:06,000 on the solutions to get a neater looking answer, 154 00:11:03,000 --> 00:11:09,000 one without so many constants in it. 155 00:11:07,000 --> 00:11:13,000 Okay, let's now go to serious stuff, where we are actually 156 00:11:11,000 --> 00:11:17,000 going to make changes of variables which we hope will 157 00:11:16,000 --> 00:11:22,000 render unsolvable equations suddenly solvable. 158 00:11:20,000 --> 00:11:26,000 Now, I'm going to do that by making substitutions. 159 00:11:24,000 --> 00:11:30,000 But, it's, I think, quite important to watch up 160 00:11:28,000 --> 00:11:34,000 there are two kinds of substitutions. 161 00:11:31,000 --> 00:11:37,000 There are direct substitutions. That's where you introduce a 162 00:11:37,000 --> 00:11:43,000 new variable. I don't know how to write this 163 00:11:41,000 --> 00:11:47,000 on the board. I'll just write it 164 00:11:43,000 --> 00:11:49,000 schematically. So, it's one which says that 165 00:11:47,000 --> 00:11:53,000 the new variable is equal to some combination of the old 166 00:11:52,000 --> 00:11:58,000 variables. The other kind of substitution 167 00:11:56,000 --> 00:12:02,000 is inverse. It's just the reverse. 168 00:12:00,000 --> 00:12:06,000 Here, you say that the old variables are some combination 169 00:12:04,000 --> 00:12:10,000 of the new. Now, often you'll have to stick 170 00:12:07,000 --> 00:12:13,000 in a few old variables, too. 171 00:12:09,000 --> 00:12:15,000 But the basic, it's what appears on the 172 00:12:12,000 --> 00:12:18,000 left-hand side. Is it a new variable that 173 00:12:15,000 --> 00:12:21,000 appears on the left-hand side by itself, or is it the old 174 00:12:19,000 --> 00:12:25,000 variable that appears on the left-hand side? 175 00:12:23,000 --> 00:12:29,000 Now, right here, we have an example. 176 00:12:25,000 --> 00:12:31,000 If I did it as a direct substitution, 177 00:12:28,000 --> 00:12:34,000 I would have written T1 equals T over M. 178 00:12:34,000 --> 00:12:40,000 That's the way I define the new variable, which of course you 179 00:12:38,000 --> 00:12:44,000 have to do if you're introducing it. 180 00:12:41,000 --> 00:12:47,000 But when I actually did the substitution, 181 00:12:44,000 --> 00:12:50,000 I did the inverse substitution. Namely, I used T equals T1, 182 00:12:48,000 --> 00:12:54,000 M times T1. And, 183 00:12:50,000 --> 00:12:56,000 the reason for doing that was because it was the capital T's 184 00:12:55,000 --> 00:13:01,000 that faced me in the equation and I had to have something to 185 00:12:59,000 --> 00:13:05,000 replace them with. Now, you see this already in 186 00:13:03,000 --> 00:13:09,000 calculus, this distinction. But that might have been a year 187 00:13:07,000 --> 00:13:13,000 and a half ago. Just let me remind you, 188 00:13:10,000 --> 00:13:16,000 typically in calculus, for example, 189 00:13:12,000 --> 00:13:18,000 when you want to do this kind of integral, let's say, 190 00:13:15,000 --> 00:13:21,000 x times the square root of one minus x squared dx, 191 00:13:19,000 --> 00:13:25,000 the substitution you'd use for 192 00:13:23,000 --> 00:13:29,000 that is u equals one minus x squared, 193 00:13:26,000 --> 00:13:32,000 right? And then, you calculate, 194 00:13:28,000 --> 00:13:34,000 and then you would observe that this, the x dx, 195 00:13:31,000 --> 00:13:37,000 more or less makes up du, apart from a constant factor, 196 00:13:35,000 --> 00:13:41,000 there. So, this would be an example of 197 00:13:39,000 --> 00:13:45,000 direct substitution. You put it in and convert the 198 00:13:42,000 --> 00:13:48,000 integral into an integral of u. What would be an example of 199 00:13:46,000 --> 00:13:52,000 inverse substitution? Well, if I take away the x and 200 00:13:50,000 --> 00:13:56,000 ask you, instead, to do this integral, 201 00:13:52,000 --> 00:13:58,000 then you know that the right thing to do is not to start with 202 00:13:56,000 --> 00:14:02,000 u, but to start with the x and write x equals sine or cosine u. 203 00:14:02,000 --> 00:14:08,000 So, this is a direct substitution in that integral, 204 00:14:05,000 --> 00:14:11,000 but this integral calls for an inverse substitution in order to 205 00:14:09,000 --> 00:14:15,000 be able to do it. And notice, they would look 206 00:14:12,000 --> 00:14:18,000 practically the same. But, of course, 207 00:14:15,000 --> 00:14:21,000 as you know from your experience, they are not. 208 00:14:18,000 --> 00:14:24,000 They're very different. Okay, so I'm going to watch for 209 00:14:21,000 --> 00:14:27,000 that distinction as I do these examples. 210 00:14:24,000 --> 00:14:30,000 The first one I want to do is an example as a direct 211 00:14:28,000 --> 00:14:34,000 substitution. 212 00:14:47,000 --> 00:14:53,000 So, it applies to the equation of the form y prime equals, 213 00:14:52,000 --> 00:14:58,000 there are two kinds of terms on the right-hand side. 214 00:14:56,000 --> 00:15:02,000 Let's use p of x, p of x times y plus q of x 215 00:15:00,000 --> 00:15:06,000 times any power whatsoever of y. 216 00:15:05,000 --> 00:15:11,000 Well, notice, for example, 217 00:15:07,000 --> 00:15:13,000 if n were zero, what kind of equation would 218 00:15:11,000 --> 00:15:17,000 this be? y to the n would be 219 00:15:14,000 --> 00:15:20,000 one, and this would be a linear equation, which you know how to 220 00:15:20,000 --> 00:15:26,000 solve. So, n equals zero we already 221 00:15:23,000 --> 00:15:29,000 know how to do. So, let's assume that n is not 222 00:15:27,000 --> 00:15:33,000 zero, so that we're in new territory. 223 00:15:33,000 --> 00:15:39,000 Well, if n were equal to one, you could separate variables. 224 00:15:38,000 --> 00:15:44,000 So, that too is not exciting. But, nonetheless, 225 00:15:42,000 --> 00:15:48,000 it will be included in what I'm going to say now. 226 00:15:46,000 --> 00:15:52,000 If n is two or three, or n could be one half. 227 00:15:50,000 --> 00:15:56,000 So anything: even zero is all right. 228 00:15:53,000 --> 00:15:59,000 It's just silly. Any number: it could be 229 00:15:57,000 --> 00:16:03,000 negative. n equals minus five. 230 00:15:59,000 --> 00:16:05,000 That would be fine also. This kind of equation, 231 00:16:04,000 --> 00:16:10,000 to give it its name, is called the Bernoulli 232 00:16:08,000 --> 00:16:14,000 equation, named after which Bernoulli, I haven't the 233 00:16:11,000 --> 00:16:17,000 faintest idea. There were, I think, 234 00:16:14,000 --> 00:16:20,000 three or four of them. And, they fought with each 235 00:16:18,000 --> 00:16:24,000 other. But, they were all smart. 236 00:16:20,000 --> 00:16:26,000 Now, the key trick, if you like, 237 00:16:23,000 --> 00:16:29,000 method, to solving any Bernoulli equation, 238 00:16:26,000 --> 00:16:32,000 let me call another thing. Most important is what's 239 00:16:30,000 --> 00:16:36,000 missing. It must not have a pure x term 240 00:16:34,000 --> 00:16:40,000 in it. And that goes for a constant 241 00:16:37,000 --> 00:16:43,000 term. In other words, 242 00:16:38,000 --> 00:16:44,000 it must look exactly like this. Everything multiplied by y, 243 00:16:43,000 --> 00:16:49,000 or a power of y, two terms. 244 00:16:45,000 --> 00:16:51,000 So, for example, if I add one to this, 245 00:16:48,000 --> 00:16:54,000 the equation becomes non-doable. 246 00:16:51,000 --> 00:16:57,000 Right, it's very easy to contaminate it into an equation 247 00:16:55,000 --> 00:17:01,000 that's unsolvable. It's got to look just like 248 00:16:59,000 --> 00:17:05,000 that. Now, you've got one on your 249 00:17:03,000 --> 00:17:09,000 homework. You've got several. 250 00:17:05,000 --> 00:17:11,000 Both part one and part two have Bernoulli equations on them. 251 00:17:10,000 --> 00:17:16,000 So, this is practical, in some sense. 252 00:17:13,000 --> 00:17:19,000 What do we got? The idea is to divide by y to 253 00:17:17,000 --> 00:17:23,000 the n. Ignore all formulas that you're 254 00:17:20,000 --> 00:17:26,000 given. Just remember that when you see 255 00:17:23,000 --> 00:17:29,000 something that looks like this, or something that you can turn 256 00:17:28,000 --> 00:17:34,000 into something that looks like this, divide through by y to the 257 00:17:34,000 --> 00:17:40,000 nth power, no matter what n is. All right, so y prime over y to 258 00:17:40,000 --> 00:17:46,000 the n is equal to p of x times one over y to the n minus one, 259 00:17:44,000 --> 00:17:50,000 right, plus q of x. 260 00:17:49,000 --> 00:17:55,000 Well, that certainly doesn't look any better than what I 261 00:17:53,000 --> 00:17:59,000 started with. And, in your terms, 262 00:17:55,000 --> 00:18:01,000 it probably looks somewhat worse because it's got all those 263 00:17:59,000 --> 00:18:05,000 Y's at the denominator, and who wants to see them 264 00:18:03,000 --> 00:18:09,000 there? But, look at it. 265 00:18:06,000 --> 00:18:12,000 In this very slightly transformed Bernoulli equation 266 00:18:10,000 --> 00:18:16,000 is a linear equation struggling to be free. 267 00:18:14,000 --> 00:18:20,000 Where is it? Why is it trying to be a linear 268 00:18:17,000 --> 00:18:23,000 equation? Make a new variable, 269 00:18:20,000 --> 00:18:26,000 call this hunk of it in new variable. 270 00:18:23,000 --> 00:18:29,000 Let's call it V. So, V is equal to one over y to 271 00:18:27,000 --> 00:18:33,000 the n minus one. 272 00:18:30,000 --> 00:18:36,000 Or, if you like, you can think of that as y to 273 00:18:34,000 --> 00:18:40,000 the one minus n. What's V prime? 274 00:18:39,000 --> 00:18:45,000 So, this is the direct substitution I am going to use, 275 00:18:44,000 --> 00:18:50,000 but of course, the problem is, 276 00:18:46,000 --> 00:18:52,000 what am I going to use on this? Well, the little miracle 277 00:18:51,000 --> 00:18:57,000 happens. What's the derivative of this? 278 00:18:54,000 --> 00:19:00,000 It is one minus n times y to the negative n times y prime 279 00:18:59,000 --> 00:19:05,000 In other words, 280 00:19:04,000 --> 00:19:10,000 up to a constant, this constant factor, 281 00:19:07,000 --> 00:19:13,000 one minus n, it's exactly the left-hand side 282 00:19:11,000 --> 00:19:17,000 of the equation. Well, let's make N not equal 283 00:19:15,000 --> 00:19:21,000 one either. As I said, you could separate 284 00:19:18,000 --> 00:19:24,000 variables if n equals one. What's the equation, 285 00:19:22,000 --> 00:19:28,000 then, turned into? A Bernoulli equation, 286 00:19:28,000 --> 00:19:34,000 divided through in this way, is then turned into the 287 00:19:36,000 --> 00:19:42,000 equation one minus n, sorry, V prime divided by one 288 00:19:44,000 --> 00:19:50,000 minus n is equal to p of x times V plus q of x. 289 00:19:55,000 --> 00:20:01,000 It's linear. 290 00:20:01,000 --> 00:20:07,000 And now, solve it as a linear equation. 291 00:20:03,000 --> 00:20:09,000 Solve it as a linear equation. You notice, it's not in 292 00:20:06,000 --> 00:20:12,000 standard form, not in standard linear form. 293 00:20:09,000 --> 00:20:15,000 To do that, you're going to have to put the p on the other 294 00:20:13,000 --> 00:20:19,000 side. That's okay, 295 00:20:14,000 --> 00:20:20,000 that term, on the other side, solve it, and at the end, 296 00:20:17,000 --> 00:20:23,000 don't forget that you put in the V. 297 00:20:19,000 --> 00:20:25,000 It wasn't in the original problem. 298 00:20:22,000 --> 00:20:28,000 So, you have to convert the problem, the answer, 299 00:20:25,000 --> 00:20:31,000 back in terms of y. It'll come out in terms of V, 300 00:20:28,000 --> 00:20:34,000 but you must put it back in terms of y. 301 00:20:32,000 --> 00:20:38,000 Let's do a really simple example just to illustrate the 302 00:20:38,000 --> 00:20:44,000 method, and to illustrate the fact that I don't want you to 303 00:20:45,000 --> 00:20:51,000 memorize formulas. Learn methods, 304 00:20:49,000 --> 00:20:55,000 not final formulas. So, suppose the equation is, 305 00:20:54,000 --> 00:21:00,000 let's say, y prime equals y over x minus y squared. 306 00:21:01,000 --> 00:21:07,000 That's a Bernoulli equation. 307 00:21:06,000 --> 00:21:12,000 I could, of course, have concealed it by writing xy 308 00:21:09,000 --> 00:21:15,000 prime plus xy prime minus xy equals negative y squared. 309 00:21:13,000 --> 00:21:19,000 Then, it wouldn't look instantly like a Bernoulli 310 00:21:16,000 --> 00:21:22,000 equation. You would have to stare at it a 311 00:21:19,000 --> 00:21:25,000 while and say, hey, that's a Bernoulli 312 00:21:22,000 --> 00:21:28,000 equation. Okay, but so I'm handing it to 313 00:21:25,000 --> 00:21:31,000 you a silver platter, as it were. 314 00:21:27,000 --> 00:21:33,000 So, what do we do? Divide through by y squared. 315 00:21:32,000 --> 00:21:38,000 So, it's y prime over y squared equals one over x times one over 316 00:21:38,000 --> 00:21:44,000 y minus one. 317 00:21:41,000 --> 00:21:47,000 And now, the substitution, then, I'm going to make, 318 00:21:46,000 --> 00:21:52,000 is for this thing. V equals one over y. 319 00:21:51,000 --> 00:21:57,000 It's a direct substitution. 320 00:21:53,000 --> 00:21:59,000 V prime is going to be negative one over y squared 321 00:21:59,000 --> 00:22:05,000 times y prime. Don't forget to use the chain 322 00:22:05,000 --> 00:22:11,000 rule when you differentiate with respect-- because the 323 00:22:08,000 --> 00:22:14,000 differentiation is with respect to x, of course, 324 00:22:12,000 --> 00:22:18,000 not with respect to y. Okay, so what's this thing? 325 00:22:15,000 --> 00:22:21,000 That's the left-hand side. The only thing is it's got a 326 00:22:19,000 --> 00:22:25,000 negative sign. So, this is minus V prime 327 00:22:22,000 --> 00:22:28,000 equals, one over x stays one over x, one over y. 328 00:22:26,000 --> 00:22:32,000 So, it's V over x minus one. 329 00:22:30,000 --> 00:22:36,000 So, let's put that in standard form. 330 00:22:32,000 --> 00:22:38,000 In standard form, it will look like, 331 00:22:35,000 --> 00:22:41,000 first imagine multiplying it through by negative one, 332 00:22:39,000 --> 00:22:45,000 and then putting this term on the other side. 333 00:22:42,000 --> 00:22:48,000 And, it will turn into V prime plus V over X is equal to one. 334 00:22:47,000 --> 00:22:53,000 So, that's the linear equation 335 00:22:51,000 --> 00:22:57,000 in standard linear form that we are asked to solve. 336 00:22:54,000 --> 00:23:00,000 And, the solution isn't very hard. 337 00:22:57,000 --> 00:23:03,000 The integrating factor is, well, I integrate one over x. 338 00:23:03,000 --> 00:23:09,000 That makes log x. And, e to the log x, 339 00:23:05,000 --> 00:23:11,000 so, it's e to the log x, which is, of course, 340 00:23:09,000 --> 00:23:15,000 just x itself. So, I should multiply this 341 00:23:12,000 --> 00:23:18,000 through by x squared, be able to integrate it. 342 00:23:15,000 --> 00:23:21,000 Now, some of you, I would hope, 343 00:23:17,000 --> 00:23:23,000 just can see that right away, that if you multiply this 344 00:23:21,000 --> 00:23:27,000 through by x, it's going to look good. 345 00:23:24,000 --> 00:23:30,000 So, after we multiply through by x, which I get? 346 00:23:27,000 --> 00:23:33,000 (xV) prime for the-- maybe I shouldn't skip a step. 347 00:23:33,000 --> 00:23:39,000 You are still learning this stuff, so let's not skip a step. 348 00:23:38,000 --> 00:23:44,000 So, it becomes x V prime plus V equals x, 349 00:23:44,000 --> 00:23:50,000 okay? After I multiplied through by 350 00:23:47,000 --> 00:23:53,000 the integrating factor, this now says this is xV prime, 351 00:23:53,000 --> 00:23:59,000 and I quickly check that that, in fact, is what it's equal to, 352 00:23:59,000 --> 00:24:05,000 equals x, and therefore xV is equal to one half x squared plus 353 00:24:05,000 --> 00:24:11,000 a constant. And, 354 00:24:08,000 --> 00:24:14,000 therefore, V is equal to one half x plus C over x. 355 00:24:14,000 --> 00:24:20,000 You can leave it at that form, 356 00:24:19,000 --> 00:24:25,000 or you can combine terms. It doesn't matter much. 357 00:24:23,000 --> 00:24:29,000 Am I done? The answer is, 358 00:24:25,000 --> 00:24:31,000 no I am not done, because nobody reading this 359 00:24:29,000 --> 00:24:35,000 answer would know what V was. V wasn't in the original 360 00:24:33,000 --> 00:24:39,000 problem. It was y that was in the 361 00:24:35,000 --> 00:24:41,000 original problem. And therefore, 362 00:24:37,000 --> 00:24:43,000 the relation is, one is the reciprocal of the 363 00:24:40,000 --> 00:24:46,000 other. And therefore, 364 00:24:41,000 --> 00:24:47,000 I have to turn this expression upside down. 365 00:24:44,000 --> 00:24:50,000 Well, if you're going to have to turn it upside down, 366 00:24:47,000 --> 00:24:53,000 you probably should make it look a little better. 367 00:24:50,000 --> 00:24:56,000 Let's rewrite it as x squared plus 2c, 368 00:24:53,000 --> 00:24:59,000 combining fractions, I think they call it in high 369 00:24:56,000 --> 00:25:02,000 school or elementary school, plus 2c. 370 00:25:00,000 --> 00:25:06,000 How's that? x squared plus 2c divided by 2x. 371 00:25:03,000 --> 00:25:09,000 Now, 2c, you don't call it 372 00:25:07,000 --> 00:25:13,000 constant 2c because this is just as arbitrary to call it c1. 373 00:25:12,000 --> 00:25:18,000 So, I'll call that, so, my answer will be y equals 374 00:25:16,000 --> 00:25:22,000 2x divided by x squared plus an arbitrary constant. 375 00:25:20,000 --> 00:25:26,000 But, to indicate it's different from that one, 376 00:25:23,000 --> 00:25:29,000 I'll call it C1. C1 is 377 00:25:27,000 --> 00:25:33,000 two times the old one, but that doesn't really matter. 378 00:25:31,000 --> 00:25:37,000 So, there's the solution. It has an arbitrary constant in 379 00:25:37,000 --> 00:25:43,000 it, but you note it's not an additive arbitrary constant. 380 00:25:40,000 --> 00:25:46,000 The arbitrary constant is tucked into the solution. 381 00:25:44,000 --> 00:25:50,000 If you had to satisfy an initial condition, 382 00:25:47,000 --> 00:25:53,000 you would take this form, and starting from this form, 383 00:25:50,000 --> 00:25:56,000 figure out what C1 was in order to satisfy that initial 384 00:25:54,000 --> 00:26:00,000 condition. Thus, Bernoulli equation is 385 00:25:57,000 --> 00:26:03,000 solved. Our first Bernoulli equation: 386 00:25:59,000 --> 00:26:05,000 isn't that exciting? So, here was the equation, 387 00:26:05,000 --> 00:26:11,000 and there is its solution. Now, the one I'm asking you to 388 00:26:11,000 --> 00:26:17,000 solve on the problem set in part two is no harder than this, 389 00:26:18,000 --> 00:26:24,000 except I ask you some hard questions about it, 390 00:26:24,000 --> 00:26:30,000 not very hard, but a little hard about it. 391 00:26:30,000 --> 00:26:36,000 I hope you will find them interesting questions. 392 00:26:33,000 --> 00:26:39,000 You already have the experimental evidence from the 393 00:26:37,000 --> 00:26:43,000 first problem set, and the problem is to explain 394 00:26:40,000 --> 00:26:46,000 the experimental evidence by actually solving the equation in 395 00:26:45,000 --> 00:26:51,000 the scene. I think you'll find it 396 00:26:47,000 --> 00:26:53,000 interesting. But, maybe that's just a pious 397 00:26:51,000 --> 00:26:57,000 hope. Okay, I like, 398 00:26:52,000 --> 00:26:58,000 now, to turn to the second method, where a second class of 399 00:26:56,000 --> 00:27:02,000 equations which require inverse substitution, 400 00:27:00,000 --> 00:27:06,000 and those are equations, which are called homogeneous, 401 00:27:04,000 --> 00:27:10,000 a highly overworked word in differential equations, 402 00:27:08,000 --> 00:27:14,000 and in mathematics in general. But, it's unfortunately just 403 00:27:14,000 --> 00:27:20,000 the right word to describe them. So, these are homogeneous, 404 00:27:19,000 --> 00:27:25,000 first-order ODE's. Now, I already used the word in 405 00:27:23,000 --> 00:27:29,000 one context in talking about the linear equations when zero is 406 00:27:28,000 --> 00:27:34,000 the right hand side. This is different, 407 00:27:32,000 --> 00:27:38,000 but nonetheless, the two uses of the word have 408 00:27:35,000 --> 00:27:41,000 the same common source. The homogeneous differential 409 00:27:39,000 --> 00:27:45,000 equation, homogeneous newspeak, is y prime equals, 410 00:27:43,000 --> 00:27:49,000 it's a question of what the right hand side looks like. 411 00:27:47,000 --> 00:27:53,000 And, now, the supposed way to say it is, you should be able to 412 00:27:52,000 --> 00:27:58,000 write the right-hand side as a function of a combined variable, 413 00:27:57,000 --> 00:28:03,000 y divided by x. In other words, 414 00:28:01,000 --> 00:28:07,000 after fooling around with the right hand side a little bit, 415 00:28:06,000 --> 00:28:12,000 you should be able to write it so that every time a variable 416 00:28:11,000 --> 00:28:17,000 appears, it's always in the combination y over x. 417 00:28:15,000 --> 00:28:21,000 Let me give some examples. For example, 418 00:28:19,000 --> 00:28:25,000 suppose y prime were, let's say, x squared y divided 419 00:28:23,000 --> 00:28:29,000 by x cubed plus y cubed. 420 00:28:29,000 --> 00:28:35,000 Well, that doesn't look in that form. 421 00:28:31,000 --> 00:28:37,000 Well, yes it is. Imagine dividing the top and 422 00:28:34,000 --> 00:28:40,000 bottom by x cubed. What would you get? 423 00:28:37,000 --> 00:28:43,000 The top would be y over x, if you divided it by x 424 00:28:40,000 --> 00:28:46,000 cubed. And, if I divide the bottom by 425 00:28:43,000 --> 00:28:49,000 x cubed, also, which, of course, 426 00:28:45,000 --> 00:28:51,000 doesn't change the value of the fraction, as they say in 427 00:28:49,000 --> 00:28:55,000 elementary school, one plus (y over x) cubed. 428 00:28:52,000 --> 00:28:58,000 So, this is the way it started 429 00:28:55,000 --> 00:29:01,000 out looking, but you just said ah-ha, that was a homogeneous 430 00:28:59,000 --> 00:29:05,000 equation because I could see it could be written that way. 431 00:29:05,000 --> 00:29:11,000 How about another homogeneous equation? 432 00:29:10,000 --> 00:29:16,000 How about x y prime? Is that a homogeneous equation? 433 00:29:18,000 --> 00:29:24,000 Of course it is: otherwise, why would I be 434 00:29:24,000 --> 00:29:30,000 talking about it? If you divide through by x, 435 00:29:29,000 --> 00:29:35,000 you can tuck it inside the radical, the square root, 436 00:29:33,000 --> 00:29:39,000 if you remember to square it when you do that. 437 00:29:36,000 --> 00:29:42,000 And, it becomes the square root of x squared over x squared, 438 00:29:41,000 --> 00:29:47,000 which is one, plus y squared over x squared. 439 00:29:44,000 --> 00:29:50,000 It's homogeneous. 440 00:29:47,000 --> 00:29:53,000 Now, you might say, hey, this looks like you had to 441 00:29:50,000 --> 00:29:56,000 be rather clever to figure out if an equation is homogeneous. 442 00:29:55,000 --> 00:30:01,000 Is there some other way? Yeah, there is another way, 443 00:29:58,000 --> 00:30:04,000 and it's the other way which explains why it's called 444 00:30:02,000 --> 00:30:08,000 homogeneous. You can think of it this way. 445 00:30:07,000 --> 00:30:13,000 It's an equation which is, in modern speak, 446 00:30:12,000 --> 00:30:18,000 invariant, invariant under the operation zoom. 447 00:30:18,000 --> 00:30:24,000 What is zoom? Zoom is, you increase the scale 448 00:30:23,000 --> 00:30:29,000 equally on both axes. So, the zoom operation is the 449 00:30:30,000 --> 00:30:36,000 one which sends x into a times x, 450 00:30:36,000 --> 00:30:42,000 and y into a times y. 451 00:30:42,000 --> 00:30:48,000 In other words, you change the scale on both 452 00:30:46,000 --> 00:30:52,000 axes by the same factor, a. 453 00:30:48,000 --> 00:30:54,000 Now, what I say is, gee, maybe you shouldn't write 454 00:30:53,000 --> 00:30:59,000 it like this. Why don't we say, 455 00:30:56,000 --> 00:31:02,000 we introduce, how about this? 456 00:31:00,000 --> 00:31:06,000 So, think of it as a change of variables. 457 00:31:02,000 --> 00:31:08,000 We will write it like that. So, you can put here an equals 458 00:31:06,000 --> 00:31:12,000 sign, if you don't know what this meaningless arrow means. 459 00:31:10,000 --> 00:31:16,000 So, I'm making this change of variables, and I'm describing it 460 00:31:14,000 --> 00:31:20,000 as an inverse substitution. But of course, 461 00:31:16,000 --> 00:31:22,000 it wouldn't make any difference. 462 00:31:19,000 --> 00:31:25,000 It's exactly the same as the direct substitution I started 463 00:31:22,000 --> 00:31:28,000 out with underscaling. The only difference is, 464 00:31:25,000 --> 00:31:31,000 I'm not using different scales on both axes. 465 00:31:28,000 --> 00:31:34,000 I'm expanding them both equally. 466 00:31:32,000 --> 00:31:38,000 That's what I mean by zoom. Now, what happens to the 467 00:31:36,000 --> 00:31:42,000 equation? Well, what happens to dy over 468 00:31:40,000 --> 00:31:46,000 dx? Well, dx is a dx1. 469 00:31:43,000 --> 00:31:49,000 dy is a dy1. 470 00:31:47,000 --> 00:31:53,000 And therefore, the ratio, dy by dx is the same 471 00:31:51,000 --> 00:31:57,000 as dy1 over dx1. 472 00:31:54,000 --> 00:32:00,000 So, the left-hand side becomes dy1 over dx1, 473 00:31:58,000 --> 00:32:04,000 and the right-hand side becomes F of, well, y over x is the same 474 00:32:04,000 --> 00:32:10,000 as y over, since I've scaled them equally, 475 00:32:08,000 --> 00:32:14,000 this is the same as y1 over x1. 476 00:32:14,000 --> 00:32:20,000 So, it's y1 over x1, and the net effect is I simply, 477 00:32:18,000 --> 00:32:24,000 everywhere I have an x, I change it to x1, 478 00:32:22,000 --> 00:32:28,000 and wherever I have a y, I change it to y1, 479 00:32:25,000 --> 00:32:31,000 which, what's in a name? It's the identical equation. 480 00:32:31,000 --> 00:32:37,000 So, I haven't changed the equation at all via zoom 481 00:32:35,000 --> 00:32:41,000 transformation. And, that's what makes it 482 00:32:38,000 --> 00:32:44,000 homogeneous. That's not an uncommon use of 483 00:32:42,000 --> 00:32:48,000 the word homogeneous. When you say space is 484 00:32:45,000 --> 00:32:51,000 homogeneous, every direction, well, that means, 485 00:32:49,000 --> 00:32:55,000 I don't know. It means, okay, 486 00:32:51,000 --> 00:32:57,000 I'm getting into trouble there. I'll let it go since I can't 487 00:32:56,000 --> 00:33:02,000 prepare any better, I haven't prepared any better 488 00:33:00,000 --> 00:33:06,000 explanation, but this is a pretty good one. 489 00:33:06,000 --> 00:33:12,000 Okay, so, suppose we've got a homogeneous equation. 490 00:33:13,000 --> 00:33:19,000 How do we solve it? So, here's our equation, 491 00:33:20,000 --> 00:33:26,000 F of y over x. Well, what substitution would 492 00:33:29,000 --> 00:33:35,000 you like to make? Obviously, we should make a 493 00:33:34,000 --> 00:33:40,000 direct substitution, z equals y over x. 494 00:33:38,000 --> 00:33:44,000 So, why did he say that this was going to be an example of 495 00:33:42,000 --> 00:33:48,000 inverse substitution? Because I wanted to confuse 496 00:33:45,000 --> 00:33:51,000 you. But look, that's fine. 497 00:33:47,000 --> 00:33:53,000 If you write it in that form, you'll know exactly what to do 498 00:33:51,000 --> 00:33:57,000 with the right-hand side. And, this is why everybody 499 00:33:55,000 --> 00:34:01,000 loves to do that. But for Charlie, 500 00:33:57,000 --> 00:34:03,000 you have to substitute into the left-hand side as well. 501 00:34:03,000 --> 00:34:09,000 And, I can testify, for many years of looking with 502 00:34:06,000 --> 00:34:12,000 sinking heart at examination papers, what happens if you try 503 00:34:10,000 --> 00:34:16,000 to make a direct substitution like this? 504 00:34:13,000 --> 00:34:19,000 You say, oh, I need z prime. 505 00:34:15,000 --> 00:34:21,000 z prime equals, well, I better use the quotient 506 00:34:18,000 --> 00:34:24,000 rule for differentiating that. And, it comes out this long, 507 00:34:22,000 --> 00:34:28,000 and then either a long pause, what do I do now? 508 00:34:26,000 --> 00:34:32,000 Because it's not at all obvious what to do at that point. 509 00:34:30,000 --> 00:34:36,000 Or, much worse, two pages of frantic 510 00:34:32,000 --> 00:34:38,000 calculations, and giving up in total despair. 511 00:34:37,000 --> 00:34:43,000 Now, the reason for that is because you tried to do it 512 00:34:40,000 --> 00:34:46,000 making a direct substitution. All you have to do instead is 513 00:34:45,000 --> 00:34:51,000 use it, treat it as an inverse substitution, 514 00:34:48,000 --> 00:34:54,000 write y equals zx. What's the motivation for doing 515 00:34:51,000 --> 00:34:57,000 that? It's clear from the equation. 516 00:34:54,000 --> 00:35:00,000 This goes through all of mathematics. 517 00:34:57,000 --> 00:35:03,000 Whenever you have to change a variable, excuse me, 518 00:35:00,000 --> 00:35:06,000 whenever you have to change a variable, look at what you have 519 00:35:05,000 --> 00:35:11,000 to substitute for, and focus your attention on 520 00:35:08,000 --> 00:35:14,000 that. I need to know what y prime is. 521 00:35:12,000 --> 00:35:18,000 Okay, well, then I better know what y is. 522 00:35:15,000 --> 00:35:21,000 If I know what y is, do I know what y prime is? 523 00:35:19,000 --> 00:35:25,000 Oh, of course. y prime is z prime x plus z 524 00:35:22,000 --> 00:35:28,000 times the derivative of this factor, which is one. 525 00:35:26,000 --> 00:35:32,000 And now, I turned with that one 526 00:35:31,000 --> 00:35:37,000 stroke, the equation has now become z prime x plus z is equal 527 00:35:36,000 --> 00:35:42,000 to F of z. Well, I don't know. 528 00:35:40,000 --> 00:35:46,000 Can I solve that? Sure. 529 00:35:42,000 --> 00:35:48,000 That can be solved because this is x times dz / dx. 530 00:35:48,000 --> 00:35:54,000 Just put the z on the other side, it's F of z minus z. 531 00:35:53,000 --> 00:35:59,000 And now, this side is just a 532 00:35:55,000 --> 00:36:01,000 function of z. Separate variables. 533 00:36:00,000 --> 00:36:06,000 And, the only thing to watch out for is, at the end, 534 00:36:03,000 --> 00:36:09,000 the z was your business. You've got to put the answer 535 00:36:07,000 --> 00:36:13,000 back in terms of z and y. Okay, let's work an example of 536 00:36:11,000 --> 00:36:17,000 this. Since I haven't done any 537 00:36:13,000 --> 00:36:19,000 modeling yet this period, let's make a little model, 538 00:36:17,000 --> 00:36:23,000 differential equations model. It's a physical situation, 539 00:36:21,000 --> 00:36:27,000 which will be solved by an equation. 540 00:36:24,000 --> 00:36:30,000 And, guess what? The equation will turn out to 541 00:36:27,000 --> 00:36:33,000 be homogeneous. Okay, so the situation is as 542 00:36:32,000 --> 00:36:38,000 follows. We are in the Caribbean 543 00:36:34,000 --> 00:36:40,000 somewhere, a little isolated island somewhere with a little 544 00:36:39,000 --> 00:36:45,000 lighthouse on it at the origin, and a beam of light shines from 545 00:36:44,000 --> 00:36:50,000 the lighthouse. The beam of light can rotate 546 00:36:48,000 --> 00:36:54,000 the way the lighthouse beams. But, this particular beam is 547 00:36:53,000 --> 00:36:59,000 being controlled by a guy in the lighthouse who can aim it 548 00:36:57,000 --> 00:37:03,000 wherever he wants. And, the reason he's interested 549 00:37:01,000 --> 00:37:07,000 in aiming it wherever he wants is there's a drug boat here, 550 00:37:06,000 --> 00:37:12,000 [LAUGHTER] which has just been caught in the beam of light. 551 00:37:13,000 --> 00:37:19,000 So, the drug boat, which has just been caught in a 552 00:37:16,000 --> 00:37:22,000 beam of light, and feels it'd a better escape. 553 00:37:20,000 --> 00:37:26,000 Now, the lighthouse keeper wants to keep the drug boat; 554 00:37:24,000 --> 00:37:30,000 the light is shining on it so that the U.S. 555 00:37:27,000 --> 00:37:33,000 Coast Guard helicopters can zoom over it and do whatever 556 00:37:31,000 --> 00:37:37,000 they do to drug boats, -- 557 00:37:34,000 --> 00:37:40,000 -- I don't know. So, the drug boat immediately 558 00:37:37,000 --> 00:37:43,000 has to follow an escape strategy. 559 00:37:39,000 --> 00:37:45,000 And, the only one that occurs to him is, well, 560 00:37:42,000 --> 00:37:48,000 he wants to go further away, of course, from the lighthouse. 561 00:37:47,000 --> 00:37:53,000 On the other hand, it doesn't seem sensible to do 562 00:37:50,000 --> 00:37:56,000 it in a straight line because the beam will keep shining on 563 00:37:54,000 --> 00:38:00,000 him. So, he fixes the boat at some 564 00:37:57,000 --> 00:38:03,000 angle, let's say, and goes off so that the angle 565 00:38:00,000 --> 00:38:06,000 stays 45 degrees. So, it goes so that the angle 566 00:38:05,000 --> 00:38:11,000 between the beam and maybe, draw the beam a little less 567 00:38:11,000 --> 00:38:17,000 like a 45 degree angle. So, the angle between the beam 568 00:38:16,000 --> 00:38:22,000 and the boat, the boat's path is always 45 569 00:38:20,000 --> 00:38:26,000 degrees, goes at a constant 45 degree angle to the beam, 570 00:38:26,000 --> 00:38:32,000 hoping thereby to escape. On the other hand, 571 00:38:30,000 --> 00:38:36,000 of course, the lighthouse guy keeps the beam always on the 572 00:38:36,000 --> 00:38:42,000 boat. So, it's not clear it's a good 573 00:38:40,000 --> 00:38:46,000 strategy, but this is a differential equations class. 574 00:38:44,000 --> 00:38:50,000 The question is, what's the path of the boat? 575 00:38:48,000 --> 00:38:54,000 What's the boat's path? Now, an obvious question is, 576 00:38:52,000 --> 00:38:58,000 why is this a problem in differential equations at all? 577 00:38:57,000 --> 00:39:03,000 In other words, looking at this, 578 00:38:59,000 --> 00:39:05,000 you might scratch your head and try to think of different ways 579 00:39:04,000 --> 00:39:10,000 to solve it. But, what suggests that it's 580 00:39:09,000 --> 00:39:15,000 going to be a problem in differential equations? 581 00:39:13,000 --> 00:39:19,000 The answer is, you're looking for a path. 582 00:39:17,000 --> 00:39:23,000 The answer is going to be a curve. 583 00:39:20,000 --> 00:39:26,000 A curve means a function. We are looking for an unknown 584 00:39:24,000 --> 00:39:30,000 function, in other words. And, what type of information 585 00:39:29,000 --> 00:39:35,000 do we have about the function? The only information we have 586 00:39:34,000 --> 00:39:40,000 about the function is something about its slope, 587 00:39:38,000 --> 00:39:44,000 that its slope makes a constant 45° angle with the lighthouse 588 00:39:44,000 --> 00:39:50,000 beam. Its slope makes a constant 589 00:39:53,000 --> 00:39:59,000 known angle to a known angle. Well, if you are trying to find 590 00:40:04,000 --> 00:40:10,000 a function, and all you know is something about its slope, 591 00:40:09,000 --> 00:40:15,000 that is a problem in differential equations. 592 00:40:13,000 --> 00:40:19,000 Well, let's try to solve it. Well, let's see. 593 00:40:16,000 --> 00:40:22,000 Well, let me draw just a little bit. 594 00:40:19,000 --> 00:40:25,000 So, here's the horizontal. Let's introduce the 595 00:40:23,000 --> 00:40:29,000 coordinates. In other words, 596 00:40:25,000 --> 00:40:31,000 there's the horizontal and here's the boat to indicate 597 00:40:30,000 --> 00:40:36,000 where I am with respect to the picture. 598 00:40:35,000 --> 00:40:41,000 So, here's the boat. Here's the beam, 599 00:40:38,000 --> 00:40:44,000 and the path of the boat is going to make a 45° angle with 600 00:40:44,000 --> 00:40:50,000 it. So, this is the path that we 601 00:40:47,000 --> 00:40:53,000 are talking about. And now, let's label what I 602 00:40:51,000 --> 00:40:57,000 know. Well, this angle is 45°. 603 00:40:54,000 --> 00:41:00,000 This angle, I don't know, but of course I can calculate 604 00:41:00,000 --> 00:41:06,000 it easily enough because it has to do with, if I know the 605 00:41:05,000 --> 00:41:11,000 coordinates of this point, (x, y), then of course that 606 00:41:11,000 --> 00:41:17,000 horizontal angle, I know the slope of this line, 607 00:41:15,000 --> 00:41:21,000 and that angle will be related to the slope. 608 00:41:22,000 --> 00:41:28,000 So, let's call this alpha. And now, what I want to know is 609 00:41:29,000 --> 00:41:35,000 what the slope of the whole path is. 610 00:41:35,000 --> 00:41:41,000 So, y prime-- let's call y equals y of x, 611 00:41:42,000 --> 00:41:48,000 the unknown function whose path, whose graph is going to be 612 00:41:50,000 --> 00:41:56,000 the boat's path, unknown graph. 613 00:41:54,000 --> 00:42:00,000 What's its slope? Well, its slope is the tangent 614 00:42:00,000 --> 00:42:06,000 of the sum of these two angles, alpha plus 45°. 615 00:42:08,000 --> 00:42:14,000 Now, what do I know? Well, I know that the tangent 616 00:42:11,000 --> 00:42:17,000 of alpha is how much? That's y over x. 617 00:42:14,000 --> 00:42:20,000 In other words, 618 00:42:16,000 --> 00:42:22,000 if this was the point, x over y, this is the angle it 619 00:42:19,000 --> 00:42:25,000 makes with a horizontal, if you think of it over here. 620 00:42:23,000 --> 00:42:29,000 So, this angle is the same as that one, and it's y over, 621 00:42:26,000 --> 00:42:32,000 its slope of that line is y over x. 622 00:42:30,000 --> 00:42:36,000 So, the tangent of the angle is y over x. 623 00:42:32,000 --> 00:42:38,000 How about the tangent of 45°? That's one, and there's a 624 00:42:36,000 --> 00:42:42,000 formula. This is the hard part. 625 00:42:38,000 --> 00:42:44,000 All you have to know is that the formula exists, 626 00:42:41,000 --> 00:42:47,000 and then you will look it up if you have forgotten it, 627 00:42:44,000 --> 00:42:50,000 relating the tangent or giving you the tangent of the sum of 628 00:42:48,000 --> 00:42:54,000 two angles, and you can, if you like, 629 00:42:50,000 --> 00:42:56,000 clever, derive it from the formula for the sign and cosine 630 00:42:54,000 --> 00:43:00,000 of the sum of two angles. But, one peak is worth a 631 00:42:57,000 --> 00:43:03,000 thousand finesses. So, it is the tangent of alpha 632 00:43:02,000 --> 00:43:08,000 plus the tangent of 45°. Let me read it out in all its 633 00:43:06,000 --> 00:43:12,000 gory details, divided by one, 634 00:43:08,000 --> 00:43:14,000 so you'll at least learn the formula, one minus tangent alpha 635 00:43:12,000 --> 00:43:18,000 times tangent 45°. 636 00:43:15,000 --> 00:43:21,000 This would work for the tangent of the sum of any two angles. 637 00:43:20,000 --> 00:43:26,000 That's the formula. So, what do I get then? 638 00:43:23,000 --> 00:43:29,000 y prime is equal to the tangent of alpha, which is y over x, 639 00:43:27,000 --> 00:43:33,000 oh, I like that combination, plus one, divided by (one minus 640 00:43:32,000 --> 00:43:38,000 y over x times one). 641 00:43:37,000 --> 00:43:43,000 Now, there is no reason for doing anything to it, 642 00:43:40,000 --> 00:43:46,000 but let's make it look a little prettier, and thereby, 643 00:43:44,000 --> 00:43:50,000 make it less obvious that it's a homogeneous equation. 644 00:43:48,000 --> 00:43:54,000 If I multiply top and bottom by x, it looks prettier. 645 00:43:52,000 --> 00:43:58,000 x plus y over x minus y equals y prime. 646 00:43:55,000 --> 00:44:01,000 That's our differential equation. 647 00:44:00,000 --> 00:44:06,000 But, notice, that let step to make it look 648 00:44:02,000 --> 00:44:08,000 pretty has undone the good work. It's fine if you immediately 649 00:44:06,000 --> 00:44:12,000 recognize this as being a homogeneous equation because you 650 00:44:10,000 --> 00:44:16,000 can divide the top and bottom by x. 651 00:44:12,000 --> 00:44:18,000 But here, it's a lot clearer that it's a homogeneous equation 652 00:44:16,000 --> 00:44:22,000 because it's already been written in the right form. 653 00:44:20,000 --> 00:44:26,000 Okay, let's solve it now, since we know what to do. 654 00:44:23,000 --> 00:44:29,000 We're going to use as the new variable, z equals y over x. 655 00:44:27,000 --> 00:44:33,000 And, as I wrote up there for y 656 00:44:33,000 --> 00:44:39,000 prime, we'll substitute z prime x plus z. 657 00:44:40,000 --> 00:44:46,000 And, with that, let's solve. 658 00:44:44,000 --> 00:44:50,000 Let's solve it. The equation becomes z prime x 659 00:44:50,000 --> 00:44:56,000 plus z is equal to z plus one over one minus z. 660 00:44:57,000 --> 00:45:03,000 We want to separate variables, 661 00:45:04,000 --> 00:45:10,000 so you have to put all the z's on one side. 662 00:45:07,000 --> 00:45:13,000 So, this is going to be x, dz / dx equals this thing minus 663 00:45:11,000 --> 00:45:17,000 z, which is (z plus one) over (one minus z) minus z. 664 00:45:14,000 --> 00:45:20,000 And now, as you realize, 665 00:45:18,000 --> 00:45:24,000 putting it on the other side, I'm going to have to turn it 666 00:45:22,000 --> 00:45:28,000 upside down. Just as before, 667 00:45:24,000 --> 00:45:30,000 if you have to turn something upside down, it's better to 668 00:45:28,000 --> 00:45:34,000 combine the terms, and make it one tiny little 669 00:45:31,000 --> 00:45:37,000 fraction. Otherwise, you are in for quite 670 00:45:35,000 --> 00:45:41,000 a lot of mess if you don't do this nicely. 671 00:45:39,000 --> 00:45:45,000 So, z plus one minus z, that gets rid of the z's. 672 00:45:43,000 --> 00:45:49,000 The numerator is one minus z squared over one minus z, 673 00:45:48,000 --> 00:45:54,000 I hope, one, is that right, 674 00:45:50,000 --> 00:45:56,000 (one plus z squared) over (one minus z). 675 00:45:56,000 --> 00:46:02,000 And so, the question is dz, 676 00:45:58,000 --> 00:46:04,000 and put this on the other side and turn it upside down. 677 00:46:05,000 --> 00:46:11,000 So, that will be (one minus z) over (one plus z squared) on the 678 00:46:10,000 --> 00:46:16,000 left-hand side and on the right-hand side, 679 00:46:14,000 --> 00:46:20,000 dx over x. Well, that's ready to be 680 00:46:17,000 --> 00:46:23,000 integrated just as it stands. The right-hand side integrates 681 00:46:23,000 --> 00:46:29,000 to be log x. The left-hand side is the sum 682 00:46:26,000 --> 00:46:32,000 of two terms. The integral of one over one 683 00:46:30,000 --> 00:46:36,000 plus z squared is the arc tangent of z, 684 00:46:34,000 --> 00:46:40,000 maybe? The derivative of this is one 685 00:46:37,000 --> 00:46:43,000 over one plus z squared. 686 00:46:40,000 --> 00:46:46,000 How about the term z over one plus z squared? 687 00:46:44,000 --> 00:46:50,000 Well, that integrates to be a 688 00:46:46,000 --> 00:46:52,000 logarithm. It is more or less the 689 00:46:48,000 --> 00:46:54,000 logarithm of one plus z squared. 690 00:46:51,000 --> 00:46:57,000 If I differentiate this, I get one over one plus z^2 691 00:46:54,000 --> 00:47:00,000 times 2z, but I wish I had negative z 692 00:46:58,000 --> 00:47:04,000 there instead. Therefore, I should put a minus 693 00:47:01,000 --> 00:47:07,000 sign, and I should multiply that by half to make it come out 694 00:47:05,000 --> 00:47:11,000 right. And, this is log x on the right 695 00:47:09,000 --> 00:47:15,000 hand side plus, put in that arbitrary constant. 696 00:47:13,000 --> 00:47:19,000 And now what? Well, let's now fool around 697 00:47:16,000 --> 00:47:22,000 with it a little bit. The arc tangent, 698 00:47:19,000 --> 00:47:25,000 I'm going to simultaneously, no, two steps. 699 00:47:22,000 --> 00:47:28,000 I have to remember your innocence, although probably a 700 00:47:26,000 --> 00:47:32,000 lot of you are better calculators than I am. 701 00:47:31,000 --> 00:47:37,000 I'm going to change this, use as many laws of logarithms 702 00:47:35,000 --> 00:47:41,000 as possible. I'm going to put this in the 703 00:47:38,000 --> 00:47:44,000 exponent, and put this on the other side. 704 00:47:41,000 --> 00:47:47,000 That's going to turn it into the log of the square root of 705 00:47:45,000 --> 00:47:51,000 one plus z squared. 706 00:47:48,000 --> 00:47:54,000 And, this is going to be plus the log of x plus c. 707 00:47:53,000 --> 00:47:59,000 And, now I'm going to make, 708 00:47:55,000 --> 00:48:01,000 go back and remember that z equals y over x. 709 00:48:00,000 --> 00:48:06,000 So, this becomes the arc tangent of y over x equals. 710 00:48:04,000 --> 00:48:10,000 Now, I combine the logarithms. 711 00:48:09,000 --> 00:48:15,000 This is the log of x times this square root, right, 712 00:48:12,000 --> 00:48:18,000 make one logarithm out of it, and then put z equals y over z. 713 00:48:16,000 --> 00:48:22,000 And, you see that if you do 714 00:48:19,000 --> 00:48:25,000 that, it'll be the log of x times the square root of one 715 00:48:23,000 --> 00:48:29,000 plus (y over x) squared, 716 00:48:27,000 --> 00:48:33,000 and what is that? Well, if I put this over x 717 00:48:30,000 --> 00:48:36,000 squared and take it out, it cancels that. 718 00:48:34,000 --> 00:48:40,000 And, what you are left with is the log of the square root of x 719 00:48:38,000 --> 00:48:44,000 squared plus y squared plus a constant. 720 00:48:42,000 --> 00:48:48,000 Now, technically, 721 00:48:43,000 --> 00:48:49,000 you have solved the equation, but not morally because, 722 00:48:47,000 --> 00:48:53,000 I mean, my God, what a mess! 723 00:48:49,000 --> 00:48:55,000 Incredible path. It tells me absolutely nothing. 724 00:48:52,000 --> 00:48:58,000 Wow, what is the screaming? Change me to polar coordinates. 725 00:48:56,000 --> 00:49:02,000 What's the arc tangent of y over x? 726 00:49:00,000 --> 00:49:06,000 Theta. In polar coordinates it's 727 00:49:02,000 --> 00:49:08,000 theta. This is r. 728 00:49:04,000 --> 00:49:10,000 So, the curve is theta equals the log of r plus a constant. 729 00:49:09,000 --> 00:49:15,000 And, I can make even that little better if I exponentiate 730 00:49:14,000 --> 00:49:20,000 everything, exponentiate both sides, combine this in the usual 731 00:49:19,000 --> 00:49:25,000 way, the and what you get is that r is equal to some other 732 00:49:24,000 --> 00:49:30,000 constant times e to the theta. 733 00:49:30,000 --> 00:49:36,000 That's the curve. It's called an exponential 734 00:49:33,000 --> 00:49:39,000 spiral, and that's what our little boat goes in. 735 00:49:37,000 --> 00:49:43,000 And notice, probably if I had set up the problem in polar 736 00:49:42,000 --> 00:49:48,000 coordinates from the beginning, nobody would have been able to 737 00:49:48,000 --> 00:49:54,000 solve it. But, anyone who did would have 738 00:49:51,000 --> 00:49:57,000 gotten that answer immediately. Thanks.