1 00:00:09,000 --> 00:00:13,000 This time, we started solving differential equations. 2 00:00:13,000 --> 00:00:16,072 This is the third lecture of the term, 3 00:00:16,072 --> 00:00:20,000 and I have yet to solve a single differential 4 00:00:20,000 --> 00:00:22,000 equation in this class. 5 00:00:22,000 --> 00:00:24,912 Well, that will be rectified from now 6 00:00:24,912 --> 00:00:27,500 until the end of the term. 7 00:00:27,500 --> 00:00:31,142 So, once you learn separation of variables, 8 00:00:31,142 --> 00:00:36,220 which is the most elementary method there is, the single, 9 00:00:36,220 --> 00:00:39,500 I think the single most important equation 10 00:00:39,500 --> 00:00:44,713 is the one that's called the first order linear equation, 11 00:00:44,713 --> 00:00:47,500 both because it occurs frequently 12 00:00:47,500 --> 00:00:52,725 in models because it's solvable, and-- I think that's enough. 13 00:00:52,725 --> 00:00:56,333 If you drop the course after today 14 00:00:56,333 --> 00:00:59,428 you will still have learned those two important methods: 15 00:00:59,428 --> 00:01:03,000 separation of variables, and first order linear equations. 16 00:01:03,000 --> 00:01:06,750 So, what does such an equation look like? 17 00:01:06,750 --> 00:01:09,000 Well, I'll write it in there. 18 00:01:09,000 --> 00:01:11,331 There are several ways of writing it, 19 00:01:11,331 --> 00:01:13,900 but I think the most basic is this. 20 00:01:13,900 --> 00:01:17,000 I'm going to use x as the independent variable 21 00:01:17,000 --> 00:01:20,000 because that's what your book does. 22 00:01:20,000 --> 00:01:22,568 But in the applications, it's often 23 00:01:22,568 --> 00:01:25,000 t, time, that is the independent variable. 24 00:01:25,000 --> 00:01:29,000 And, I'll try to give you examples which show that. 25 00:01:29,000 --> 00:01:31,000 So, the equation looks like this. 26 00:01:31,000 --> 00:01:34,328 I'll find some function of x times y 27 00:01:34,328 --> 00:01:37,535 prime plus some other function of x times y 28 00:01:37,535 --> 00:01:40,000 is equal to yet another function of x. 29 00:01:40,000 --> 00:01:46,999 Obviously, the x doesn't have the same status 30 00:01:46,999 --> 00:01:50,750 here that y does, so y is extremely limited in how 31 00:01:50,750 --> 00:01:53,000 it can appear in the equation. 32 00:01:53,000 --> 00:01:57,000 But, x can be pretty much arbitrary in those places. 33 00:01:57,000 --> 00:02:00,000 So, that's the equation we are talking about, 34 00:02:00,000 --> 00:02:02,270 and I'll put it up. 35 00:02:02,270 --> 00:02:07,496 This is the first version of it, and we'll call them purple. 36 00:02:07,496 --> 00:02:11,000 Now, why is that called the linear equation? 37 00:02:11,000 --> 00:02:15,998 The word linear is a very heavily used word 38 00:02:15,998 --> 00:02:18,800 in mathematics, science, and engineering. 39 00:02:18,800 --> 00:02:21,664 For the moment, the best simple answer 40 00:02:21,664 --> 00:02:27,571 is because it's linear in y and y prime, the variables 41 00:02:27,571 --> 00:02:31,000 y and y prime. 42 00:02:31,000 --> 00:02:33,000 Well, y prime is not a variable. 43 00:02:33,000 --> 00:02:36,000 Well, you will learn, in a certain sense, 44 00:02:36,000 --> 00:02:41,000 it helps to think of it as one, not right now perhaps, 45 00:02:41,000 --> 00:02:43,400 but think of it as linear. 46 00:02:43,400 --> 00:02:47,664 The most closely analogous thing would be a linear equation, 47 00:02:47,664 --> 00:02:53,000 a real linear equation, the kind you studied in high school, 48 00:02:53,000 --> 00:02:55,500 which would look like this. 49 00:02:55,500 --> 00:02:59,200 It would have two variables, and, I guess, 50 00:02:59,200 --> 00:03:01,428 constant coefficients, equal c. 51 00:03:01,428 --> 00:03:05,000 Now, that's a linear equation. 52 00:03:05,000 --> 00:03:09,000 And that's the sense in which this is linear. 53 00:03:09,000 --> 00:03:12,072 It's linear in y prime and y, which 54 00:03:12,072 --> 00:03:15,776 are the analogs of the variables y1 and y2. 55 00:03:15,776 --> 00:03:20,664 A little bit of terminology, if c is equal to zero, 56 00:03:20,664 --> 00:03:25,000 it's called homogeneous, the same way this equation is 57 00:03:25,000 --> 00:03:30,500 called homogeneous, as you know from 18.02, if the right hand 58 00:03:30,500 --> 00:03:32,000 side is zero. 59 00:03:32,000 --> 00:03:37,000 So, c of x I should write here, but I won't. 60 00:03:37,000 --> 00:03:38,500 That's called homogeneous. 61 00:03:38,500 --> 00:03:43,000 Now, this is a common form for the equation, 62 00:03:43,000 --> 00:03:47,142 but it's not what it's called standard form. 63 00:03:47,142 --> 00:03:51,664 The standard form for the equation, and since this 64 00:03:51,664 --> 00:03:58,332 is going to be a prime course of confusion, which is probably 65 00:03:58,332 --> 00:04:03,600 completely correct, but a prime source of confusion 66 00:04:03,600 --> 00:04:06,000 is what I meant. 67 00:04:06,000 --> 00:04:09,330 The standard linear form, and I'll 68 00:04:09,330 --> 00:04:13,912 underline linear is the first co efficient of y prime 69 00:04:13,912 --> 00:04:16,000 is taken to be one. 70 00:04:16,000 --> 00:04:20,160 So, you can always convert that to a standard form 71 00:04:20,160 --> 00:04:22,776 by simply dividing through by it. 72 00:04:22,776 --> 00:04:28,552 And if I do that, the equation will look like y prime plus, 73 00:04:28,552 --> 00:04:35,000 now, it's common to not call it b anymore, the coefficient, 74 00:04:35,000 --> 00:04:38,750 because it's really b over a. 75 00:04:38,750 --> 00:04:42,000 And, therefore, it's common to adopt, yet, 76 00:04:42,000 --> 00:04:44,000 a new letter for it. 77 00:04:44,000 --> 00:04:48,000 And, the standard one that many people use is p. 78 00:04:48,000 --> 00:04:50,496 How about the right hand side? 79 00:04:50,496 --> 00:04:53,200 We needed a letter for that, too. 80 00:04:53,200 --> 00:04:55,815 It's c over a, but we'll call it q. 81 00:04:55,815 --> 00:04:59,665 So, when I talk about the standard linear form 82 00:04:59,665 --> 00:05:03,000 for a linear first order equation, 83 00:05:03,000 --> 00:05:08,000 it's absolutely that that I'm talking about. 84 00:05:08,000 --> 00:05:11,000 Now, you immediately see that there 85 00:05:11,000 --> 00:05:14,166 is a potential for confusion here 86 00:05:14,166 --> 00:05:21,250 because what did I call the standard form for a first order 87 00:05:21,250 --> 00:05:21,875 equation? 88 00:05:21,875 --> 00:05:26,142 So, I'm going to say, not this. 89 00:05:26,142 --> 00:05:31,400 The standard first order form, what would that be? 90 00:05:31,400 --> 00:05:36,776 Well, it would be y prime equals, and everything else 91 00:05:36,776 --> 00:05:39,000 on the left hand side. 92 00:05:39,000 --> 00:05:41,304 So, it would be y prime. 93 00:05:41,304 --> 00:05:46,305 And now, if I turn this into the standard first order form, 94 00:05:46,305 --> 00:05:53,000 it would be negative p of x y plus q of x. 95 00:05:53,000 --> 00:05:57,750 But, of course, nobody would write negative p of x. 96 00:05:57,750 --> 00:06:00,384 So, now, I explicitly want to say 97 00:06:00,384 --> 00:06:05,777 that this is a form which I will never use for this equation, 98 00:06:05,777 --> 00:06:12,000 although half the books of the world do. 99 00:06:12,000 --> 00:06:14,400 In short, this poor little first order equation 100 00:06:14,400 --> 00:06:16,125 belongs to two ethnic groups. 101 00:06:16,125 --> 00:06:18,375 It's both a first order equation, 102 00:06:18,375 --> 00:06:22,125 and therefore, its standard form should be written this way, 103 00:06:22,125 --> 00:06:26,664 but it's also a linear equation, and therefore its standard form 104 00:06:26,664 --> 00:06:28,570 should be used this way. 105 00:06:28,570 --> 00:06:32,400 Well, it has to decide, and I have decided for it. 106 00:06:32,400 --> 00:06:35,875 It is, above all, a linear equation, not just 107 00:06:35,875 --> 00:06:37,600 a first order equation. 108 00:06:37,600 --> 00:06:43,000 And, in this course, this will always be the standard form. 109 00:06:43,000 --> 00:06:46,000 Now, well, what on earth is the difference? 110 00:06:46,000 --> 00:06:49,744 If you don't do it that way, the difference 111 00:06:49,744 --> 00:06:51,800 is entirely in the sin(p). 112 00:06:51,800 --> 00:06:56,816 But, if you get the sign of p wrong in the answers, 113 00:06:56,816 --> 00:07:01,000 it is just a disaster from that point on. 114 00:07:01,000 --> 00:07:05,500 A trivial little change of sign in the answer 115 00:07:05,500 --> 00:07:08,500 produces solutions and functions which have 116 00:07:08,500 --> 00:07:09,888 totally different behavior. 117 00:07:09,888 --> 00:07:14,452 And, you are going to be really lost in this course. 118 00:07:14,452 --> 00:07:17,888 So, maybe I should draw a line through it 119 00:07:17,888 --> 00:07:21,000 to indicate, please don't pay any attention 120 00:07:21,000 --> 00:07:25,856 to this whatsoever, except that we are not going to do that. 121 00:07:25,856 --> 00:07:29,998 Okay, well, what's so important about this equation? 122 00:07:29,998 --> 00:07:34,500 Well, number one, it can always be solved. 123 00:07:34,500 --> 00:07:39,000 That's a very, very big thing in differential equations. 124 00:07:39,000 --> 00:07:41,724 But, it's also the equation which 125 00:07:41,724 --> 00:07:44,500 arises in a variety of models. 126 00:07:44,500 --> 00:07:48,900 Now, I'm just going to list a few of them. 127 00:07:48,900 --> 00:07:54,227 All of them I think you will need either in part one or part 128 00:07:54,227 --> 00:08:00,500 two of problem sets over these first couple of problem sets, 129 00:08:00,500 --> 00:08:04,000 or second and third maybe. 130 00:08:04,000 --> 00:08:11,152 But, of them, I'm going to put at the very top of the list 131 00:08:11,152 --> 00:08:16,800 of what I'll call here, I'll give it two names: 132 00:08:16,800 --> 00:08:21,600 the temperature diffusion model, well, it 133 00:08:21,600 --> 00:08:27,570 would be better to call it temperature concentration 134 00:08:27,570 --> 00:08:35,000 by analogy, temperature concentration model. 135 00:08:35,000 --> 00:08:40,000 There's the mixing model, which is hardly less important. 136 00:08:40,000 --> 00:08:43,000 In other words, it's almost as important. 137 00:08:43,000 --> 00:08:46,000 You have that in your problem set. 138 00:08:46,000 --> 00:08:52,000 And then, there are other, slightly less important models. 139 00:08:52,000 --> 00:08:55,000 There is the model of radioactive decay. 140 00:08:55,000 --> 00:09:00,000 There's the model of a bank interest, bank account, 141 00:09:00,000 --> 00:09:04,428 various motion models, you know, Newton's Law type problems 142 00:09:04,428 --> 00:09:08,280 if you can figure out a way of getting 143 00:09:08,280 --> 00:09:15,000 rid of the second derivative, some motion problems. 144 00:09:15,000 --> 00:09:23,000 A classic example is the motion of a rocket being fired off, 145 00:09:23,000 --> 00:09:24,500 etc., etc., etc. 146 00:09:24,500 --> 00:09:28,625 Now, today I have to pick a model. 147 00:09:28,625 --> 00:09:37,200 And, the one I'm going to pick is this temperature 148 00:09:37,200 --> 00:09:40,000 concentration model. 149 00:09:40,000 --> 00:09:43,000 So, this is going to be today's model. 150 00:09:43,000 --> 00:09:45,500 Tomorrow's model in the recitation, 151 00:09:45,500 --> 00:09:50,000 I'm asking the recitations to, among other things, 152 00:09:50,000 --> 00:09:54,224 make sure they do a mixing problem, A) to show you 153 00:09:54,224 --> 00:10:00,000 how to do it, and B) because it's on the problem sets. 154 00:10:00,000 --> 00:10:04,000 That's not a good reason, but it's not a bad one. 155 00:10:04,000 --> 00:10:06,688 The others are either in part one 156 00:10:06,688 --> 00:10:10,452 or we will take them up later in the term. 157 00:10:10,452 --> 00:10:16,000 This is not going to be the only lecture on the linear equation. 158 00:10:16,000 --> 00:10:21,000 There will be another one next week of equal importance. 159 00:10:21,000 --> 00:10:24,000 But, we can't do everything today. 160 00:10:24,000 --> 00:10:27,500 So, let's talk about the temperature concentration 161 00:10:27,500 --> 00:10:32,000 model, except I'm going to change its name. 162 00:10:32,000 --> 00:10:39,000 I'm going to change its name to the conduction diffusion model. 163 00:10:39,000 --> 00:10:44,000 I'll put conduction over there, and diffusion over here, 164 00:10:44,000 --> 00:10:48,428 let's say, since, as you will see, the similarities, 165 00:10:48,428 --> 00:10:52,428 they are practically the same model. 166 00:10:52,428 --> 00:10:56,250 All that's changed from one to the other 167 00:10:56,250 --> 00:11:00,000 is the name of the ideas. 168 00:11:00,000 --> 00:11:03,000 In one case, you call it temperature, 169 00:11:03,000 --> 00:11:06,800 and the other, you should call it concentration. 170 00:11:06,800 --> 00:11:11,332 But, the actual mathematics isn't identical. 171 00:11:11,332 --> 00:11:14,500 So, let's begin with conduction. 172 00:11:14,500 --> 00:11:19,330 All right, so, I need a simple physical situation 173 00:11:19,330 --> 00:11:21,000 that I'm modeling. 174 00:11:21,000 --> 00:11:24,000 So, imagine a tank of some liquid. 175 00:11:24,000 --> 00:11:27,000 Water will do as well as anything. 176 00:11:27,000 --> 00:11:30,500 And, in the inside is a suspended, 177 00:11:30,500 --> 00:11:33,250 somehow, is a chamber. 178 00:11:33,250 --> 00:11:38,665 A metal cube will do, and let's suppose 179 00:11:38,665 --> 00:11:43,332 that its walls are partly insulated, not so much 180 00:11:43,332 --> 00:11:46,000 that no heat can get through. 181 00:11:46,000 --> 00:11:50,440 There is no such thing as perfect insulation 182 00:11:50,440 --> 00:11:56,000 anyway, except maybe an absolute perfect vacuum. 183 00:11:56,000 --> 00:11:59,500 Now, inside, so here on the outside is liquid. 184 00:11:59,500 --> 00:12:04,400 Okay, on the inside is, what I'm interested 185 00:12:04,400 --> 00:12:10,000 in is the temperature of this thing. 186 00:12:10,000 --> 00:12:13,666 I'll call that T. Now, that's different from the temperature 187 00:12:13,666 --> 00:12:17,000 of the external water bath. 188 00:12:17,000 --> 00:12:21,000 So, I'll call that T sub e, T for temperature 189 00:12:21,000 --> 00:12:25,500 measured in Celsius, let's say, for the sake of definiteness. 190 00:12:25,500 --> 00:12:28,625 But, this is the external temperature. 191 00:12:28,625 --> 00:12:33,000 So, I'll indicate it with an e. 192 00:12:33,000 --> 00:12:35,775 Now, what is the model? 193 00:12:35,775 --> 00:12:43,328 Well, in other words, how do I set up a differential equation 194 00:12:43,328 --> 00:12:46,000 to model the situation? 195 00:12:46,000 --> 00:12:53,000 Well, it's based on a physical law, which I think you know, 196 00:12:53,000 --> 00:12:57,362 you've had simple examples like this, 197 00:12:57,362 --> 00:13:03,000 the so called Newton's Law of cooling, -- -- 198 00:13:03,000 --> 00:13:06,545 which says that the rate of change, 199 00:13:06,545 --> 00:13:13,000 the temperature of the heat goes from the outside to the inside 200 00:13:13,000 --> 00:13:14,500 by conduction only. 201 00:13:14,500 --> 00:13:18,140 Heat, of course, can travel in various ways, 202 00:13:18,140 --> 00:13:23,000 by convection, by conduction, as here, or by radiation, 203 00:13:23,000 --> 00:13:25,270 are the three most common. 204 00:13:25,270 --> 00:13:31,332 Of these, I only want one, namely transmission of heat 205 00:13:31,332 --> 00:13:33,000 by conduction. 206 00:13:33,000 --> 00:13:36,267 And, that's the way it's probably a little better 207 00:13:36,267 --> 00:13:41,000 to call it the conduction model, rather than the temperature 208 00:13:41,000 --> 00:13:45,086 model, which might involve other ways for the heat 209 00:13:45,086 --> 00:13:46,600 to be traveling. 210 00:13:46,600 --> 00:13:49,333 So, dt, the independent variable, 211 00:13:49,333 --> 00:13:53,000 is not going to be x, as it was over there. 212 00:13:53,000 --> 00:13:55,688 It's going to be t for time. 213 00:13:55,688 --> 00:13:59,000 So, maybe I should write that down. t equals time. 214 00:13:59,000 --> 00:14:05,000 Capital T equals temperature in degrees Celsius. 215 00:14:05,000 --> 00:14:08,000 So, you can put in the degrees Celsius if you want. 216 00:14:08,000 --> 00:14:12,375 So, it's proportional to the temperature difference 217 00:14:12,375 --> 00:14:13,750 between these two. 218 00:14:13,750 --> 00:14:16,428 Now, how shall I write the difference? 219 00:14:16,428 --> 00:14:21,000 Write it this way because if you don't you will be in trouble. 220 00:14:21,000 --> 00:14:24,000 Now, why do I write it that way? 221 00:14:24,000 --> 00:14:26,664 Well, I write it that way because I 222 00:14:26,664 --> 00:14:32,000 want this constant to be positive, a positive constant. 223 00:14:32,000 --> 00:14:35,800 In general, any constant, so, parameters which are physical, 224 00:14:35,800 --> 00:14:39,726 have some physical significance, one always 225 00:14:39,726 --> 00:14:44,142 wants to arrange the equation so that they are positive numbers, 226 00:14:44,142 --> 00:14:47,855 the way people normally think of these things. 227 00:14:47,855 --> 00:14:49,600 This is called the conductivity. 228 00:14:49,600 --> 00:14:52,000 The conductivity of what? 229 00:14:52,000 --> 00:14:54,800 Well, I don't know, of the system 230 00:14:54,800 --> 00:14:58,080 of the situation, the conductivity of the wall, 231 00:14:58,080 --> 00:15:01,999 or the wall if the metal were just by itself. 232 00:15:01,999 --> 00:15:04,332 At any rate, it's a constant. 233 00:15:04,332 --> 00:15:07,000 It's thought of as a constant. 234 00:15:07,000 --> 00:15:11,142 And, why positive, well, because if the external temperature is 235 00:15:11,142 --> 00:15:14,000 bigger than the internal temperature, 236 00:15:14,000 --> 00:15:18,666 I expect T to rise, the internal temperature to rise. 237 00:15:18,666 --> 00:15:24,220 That means dT / dt, its slope, should be positive. 238 00:15:24,220 --> 00:15:27,500 So, in other words, if Te is bigger than T, 239 00:15:27,500 --> 00:15:29,375 I expect this number to be positive. 240 00:15:29,375 --> 00:15:33,452 And, that tells you that k must be a positive constant. 241 00:15:33,452 --> 00:15:37,000 If I had turned it the other way, expressed 242 00:15:37,000 --> 00:15:40,000 the difference in the reverse order, 243 00:15:40,000 --> 00:15:44,000 K would then be negative, have to be negative in order 244 00:15:44,000 --> 00:15:47,000 that this turn out to be positive in that situation I 245 00:15:47,000 --> 00:15:47,600 described. 246 00:15:47,600 --> 00:15:51,200 And, since nobody wants negative values of k, 247 00:15:51,200 --> 00:15:53,999 you have to write the equation in this form 248 00:15:53,999 --> 00:15:56,000 rather than the other way around. 249 00:15:56,000 --> 00:15:59,000 So, there's our differential equation. 250 00:15:59,000 --> 00:16:02,000 It will probably have an initial condition. 251 00:16:02,000 --> 00:16:05,663 So, it could be the temperature at the starting time should 252 00:16:05,663 --> 00:16:10,000 be some given number, T zero. 253 00:16:10,000 --> 00:16:13,000 But, the condition could be given in other ways. 254 00:16:13,000 --> 00:16:15,000 One can ask, what's the temperature as time 255 00:16:15,000 --> 00:16:17,000 goes to infinity, for example? 256 00:16:17,000 --> 00:16:21,000 There are different ways of getting that initial condition. 257 00:16:21,000 --> 00:16:23,000 Okay, that's the conduction model. 258 00:16:23,000 --> 00:16:25,000 What would the diffusion model be? 259 00:16:25,000 --> 00:16:28,142 The diffusion model, mathematically, would be, 260 00:16:28,142 --> 00:16:31,000 word for word, the same. 261 00:16:31,000 --> 00:16:33,912 The only difference is that now, what 262 00:16:33,912 --> 00:16:38,270 I imagine is I'll draw the picture the same way, 263 00:16:38,270 --> 00:16:43,541 except now I'm going to put, label the inside not with a T 264 00:16:43,541 --> 00:16:46,665 but with a C, C for concentration. 265 00:16:46,665 --> 00:16:50,856 It's in an external water bath, let's say. 266 00:16:50,856 --> 00:16:53,571 So, there is an external concentration. 267 00:16:53,571 --> 00:16:57,888 And, what I'm talking about is some chemical, 268 00:16:57,888 --> 00:17:02,250 let's say salt will do as well as anything. 269 00:17:02,250 --> 00:17:07,500 So, C is equal to salt concentration inside, 270 00:17:07,500 --> 00:17:13,714 and Ce would be the salt concentration outside, 271 00:17:13,714 --> 00:17:18,000 outside in the water bath. 272 00:17:18,000 --> 00:17:26,000 Now, I imagine some mechanism, so this is a salt solution. 273 00:17:26,000 --> 00:17:28,664 That's a salt solution. 274 00:17:28,664 --> 00:17:34,178 And, I imagine some mechanism by which the salt can diffuse, 275 00:17:34,178 --> 00:17:36,999 it's a diffusion model now, diffuse from here 276 00:17:36,999 --> 00:17:40,000 into the air or possibly out the other way. 277 00:17:40,000 --> 00:17:43,500 And that's usually done by vaguely referring 278 00:17:43,500 --> 00:17:47,140 to the outside as a semi permeable membrane, 279 00:17:47,140 --> 00:17:50,080 semi permeable, so that the salt will 280 00:17:50,080 --> 00:17:53,750 have a little hard time getting through but permeable, 281 00:17:53,750 --> 00:17:56,800 so that it won't be blocked completely. 282 00:17:56,800 --> 00:18:00,000 So, there's a membrane. 283 00:18:00,000 --> 00:18:06,000 You write the semi permeable membrane outside, 284 00:18:06,000 --> 00:18:07,713 outside the inside. 285 00:18:07,713 --> 00:18:10,000 Well, I give up. 286 00:18:10,000 --> 00:18:14,000 You know, membrane somewhere. 287 00:18:14,000 --> 00:18:17,000 Sorry, membrane wall. 288 00:18:17,000 --> 00:18:18,332 How's that? 289 00:18:18,332 --> 00:18:21,000 Now, what's the equation? 290 00:18:21,000 --> 00:18:26,600 Well, the equation is the same, except it's 291 00:18:26,600 --> 00:18:29,500 called the diffusion equation. 292 00:18:29,500 --> 00:18:35,284 I don't think Newton got his name on this. 293 00:18:35,284 --> 00:18:40,178 The diffusion equation says that the rate at which the salt 294 00:18:40,178 --> 00:18:42,384 diffuses across the membrane, which 295 00:18:42,384 --> 00:18:47,000 is the same up to a constant as the rate at which 296 00:18:47,000 --> 00:18:51,000 the concentration inside changes, is some constant, 297 00:18:51,000 --> 00:18:54,000 usually called k still, okay. 298 00:18:54,000 --> 00:18:55,332 Do I contradict? 299 00:18:55,332 --> 00:18:58,000 Okay, let's keep calling it k1. 300 00:18:58,000 --> 00:19:01,000 Now it's different, times Ce minus C. 301 00:19:01,000 --> 00:19:03,800 And, for the same reason as before, 302 00:19:03,800 --> 00:19:07,142 if the external concentration is bigger 303 00:19:07,142 --> 00:19:12,724 than the internal concentration, we expect salt to flow in. 304 00:19:12,724 --> 00:19:15,000 That will make C rise. 305 00:19:15,000 --> 00:19:18,307 It will make this positive, and therefore, we 306 00:19:18,307 --> 00:19:21,070 want k to be positive, just k1 to be 307 00:19:21,070 --> 00:19:25,000 positive for the same reason it had to be positive before. 308 00:19:25,000 --> 00:19:28,600 So, in each case, the model that I'm talking about 309 00:19:28,600 --> 00:19:31,000 is the differential equation. 310 00:19:31,000 --> 00:19:34,500 So, maybe I should, let's put that, make that clear. 311 00:19:34,500 --> 00:19:39,713 Or, I would say that this first order differential equation 312 00:19:39,713 --> 00:19:43,332 models this physical situation, and the same thing 313 00:19:43,332 --> 00:19:46,000 is true on the other side over here. 314 00:19:46,000 --> 00:19:48,800 This is the diffusion equation, and this 315 00:19:48,800 --> 00:19:50,375 is the conduction equation. 316 00:19:50,375 --> 00:19:55,220 Now, if you are in any doubt about the power of differential 317 00:19:55,220 --> 00:19:58,842 equations, the point is, when I talk about this thing, 318 00:19:58,842 --> 00:20:02,875 I don't have to say which of these I'm following. 319 00:20:02,875 --> 00:20:07,284 I'll use neutral variables like Y and X 320 00:20:07,284 --> 00:20:09,000 to solve these equations. 321 00:20:09,000 --> 00:20:12,600 But, with a single stroke, I will be handling 322 00:20:12,600 --> 00:20:13,750 those situations together. 323 00:20:13,750 --> 00:20:16,500 And, that's the power of the method. 324 00:20:16,500 --> 00:20:20,200 Now, you obviously must be wondering, look, 325 00:20:20,200 --> 00:20:22,666 these look very, very special. 326 00:20:22,666 --> 00:20:27,140 He said he was going to talk about the first, general first 327 00:20:27,140 --> 00:20:28,000 order equation. 328 00:20:28,000 --> 00:20:31,000 But, these look rather special to me. 329 00:20:31,000 --> 00:20:33,220 Well, not too special. 330 00:20:33,220 --> 00:20:36,000 How should we write it? 331 00:20:36,000 --> 00:20:41,332 Suppose I write, let's take the temperature equation just 332 00:20:41,332 --> 00:20:44,000 to have something definite. 333 00:20:44,000 --> 00:20:48,000 Notice that it's in a form corresponding to Newton's Law. 334 00:20:48,000 --> 00:20:52,000 But it is not in the standard linear form. 335 00:20:52,000 --> 00:20:54,688 Let's put it in standard linear form, 336 00:20:54,688 --> 00:20:59,080 so at least you could see that it's a linear equation. 337 00:20:59,080 --> 00:21:02,384 So, if I put it in standard form, 338 00:21:02,384 --> 00:21:06,224 it's going to look like DTDTD little t plus KT 339 00:21:06,224 --> 00:21:09,220 is equal to K times TE. 340 00:21:09,220 --> 00:21:13,200 Now, compare that with the general, the way 341 00:21:13,200 --> 00:21:16,000 the general equation is supposed to look, 342 00:21:16,000 --> 00:21:20,000 the yellow box over there, the standard linear form. 343 00:21:20,000 --> 00:21:21,998 How are they going to compare? 344 00:21:21,998 --> 00:21:24,500 Well, this is a pretty general function. 345 00:21:24,500 --> 00:21:26,000 This is general. 346 00:21:26,000 --> 00:21:29,840 This is a general function of T because I can 347 00:21:29,840 --> 00:21:31,428 make the external temperature. 348 00:21:31,428 --> 00:21:37,000 I could suppose it behaves in anyway I like, steadily rising, 349 00:21:37,000 --> 00:21:39,000 decaying exponentially, maybe oscillating 350 00:21:39,000 --> 00:21:42,180 back and forth for some reason. 351 00:21:42,180 --> 00:21:46,500 The only way in which it's not general 352 00:21:46,500 --> 00:21:50,000 is that this K is a constant. 353 00:21:50,000 --> 00:21:53,000 So, I will ask you to be generous. 354 00:21:53,000 --> 00:21:57,000 Let's imagine the conductivity is changing over time. 355 00:21:57,000 --> 00:22:00,815 So, this is usually constant, but there's 356 00:22:00,815 --> 00:22:05,220 no law which says it has to be. 357 00:22:05,220 --> 00:22:09,142 How could a conductivity change over time? 358 00:22:09,142 --> 00:22:12,888 Well, we could suppose that this wall 359 00:22:12,888 --> 00:22:17,142 was made of slowly congealing Jell O, for instance. 360 00:22:17,142 --> 00:22:23,000 It starts out as liquid, and then it gets solid. 361 00:22:23,000 --> 00:22:26,444 And, Jell O doesn't transmit heat, 362 00:22:26,444 --> 00:22:32,180 I believe, quite as well as liquid does, as a liquid would. 363 00:22:32,180 --> 00:22:36,000 Is Jell O a solid or liquid? 364 00:22:36,000 --> 00:22:37,713 I don't know. 365 00:22:37,713 --> 00:22:40,000 Let's forget about that. 366 00:22:40,000 --> 00:22:46,000 So, with this understanding, so let's say not necessarily 367 00:22:46,000 --> 00:22:51,500 here, but not necessarily, I can think of this, therefore, 368 00:22:51,500 --> 00:22:55,000 by allowing K to vary with time. 369 00:22:55,000 --> 00:23:00,000 And the external temperature to vary with time. 370 00:23:00,000 --> 00:23:07,000 I can think of it as a general, linear equation. 371 00:23:07,000 --> 00:23:09,000 So, these models are not special. 372 00:23:09,000 --> 00:23:10,452 They are fairly general. 373 00:23:10,452 --> 00:23:14,284 Well, I did promise you I would solve an equation, 374 00:23:14,284 --> 00:23:18,000 and that this lecture, I still have not solved any equations. 375 00:23:18,000 --> 00:23:21,000 OK, time to stop temporizing and solve. 376 00:23:21,000 --> 00:23:24,000 So, I'm going to, in order not to play favorites 377 00:23:24,000 --> 00:23:27,000 with these two models, I'll go back to, 378 00:23:27,000 --> 00:23:31,000 and to get you used to thinking of the variables all the time, 379 00:23:31,000 --> 00:23:35,444 that is, you know, be eclectic switching from one variable 380 00:23:35,444 --> 00:23:38,552 to another according to which particular lecture 381 00:23:38,552 --> 00:23:42,635 you happened to be sitting in. 382 00:23:42,635 --> 00:23:52,712 So, let's take our equation in the form, Y prime plus P of XY, 383 00:23:52,712 --> 00:23:58,000 the general form using the old variables 384 00:23:58,000 --> 00:24:04,000 equals Q of X. Solve me. 385 00:24:04,000 --> 00:24:06,928 Well, there are different ways of describing the solution 386 00:24:06,928 --> 00:24:07,428 process. 387 00:24:07,428 --> 00:24:10,614 No matter how you do it, it amounts 388 00:24:10,614 --> 00:24:13,684 to the same amount of work and there is always 389 00:24:13,684 --> 00:24:16,331 a trick involved at each one of them 390 00:24:16,331 --> 00:24:19,125 since you can't suppress a trick by doing 391 00:24:19,125 --> 00:24:21,000 the problem some other way. 392 00:24:21,000 --> 00:24:24,000 The way I'm going to do it, I think, is the best. 393 00:24:24,000 --> 00:24:26,000 That's why I'm giving it to you. 394 00:24:26,000 --> 00:24:27,815 It's the easiest to remember. 395 00:24:27,815 --> 00:24:31,000 It leads to the least work, but I 396 00:24:31,000 --> 00:24:36,000 have colleagues who would fight with me about that point. 397 00:24:36,000 --> 00:24:39,330 So, since they are not here to fight with me 398 00:24:39,330 --> 00:24:42,332 I am free to do whatever I like. 399 00:24:42,332 --> 00:24:45,400 One of the main reasons for doing 400 00:24:45,400 --> 00:24:49,400 it the way I'm going to do is because I 401 00:24:49,400 --> 00:24:55,555 want you to get what our word into your consciousness, two 402 00:24:55,555 --> 00:24:57,220 words, integrating factor. 403 00:24:57,220 --> 00:25:02,500 I'm going to solve this equation by finding and integrating 404 00:25:02,500 --> 00:25:09,000 factor of the form U of X. What's an integrating factor? 405 00:25:09,000 --> 00:25:13,000 Well, I'll show you not by writing an elaborate definition 406 00:25:13,000 --> 00:25:16,666 on the board, but showing you what its function is. 407 00:25:16,666 --> 00:25:19,332 It's a certain function, U of X, I 408 00:25:19,332 --> 00:25:23,200 don't know what it is, but here's what I wanted to do. 409 00:25:23,200 --> 00:25:26,763 I want to multiply, I'm going to drop the X's a just so 410 00:25:26,763 --> 00:25:29,110 that the thing looks less complicated. 411 00:25:29,110 --> 00:25:34,125 So, what I want to do is multiply this equation 412 00:25:34,125 --> 00:25:36,000 through by U of X. 413 00:25:36,000 --> 00:25:40,360 That's why it's called a factor because you're 414 00:25:40,360 --> 00:25:43,816 going to multiply everything through by it. 415 00:25:43,816 --> 00:25:49,220 So, it's going to look like UY prime plus PUY equals QU, 416 00:25:49,220 --> 00:25:52,776 and now, so far, it's just a factor. 417 00:25:52,776 --> 00:25:55,375 What makes it an integrating factor 418 00:25:55,375 --> 00:26:00,766 is that this, after I do that, I want this to turn out 419 00:26:00,766 --> 00:26:04,500 to be the derivative of something with respect 420 00:26:04,500 --> 00:26:08,250 to X. You see the motivation for that. 421 00:26:08,250 --> 00:26:11,125 If this turns out to be the derivative of something, 422 00:26:11,125 --> 00:26:13,363 because I've chosen U so cleverly, 423 00:26:13,363 --> 00:26:17,000 then I will be able to solve the equation immediately 424 00:26:17,000 --> 00:26:19,000 just by integrating this with respect to X, 425 00:26:19,000 --> 00:26:21,000 and integrating that with respect to X. 426 00:26:21,000 --> 00:26:24,750 You just, then, integrate both sides with respect to X, 427 00:26:24,750 --> 00:26:26,000 and the equation is solved. 428 00:26:26,000 --> 00:26:31,000 Now, the only question is, what should I choose for U? 429 00:26:31,000 --> 00:26:34,000 Well, if you think of the product formula, 430 00:26:34,000 --> 00:26:37,000 there might be many things to try here. 431 00:26:37,000 --> 00:26:40,000 But there's only one reasonable thing to try. 432 00:26:40,000 --> 00:26:45,000 Try to pick U so that it's the derivative of U times Y. 433 00:26:45,000 --> 00:26:46,665 See how reasonable that is? 434 00:26:46,665 --> 00:26:49,333 If I use the product rule on this, 435 00:26:49,333 --> 00:26:52,000 the first term is U times Y prime. 436 00:26:52,000 --> 00:26:56,000 The second term would be U prime times Y. 437 00:26:56,000 --> 00:26:59,000 Well, I've got the Y there. 438 00:26:59,000 --> 00:27:01,000 So, this will work. 439 00:27:01,000 --> 00:27:05,888 It works if, what's the condition that you must satisfy 440 00:27:05,888 --> 00:27:09,000 in order for that to be true? 441 00:27:09,000 --> 00:27:15,000 Well, it must be that after it to the differentiation, 442 00:27:15,000 --> 00:27:19,000 U prime turns out to be P times U. 443 00:27:19,000 --> 00:27:20,816 So, is it clear? 444 00:27:20,816 --> 00:27:26,625 This is something we want to be equal to, and the thing I will 445 00:27:26,625 --> 00:27:32,331 try to do it is by choosing U in such a way 446 00:27:32,331 --> 00:27:37,000 that this equality will take place. 447 00:27:37,000 --> 00:27:40,000 And then I will be able to solve the equation. 448 00:27:40,000 --> 00:27:43,000 And so, here's what my U prime must satisfy. 449 00:27:43,000 --> 00:27:44,815 Hey, we can solve that. 450 00:27:44,815 --> 00:27:47,222 But please don't forget that P is 451 00:27:47,222 --> 00:27:49,000 P of X. It's a function of X. 452 00:27:49,000 --> 00:27:53,000 So, if you separate variables, I'm going to do this. 453 00:27:53,000 --> 00:27:56,600 So, what is it, DU over U equals P of X times DX. 454 00:27:56,600 --> 00:28:00,500 If I integrate that, so, separate variables, 455 00:28:00,500 --> 00:28:04,665 integrate, and you're going to get DU over U integrates 456 00:28:04,665 --> 00:28:08,428 to the be the log of U, and the other side 457 00:28:08,428 --> 00:28:12,000 integrates to be the integral of P of X DX. 458 00:28:12,000 --> 00:28:16,000 Now, you can put an arbitrary constant there, 459 00:28:16,000 --> 00:28:18,448 or you can think of it as already implied 460 00:28:18,448 --> 00:28:20,142 by the indefinite integral. 461 00:28:20,142 --> 00:28:24,500 Well, that doesn't tell us, yet, what U is. 462 00:28:24,500 --> 00:28:26,000 What should U be? 463 00:28:26,000 --> 00:28:30,250 Notice, I don't have to find every possible U, which works. 464 00:28:30,250 --> 00:28:34,000 All I'm looking for is one. 465 00:28:34,000 --> 00:28:38,000 All I want is a single view which satisfies that equation. 466 00:28:38,000 --> 00:28:42,000 Well, U equals the integral, E to the integral of PDX. 467 00:28:42,000 --> 00:28:44,500 That's not too beautiful looking, 468 00:28:44,500 --> 00:28:46,726 but by differential equations, things 469 00:28:46,726 --> 00:28:50,454 can get so complicated that in a week or two, 470 00:28:50,454 --> 00:28:55,000 you will think of this as an extremely simple formula. 471 00:28:55,000 --> 00:29:00,000 So, there is a formula for our integrating factor. 472 00:29:00,000 --> 00:29:01,500 We found it. 473 00:29:01,500 --> 00:29:07,250 We will always be able to write an integrating factor. 474 00:29:07,250 --> 00:29:14,750 Don't worry about the arbitrary constant because you only need 475 00:29:14,750 --> 00:29:17,000 one such U. 476 00:29:17,000 --> 00:29:23,000 So: no arbitrary constant since only one U needed. 477 00:29:23,000 --> 00:29:26,600 And, that's the solution, the way 478 00:29:26,600 --> 00:29:30,200 we solve the linear equation. 479 00:29:30,200 --> 00:29:37,544 OK, let's take over, and actually do it. 480 00:29:37,544 --> 00:29:43,714 I think it would be better to summarize it 481 00:29:43,714 --> 00:29:48,000 as a clear cut method. 482 00:29:48,000 --> 00:29:51,000 So, let's do that. 483 00:29:51,000 --> 00:29:54,000 So, what's our method? 484 00:29:54,000 --> 00:30:04,000 It's the method for solving Y prime plus PY equals Q. 485 00:30:04,000 --> 00:30:07,800 Well, the first place, make sure it's in standard linear form. 486 00:30:07,800 --> 00:30:11,331 If it isn't, you must put it in that form. 487 00:30:11,331 --> 00:30:14,712 Notice, the formula for the integrating factor, the formula 488 00:30:14,712 --> 00:30:16,856 for the integrating factor involves 489 00:30:16,856 --> 00:30:19,000 P, the integral of PDX. 490 00:30:19,000 --> 00:30:21,000 So, you'd better get the right P. 491 00:30:21,000 --> 00:30:22,600 Otherwise, you are sunk. 492 00:30:22,600 --> 00:30:25,750 OK, so put it in standard linear form. 493 00:30:25,750 --> 00:30:29,332 That way, you will have the right P. Notice 494 00:30:29,332 --> 00:30:32,544 that if you wrote it in that form, 495 00:30:32,544 --> 00:30:35,375 and all you remembered was E to the integral PDX, 496 00:30:35,375 --> 00:30:38,000 the P would have the wrong sign. 497 00:30:38,000 --> 00:30:41,070 If you're going to write, that P should have a negative sign 498 00:30:41,070 --> 00:30:41,570 there. 499 00:30:41,570 --> 00:30:44,500 So, do it this way, and no other way. 500 00:30:44,500 --> 00:30:47,665 Otherwise, you will get confused and get wrong signs. 501 00:30:47,665 --> 00:30:51,331 And, as I say, that will produce wrong answers, and not 502 00:30:51,331 --> 00:30:54,000 just slightly wrong answers, but disastrously 503 00:30:54,000 --> 00:30:57,664 wrong answers from the point of view of the modeling 504 00:30:57,664 --> 00:31:02,108 if you really want answers to physical problems. 505 00:31:02,108 --> 00:31:06,600 So, here's a standard linear form. 506 00:31:06,600 --> 00:31:09,666 Then, find the integrating factor. 507 00:31:09,666 --> 00:31:17,000 So, calculate E to the integral, PDX, the integrating factor, 508 00:31:17,000 --> 00:31:22,000 and that multiply both, I'm putting this as both, 509 00:31:22,000 --> 00:31:31,000 underlined that as many times as you have room in your notes. 510 00:31:31,000 --> 00:31:38,362 Multiply both sides by this integrating factor by E 511 00:31:38,362 --> 00:31:42,400 to the integral PDX. 512 00:31:42,400 --> 00:31:46,000 And then, integrate. 513 00:31:46,000 --> 00:31:51,000 OK, let's take a simple example. 514 00:31:51,000 --> 00:31:57,400 Suppose we started with the equation XY prime 515 00:31:57,400 --> 00:32:06,000 minus Y equals, I had X2, X3, something like that, 516 00:32:06,000 --> 00:32:12,000 X3, I think, yeah, X2. 517 00:32:12,000 --> 00:32:16,000 OK, what's the first thing to do? 518 00:32:16,000 --> 00:32:18,915 Put it in standard form. 519 00:32:18,915 --> 00:32:25,915 So, step zero will be to write it as Y prime minus 520 00:32:25,915 --> 00:32:30,000 one over X times Y equals X2. 521 00:32:30,000 --> 00:32:34,833 Let's do the work first, and then I'll talk about mistakes. 522 00:32:34,833 --> 00:32:39,888 Well, we now calculate the integrating factor. 523 00:32:39,888 --> 00:32:43,000 So, I would do it in steps. 524 00:32:43,000 --> 00:32:48,000 You can integrate negative one over X, right? 525 00:32:48,000 --> 00:32:51,500 That integrates to minus log X. So, 526 00:32:51,500 --> 00:32:57,000 the integrating factor is E to the integral of this, DX. 527 00:32:57,000 --> 00:33:02,000 So, it's E to the negative log X. 528 00:33:02,000 --> 00:33:06,500 Now, in real life, that's not the way to leave that. 529 00:33:06,500 --> 00:33:10,333 What is E to the negative log X? 530 00:33:10,333 --> 00:33:16,000 Well, think of it as E to the log X to the minus one. 531 00:33:16,000 --> 00:33:21,570 Or, in other words, it is E to the log X is X. So, 532 00:33:21,570 --> 00:33:23,000 it's one over X. 533 00:33:23,000 --> 00:33:27,000 So, the integrating factor is one over X. 534 00:33:27,000 --> 00:33:34,000 OK, multiply both sides by the integrating factor. 535 00:33:34,000 --> 00:33:35,500 Both sides of what? 536 00:33:35,500 --> 00:33:41,000 Both sides of this: the equation written in standard form, 537 00:33:41,000 --> 00:33:42,200 and both sides. 538 00:33:42,200 --> 00:33:47,800 So, it's going to be one over XY prime minus one over X2 Y 539 00:33:47,800 --> 00:33:52,815 is equal to X2 times one over X, which is simply X. Now, 540 00:33:52,815 --> 00:33:55,545 if you have done the work correctly, 541 00:33:55,545 --> 00:34:01,000 you should be able, now, to integrate the left hand 542 00:34:01,000 --> 00:34:02,110 side directly. 543 00:34:02,110 --> 00:34:06,428 So, I'm going to write it this way. 544 00:34:06,428 --> 00:34:10,815 I always recommend that you put it as extra step, well, 545 00:34:10,815 --> 00:34:14,142 put it as an extra step the reason 546 00:34:14,142 --> 00:34:17,000 for using that integrating factor, 547 00:34:17,000 --> 00:34:21,200 in other words, that the left hand side is supposed to be, 548 00:34:21,200 --> 00:34:24,000 now, one over X times Y prime. 549 00:34:24,000 --> 00:34:27,328 I always put it that because there's always 550 00:34:27,328 --> 00:34:31,775 a chance you made a mistake or forgot something. 551 00:34:31,775 --> 00:34:34,555 Look at it, mentally differentiated 552 00:34:34,555 --> 00:34:39,855 using the product rule just to check that, in fact, it 553 00:34:39,855 --> 00:34:43,000 turns out to be the same as the left hand side. 554 00:34:43,000 --> 00:34:44,535 So, what do we get? 555 00:34:44,535 --> 00:34:48,500 One over X times Y prime plus Y times the derivative 556 00:34:48,500 --> 00:34:53,800 of one over X, which indeed is negative one over X2. 557 00:34:53,800 --> 00:34:58,000 And now, finally, that's 3A, continue, do the integration. 558 00:34:58,000 --> 00:35:00,149 So, you're going to get, let's see 559 00:35:00,149 --> 00:35:04,904 if we can do it all on one board, one over X times Y 560 00:35:04,904 --> 00:35:07,904 is equal to X plus a constant, X, sorry, X2 561 00:35:07,904 --> 00:35:09,857 over two plus a constant. 562 00:35:09,857 --> 00:35:16,332 And, the final step will be, therefore, now 563 00:35:16,332 --> 00:35:21,000 I want to isolate Y by itself. 564 00:35:21,000 --> 00:35:26,000 So, Y will be equal to multiply through by X. 565 00:35:26,000 --> 00:35:34,000 X3 over two plus C times X. And, that's the solution. 566 00:35:34,000 --> 00:35:40,000 OK, let's do one a little slightly more complicated. 567 00:35:40,000 --> 00:35:41,816 Let's try this one. 568 00:35:41,816 --> 00:35:45,545 Now, my equation is going to be one, 569 00:35:45,545 --> 00:35:51,000 I'll still keep two, Y and X, as the variables. 570 00:35:51,000 --> 00:35:57,000 I'll use T and F for a minute or two. 571 00:35:57,000 --> 00:36:01,360 One plus cosine X, so, I'm not going 572 00:36:01,360 --> 00:36:07,000 to give you this one in standard form either. 573 00:36:07,000 --> 00:36:09,332 It's a trick question. 574 00:36:09,332 --> 00:36:17,332 Y prime minus sine X times Y is equal to anything reasonable, 575 00:36:17,332 --> 00:36:19,000 I guess. 576 00:36:19,000 --> 00:36:24,000 I think X, 2X, make it more exciting. 577 00:36:24,000 --> 00:36:28,920 OK, now, I think I should warn you 578 00:36:28,920 --> 00:36:38,000 where the mistakes are just so that you can make all of them. 579 00:36:38,000 --> 00:36:41,000 So, this is mistake number one. 580 00:36:41,000 --> 00:36:44,750 You don't put it in standard form. 581 00:36:44,750 --> 00:36:51,000 Mistake number two: generally people can do step one fine. 582 00:36:51,000 --> 00:36:57,000 Mistake number two is, this is my most common mistake, 583 00:36:57,000 --> 00:37:00,000 so I'm very sensitive to it. 584 00:37:00,000 --> 00:37:03,750 But that doesn't mean if you make it, 585 00:37:03,750 --> 00:37:06,000 you'll get any sympathy from me. 586 00:37:06,000 --> 00:37:08,000 I don't give sympathy to myself. 587 00:37:08,000 --> 00:37:10,400 You are so intense, so happy at having 588 00:37:10,400 --> 00:37:12,332 found the integrating factor, you 589 00:37:12,332 --> 00:37:16,332 forget to multiply Q by the integrating factor also. 590 00:37:16,332 --> 00:37:20,332 You just handle the left hand side of the equation, 591 00:37:20,332 --> 00:37:22,999 if you forget about the right hand side. 592 00:37:22,999 --> 00:37:27,220 So, the emphasis on the both here is the right hand, 593 00:37:27,220 --> 00:37:29,000 please include the Q. 594 00:37:29,000 --> 00:37:33,000 Please include the right hand side. 595 00:37:33,000 --> 00:37:34,332 Any other mistakes? 596 00:37:34,332 --> 00:37:37,250 Well, nothing that I can think of. 597 00:37:37,250 --> 00:37:39,664 Well, maybe only, anyway, we are not 598 00:37:39,664 --> 00:37:43,570 going to make any mistakes the rest of this lecture. 599 00:37:43,570 --> 00:37:45,000 So, what do we do? 600 00:37:45,000 --> 00:37:47,304 We write this in standard form. 601 00:37:47,304 --> 00:37:51,600 So, it's going to look like Y prime minus sine X, 602 00:37:51,600 --> 00:37:55,816 sine X divided by one plus cosine X times Y 603 00:37:55,816 --> 00:37:59,000 equals, my heart sinks because I know 604 00:37:59,000 --> 00:38:04,000 I'm supposed to integrate something like this. 605 00:38:04,000 --> 00:38:07,000 And, boy, that's going to give me problems. 606 00:38:07,000 --> 00:38:08,713 Well, not yet. 607 00:38:08,713 --> 00:38:11,000 With the integrating factor? 608 00:38:11,000 --> 00:38:14,000 The integrating factor is, well, we 609 00:38:14,000 --> 00:38:18,220 want to calculate the integral of negative sine X 610 00:38:18,220 --> 00:38:20,000 over one plus cosine. 611 00:38:20,000 --> 00:38:21,875 That's the integral of PDX. 612 00:38:21,875 --> 00:38:25,500 And, after that, we have to exponentiate it. 613 00:38:25,500 --> 00:38:28,000 Well, can you do this? 614 00:38:28,000 --> 00:38:31,000 Yeah, but if you stare at it a little while, 615 00:38:31,000 --> 00:38:38,000 you can see that the top is the derivative of the bottom. 616 00:38:38,000 --> 00:38:39,332 That is great. 617 00:38:39,332 --> 00:38:43,600 That means it integrates to be the log of one 618 00:38:43,600 --> 00:38:47,362 plus cosine X. Is that right, one over one 619 00:38:47,362 --> 00:38:51,625 plus cosine X times the derivative of this, which 620 00:38:51,625 --> 00:38:56,375 is negative cosine X. Therefore, the integrating factor 621 00:38:56,375 --> 00:38:57,875 is E to that. 622 00:38:57,875 --> 00:39:02,270 In other words, it is one plus cosine X. 623 00:39:02,270 --> 00:39:05,500 Therefore, so this was step zero. 624 00:39:05,500 --> 00:39:09,332 Step one, we found the integrating factor. 625 00:39:09,332 --> 00:39:17,000 And now, step two, we multiply through the integrating factor. 626 00:39:17,000 --> 00:39:19,220 And what do we get? 627 00:39:19,220 --> 00:39:23,142 We multiply through the standard for equation 628 00:39:23,142 --> 00:39:29,000 by the integrating factor, if you do that, what you get is, 629 00:39:29,000 --> 00:39:35,000 well, Y prime gets the coefficient one plus cosine X, 630 00:39:35,000 --> 00:39:38,885 Y prime minus sign X equals 2X. 631 00:39:38,885 --> 00:39:40,000 Oh, dear. 632 00:39:40,000 --> 00:39:45,000 Well, I hope somebody would giggle at this point. 633 00:39:45,000 --> 00:39:47,270 What's giggle able about it? 634 00:39:47,270 --> 00:39:51,000 Well, that all this was totally wasted work. 635 00:39:51,000 --> 00:39:53,500 It's called spinning your wheels. 636 00:39:53,500 --> 00:39:56,500 No, it's not spinning your wheels. 637 00:39:56,500 --> 00:40:00,000 It's doing what you're supposed to do, 638 00:40:00,000 --> 00:40:05,000 and finding out that you wasted the entire time doing 639 00:40:05,000 --> 00:40:08,000 what you were supposed to do. 640 00:40:08,000 --> 00:40:12,140 Well, in other words, that net effect of this 641 00:40:12,140 --> 00:40:16,856 is to end up with the same equation we started with. 642 00:40:16,856 --> 00:40:19,000 But, what is the point? 643 00:40:19,000 --> 00:40:22,178 The point of having done all this 644 00:40:22,178 --> 00:40:26,775 was because now the left hand side is exactly 645 00:40:26,775 --> 00:40:32,270 the derivative of something, and the left hand side should 646 00:40:32,270 --> 00:40:35,000 be the derivative of what? 647 00:40:35,000 --> 00:40:37,904 Well, it should be the derivative of one 648 00:40:37,904 --> 00:40:40,600 plus cosine X times Y, all prime. 649 00:40:40,600 --> 00:40:44,776 Now, you can check that that's in fact the case. 650 00:40:44,776 --> 00:40:48,815 It's one plus cosine X, Y prime, plus minus sine 651 00:40:48,815 --> 00:40:51,856 X, the derivative of this side times Y. 652 00:40:51,856 --> 00:40:56,220 So, if you had thought, in looking at the equation, 653 00:40:56,220 --> 00:41:00,724 to say to yourself, this is a derivative of that, 654 00:41:00,724 --> 00:41:04,248 maybe I'll just check right away to see 655 00:41:04,248 --> 00:41:08,571 if it's the derivative of one plus cosine X sine. 656 00:41:08,571 --> 00:41:12,000 You would have saved that work. 657 00:41:12,000 --> 00:41:16,000 Well, you don't have to be brilliant or clever, 658 00:41:16,000 --> 00:41:17,332 or anything like that. 659 00:41:17,332 --> 00:41:20,200 You can follow your nose, and it's just, 660 00:41:20,200 --> 00:41:24,875 I want to give you a positive experience in solving 661 00:41:24,875 --> 00:41:28,000 linear equations, not too negative. 662 00:41:28,000 --> 00:41:31,000 Anyway, so we got to this point. 663 00:41:31,000 --> 00:41:37,000 So, now this is 2X, and now we are ready to solve 664 00:41:37,000 --> 00:41:43,500 the equation, which is the solution now will be one plus 665 00:41:43,500 --> 00:41:49,000 cosine X times Y is equal to X2 plus a constant, 666 00:41:49,000 --> 00:41:55,524 and so Y is equal to X2 divided by X2 plus a constant divided 667 00:41:55,524 --> 00:42:01,400 by one plus cosine X. Suppose I have given you an initial 668 00:42:01,400 --> 00:42:03,000 condition, which I didn't. 669 00:42:03,000 --> 00:42:07,000 But, suppose the initial condition said that Y of zero 670 00:42:07,000 --> 00:42:08,332 were one, for instance. 671 00:42:08,332 --> 00:42:11,665 Then, the solution would be, so, this is an if, 672 00:42:11,665 --> 00:42:15,498 I'm throwing in at the end just to make it a little bit more 673 00:42:15,498 --> 00:42:17,400 of a problem, how would I put, then 674 00:42:17,400 --> 00:42:21,400 I could evaluate the constant by using the initial condition. 675 00:42:21,400 --> 00:42:23,000 What would it be? 676 00:42:23,000 --> 00:42:25,454 This would be, on the left hand side, one, 677 00:42:25,454 --> 00:42:30,000 on the right hand side would be C over two. 678 00:42:30,000 --> 00:42:34,000 So, I would get one equals C over two. 679 00:42:34,000 --> 00:42:35,875 Is that correct? 680 00:42:35,875 --> 00:42:42,180 Cosine of zero is one, so that's two down below. 681 00:42:42,180 --> 00:42:48,856 Therefore, C is equal to two, and that would then 682 00:42:48,856 --> 00:42:51,000 complete the solution. 683 00:42:51,000 --> 00:42:57,000 We would be X2 plus two over one plus cosine X. 684 00:42:57,000 --> 00:43:03,000 Now, you can do this in general, of course, 685 00:43:03,000 --> 00:43:06,180 and get a general formula. 686 00:43:06,180 --> 00:43:11,712 And, we will have occasion to use that next week. 687 00:43:11,712 --> 00:43:15,220 But for now, why don't we concentrate 688 00:43:15,220 --> 00:43:18,555 on the most interesting case, namely 689 00:43:18,555 --> 00:43:21,885 that of the most linear equation, 690 00:43:21,885 --> 00:43:24,332 with constant coefficient, that is, 691 00:43:24,332 --> 00:43:27,714 so let's look at the linear equation 692 00:43:27,714 --> 00:43:31,284 with constant coefficient, because that's 693 00:43:31,284 --> 00:43:37,000 the one that most closely models the conduction and diffusion 694 00:43:37,000 --> 00:43:37,666 equations. 695 00:43:37,666 --> 00:43:43,000 So, what I'm interested in, is since this is the, of them all, 696 00:43:43,000 --> 00:43:45,400 probably it's the most important case 697 00:43:45,400 --> 00:43:48,500 is the one where P is a constant because 698 00:43:48,500 --> 00:43:50,000 of its application to that. 699 00:43:50,000 --> 00:43:53,500 And, many of the other, the bank account, for example, 700 00:43:53,500 --> 00:43:55,666 all of those will use a constant coefficient. 701 00:43:55,666 --> 00:43:58,428 So, how is the thing going to look? 702 00:43:58,428 --> 00:44:01,000 Well, I will use the cooling. 703 00:44:01,000 --> 00:44:05,000 Let's use the temperature model, for example. 704 00:44:05,000 --> 00:44:07,220 The temperature model, the equation 705 00:44:07,220 --> 00:44:10,600 will be DTDT plus KT is equal to. 706 00:44:10,600 --> 00:44:14,800 Now, notice on the right hand side, this is a common error. 707 00:44:14,800 --> 00:44:16,000 You don't put TE. 708 00:44:16,000 --> 00:44:21,000 You have to put KTE because that's what the equation says. 709 00:44:21,000 --> 00:44:25,000 If you think units, you won't have any trouble. 710 00:44:25,000 --> 00:44:30,000 Units have to be compatible on both sides of a differential 711 00:44:30,000 --> 00:44:30,666 equation. 712 00:44:30,666 --> 00:44:34,331 And therefore, whatever the units were for capital KT, 713 00:44:34,331 --> 00:44:38,541 I'd have to have the same units on the right hand side, 714 00:44:38,541 --> 00:44:42,664 which indicates I cannot have KT on the left of the differential 715 00:44:42,664 --> 00:44:44,998 equation, and just T on the right, 716 00:44:44,998 --> 00:44:47,500 and expect the units to be compatible. 717 00:44:47,500 --> 00:44:49,000 That's not possible. 718 00:44:49,000 --> 00:44:51,100 So, that's a good way of remembering 719 00:44:51,100 --> 00:44:54,000 that if you're modeling temperature or concentration, 720 00:44:54,000 --> 00:44:57,000 you have to have the K on both sides. 721 00:44:57,000 --> 00:45:02,080 OK, let's do, now, a lot of this we are going to do in our head 722 00:45:02,080 --> 00:45:05,000 now because this is really too easy. 723 00:45:05,000 --> 00:45:07,220 What's the integrating factor? 724 00:45:07,220 --> 00:45:13,227 Well, the integrating factor is going to be the integral of K, 725 00:45:13,227 --> 00:45:16,000 the coefficient now is just K. 726 00:45:16,000 --> 00:45:21,284 P is a constant, K, and if I integrate KDT, I get KT, 727 00:45:21,284 --> 00:45:23,000 and I exponentiate that. 728 00:45:23,000 --> 00:45:28,000 So, the integrating factor is E to the KT. 729 00:45:28,000 --> 00:45:34,000 I multiply through both sides, multiply by E to the KT, 730 00:45:34,000 --> 00:45:38,000 and what's the resulting equation? 731 00:45:38,000 --> 00:45:44,333 Well, it's going to be , I'll write it in the compact form. 732 00:45:44,333 --> 00:45:50,000 It's going to be E to the KT times T, all prime. 733 00:45:50,000 --> 00:45:55,284 The differentiation is now, of course, with respect 734 00:45:55,284 --> 00:45:57,000 to the time. 735 00:45:57,000 --> 00:46:00,750 And, that's equal to KTE, whatever 736 00:46:00,750 --> 00:46:04,725 that is, times E to the KT. 737 00:46:04,725 --> 00:46:09,000 This is a function of T, of course, 738 00:46:09,000 --> 00:46:13,500 the function of little time, sorry, little T time. 739 00:46:13,500 --> 00:46:18,000 OK, and now, finally, we are going to integrate. 740 00:46:18,000 --> 00:46:19,332 What's the answer? 741 00:46:19,332 --> 00:46:26,224 Well, it is E to the, so, are we going to get E to the KT times 742 00:46:26,224 --> 00:46:31,750 T is, sorry, K little t, K times time times the temperature 743 00:46:31,750 --> 00:46:37,000 is equal to the integral of KTE. 744 00:46:37,000 --> 00:46:40,840 I'll put the fact that it's a function of T 745 00:46:40,840 --> 00:46:43,998 inside just to remind you, E to the KT, 746 00:46:43,998 --> 00:46:46,500 and now I'll put the arbitrary constant. 747 00:46:46,500 --> 00:46:50,142 Let's put in the arbitrary constant explicitly. 748 00:46:50,142 --> 00:46:53,000 So, what will T be? 749 00:46:53,000 --> 00:46:56,000 OK, T will look like this, finally. 750 00:46:56,000 --> 00:46:59,000 It will be E to the negative KT. 751 00:46:59,000 --> 00:47:01,500 That's on the outside. 752 00:47:01,500 --> 00:47:04,000 Then, you will integrate. 753 00:47:04,000 --> 00:47:07,552 Of course, the difficulty of doing this integral 754 00:47:07,552 --> 00:47:12,000 depends entirely upon how this external temperature varies. 755 00:47:12,000 --> 00:47:16,726 But anyways, it's going to be K times that function, which 756 00:47:16,726 --> 00:47:20,000 I haven't specified, E to the KT plus C 757 00:47:20,000 --> 00:47:22,664 times E to the negative KT. 758 00:47:22,664 --> 00:47:27,000 Now, some people, many, in fact, that almost always, 759 00:47:27,000 --> 00:47:30,750 in the engineering literature, almost never 760 00:47:30,750 --> 00:47:36,285 write indefinite integrals because an indefinite integral 761 00:47:36,285 --> 00:47:38,000 is indefinite. 762 00:47:38,000 --> 00:47:40,400 In other words, this covers not just one function, 763 00:47:40,400 --> 00:47:42,800 but a whole multitude of functions 764 00:47:42,800 --> 00:47:46,000 which differ from each other by an arbitrary constant. 765 00:47:46,000 --> 00:47:49,000 So, in a formula like this, there's a certain vagueness, 766 00:47:49,000 --> 00:47:51,541 and it's further compounded by the fact 767 00:47:51,541 --> 00:47:55,000 that I don't know whether the arbitrary constant is here. 768 00:47:55,000 --> 00:47:58,571 I seem to have put it explicitly on the outside the way 769 00:47:58,571 --> 00:48:02,000 you're used to doing from calculus. 770 00:48:02,000 --> 00:48:04,570 Many people, therefore, prefer, and I 771 00:48:04,570 --> 00:48:06,565 think you should learn this, to do 772 00:48:06,565 --> 00:48:11,885 what is done in the very first section of the notes called 773 00:48:11,885 --> 00:48:13,500 definite integral solutions. 774 00:48:13,500 --> 00:48:16,400 If there's an initial condition saying 775 00:48:16,400 --> 00:48:19,200 that the internal temperature at time zero 776 00:48:19,200 --> 00:48:22,912 is some given value, what they like to do 777 00:48:22,912 --> 00:48:26,248 is make this thing definite by integrating here 778 00:48:26,248 --> 00:48:30,625 from zero to T, and making this a dummy variable. 779 00:48:30,625 --> 00:48:36,272 You see, what that does is it gives you 780 00:48:36,272 --> 00:48:39,452 a particular function, whereas, I'm 781 00:48:39,452 --> 00:48:44,766 sorry I didn't put in the DT one minus two. 782 00:48:44,766 --> 00:48:49,554 What it does is that when time is zero, 783 00:48:49,554 --> 00:48:52,662 all this automatically disappears, 784 00:48:52,662 --> 00:49:00,000 and the arbitrary constant will then be, it's T. 785 00:49:00,000 --> 00:49:03,125 So, in other words, C times this, which is one, 786 00:49:03,125 --> 00:49:05,000 is that equal to [T?]. 787 00:49:05,000 --> 00:49:07,000 In other words, if I make this zero, 788 00:49:07,000 --> 00:49:12,000 that I can write C as equal to this arbitrary starting value. 789 00:49:12,000 --> 00:49:15,000 Now, when you do this, the essential thing, 790 00:49:15,000 --> 00:49:18,000 and we're going to come back to this next week, 791 00:49:18,000 --> 00:49:21,000 but right away, because K is positive, 792 00:49:21,000 --> 00:49:25,400 I want to emphasize that so much at the beginning of the period, 793 00:49:25,400 --> 00:49:30,428 I want to conclude by showing you what its significance is. 794 00:49:30,428 --> 00:49:35,332 This part disappears because K is positive. 795 00:49:35,332 --> 00:49:38,000 The conductivity is positive. 796 00:49:38,000 --> 00:49:41,000 This part disappears as T goes to zero. 797 00:49:41,000 --> 00:49:45,000 This goes to zero as T goes to infinity. 798 00:49:45,000 --> 00:49:48,000 So, this is a solution that remains. 799 00:49:48,000 --> 00:49:52,625 This, therefore, is called the steady state solution, 800 00:49:52,625 --> 00:49:57,000 the thing which the temperature behaves like, 801 00:49:57,000 --> 00:49:59,000 as T goes to infinity. 802 00:49:59,000 --> 00:50:01,000 This is called the transient. 803 00:50:01,000 --> 00:50:07,000 because it disappears as T goes to infinity. 804 00:50:07,000 --> 00:50:09,664 It depends on the initial condition, 805 00:50:09,664 --> 00:50:12,200 but it disappears, which shows you, 806 00:50:12,200 --> 00:50:16,665 then, in the long run for this type of problem 807 00:50:16,665 --> 00:50:20,000 the initial condition makes no difference. 808 00:50:20,000 --> 00:50:24,550 The function behaves always the same way as T goes to infinity.