WEBVTT 00:00:09.000 --> 00:00:13.000 This time, we started solving differential equations. 00:00:13.000 --> 00:00:16.072 This is the third lecture of the term, 00:00:16.072 --> 00:00:20.000 and I have yet to solve a single differential 00:00:20.000 --> 00:00:22.000 equation in this class. 00:00:22.000 --> 00:00:24.912 Well, that will be rectified from now 00:00:24.912 --> 00:00:27.500 until the end of the term. 00:00:27.500 --> 00:00:31.142 So, once you learn separation of variables, 00:00:31.142 --> 00:00:36.220 which is the most elementary method there is, the single, 00:00:36.220 --> 00:00:39.500 I think the single most important equation 00:00:39.500 --> 00:00:44.713 is the one that's called the first order linear equation, 00:00:44.713 --> 00:00:47.500 both because it occurs frequently 00:00:47.500 --> 00:00:52.725 in models because it's solvable, and-- I think that's enough. 00:00:52.725 --> 00:00:56.333 If you drop the course after today 00:00:56.333 --> 00:00:59.428 you will still have learned those two important methods: 00:00:59.428 --> 00:01:03.000 separation of variables, and first order linear equations. 00:01:03.000 --> 00:01:06.750 So, what does such an equation look like? 00:01:06.750 --> 00:01:09.000 Well, I'll write it in there. 00:01:09.000 --> 00:01:11.331 There are several ways of writing it, 00:01:11.331 --> 00:01:13.900 but I think the most basic is this. 00:01:13.900 --> 00:01:17.000 I'm going to use x as the independent variable 00:01:17.000 --> 00:01:20.000 because that's what your book does. 00:01:20.000 --> 00:01:22.568 But in the applications, it's often 00:01:22.568 --> 00:01:25.000 t, time, that is the independent variable. 00:01:25.000 --> 00:01:29.000 And, I'll try to give you examples which show that. 00:01:29.000 --> 00:01:31.000 So, the equation looks like this. 00:01:31.000 --> 00:01:34.328 I'll find some function of x times y 00:01:34.328 --> 00:01:37.535 prime plus some other function of x times y 00:01:37.535 --> 00:01:40.000 is equal to yet another function of x. 00:01:40.000 --> 00:01:46.999 Obviously, the x doesn't have the same status 00:01:46.999 --> 00:01:50.750 here that y does, so y is extremely limited in how 00:01:50.750 --> 00:01:53.000 it can appear in the equation. 00:01:53.000 --> 00:01:57.000 But, x can be pretty much arbitrary in those places. 00:01:57.000 --> 00:02:00.000 So, that's the equation we are talking about, 00:02:00.000 --> 00:02:02.270 and I'll put it up. 00:02:02.270 --> 00:02:07.496 This is the first version of it, and we'll call them purple. 00:02:07.496 --> 00:02:11.000 Now, why is that called the linear equation? 00:02:11.000 --> 00:02:15.998 The word linear is a very heavily used word 00:02:15.998 --> 00:02:18.800 in mathematics, science, and engineering. 00:02:18.800 --> 00:02:21.664 For the moment, the best simple answer 00:02:21.664 --> 00:02:27.571 is because it's linear in y and y prime, the variables 00:02:27.571 --> 00:02:31.000 y and y prime. 00:02:31.000 --> 00:02:33.000 Well, y prime is not a variable. 00:02:33.000 --> 00:02:36.000 Well, you will learn, in a certain sense, 00:02:36.000 --> 00:02:41.000 it helps to think of it as one, not right now perhaps, 00:02:41.000 --> 00:02:43.400 but think of it as linear. 00:02:43.400 --> 00:02:47.664 The most closely analogous thing would be a linear equation, 00:02:47.664 --> 00:02:53.000 a real linear equation, the kind you studied in high school, 00:02:53.000 --> 00:02:55.500 which would look like this. 00:02:55.500 --> 00:02:59.200 It would have two variables, and, I guess, 00:02:59.200 --> 00:03:01.428 constant coefficients, equal c. 00:03:01.428 --> 00:03:05.000 Now, that's a linear equation. 00:03:05.000 --> 00:03:09.000 And that's the sense in which this is linear. 00:03:09.000 --> 00:03:12.072 It's linear in y prime and y, which 00:03:12.072 --> 00:03:15.776 are the analogs of the variables y1 and y2. 00:03:15.776 --> 00:03:20.664 A little bit of terminology, if c is equal to zero, 00:03:20.664 --> 00:03:25.000 it's called homogeneous, the same way this equation is 00:03:25.000 --> 00:03:30.500 called homogeneous, as you know from 18.02, if the right hand 00:03:30.500 --> 00:03:32.000 side is zero. 00:03:32.000 --> 00:03:37.000 So, c of x I should write here, but I won't. 00:03:37.000 --> 00:03:38.500 That's called homogeneous. 00:03:38.500 --> 00:03:43.000 Now, this is a common form for the equation, 00:03:43.000 --> 00:03:47.142 but it's not what it's called standard form. 00:03:47.142 --> 00:03:51.664 The standard form for the equation, and since this 00:03:51.664 --> 00:03:58.332 is going to be a prime course of confusion, which is probably 00:03:58.332 --> 00:04:03.600 completely correct, but a prime source of confusion 00:04:03.600 --> 00:04:06.000 is what I meant. 00:04:06.000 --> 00:04:09.330 The standard linear form, and I'll 00:04:09.330 --> 00:04:13.912 underline linear is the first co efficient of y prime 00:04:13.912 --> 00:04:16.000 is taken to be one. 00:04:16.000 --> 00:04:20.160 So, you can always convert that to a standard form 00:04:20.160 --> 00:04:22.776 by simply dividing through by it. 00:04:22.776 --> 00:04:28.552 And if I do that, the equation will look like y prime plus, 00:04:28.552 --> 00:04:35.000 now, it's common to not call it b anymore, the coefficient, 00:04:35.000 --> 00:04:38.750 because it's really b over a. 00:04:38.750 --> 00:04:42.000 And, therefore, it's common to adopt, yet, 00:04:42.000 --> 00:04:44.000 a new letter for it. 00:04:44.000 --> 00:04:48.000 And, the standard one that many people use is p. 00:04:48.000 --> 00:04:50.496 How about the right hand side? 00:04:50.496 --> 00:04:53.200 We needed a letter for that, too. 00:04:53.200 --> 00:04:55.815 It's c over a, but we'll call it q. 00:04:55.815 --> 00:04:59.665 So, when I talk about the standard linear form 00:04:59.665 --> 00:05:03.000 for a linear first order equation, 00:05:03.000 --> 00:05:08.000 it's absolutely that that I'm talking about. 00:05:08.000 --> 00:05:11.000 Now, you immediately see that there 00:05:11.000 --> 00:05:14.166 is a potential for confusion here 00:05:14.166 --> 00:05:21.250 because what did I call the standard form for a first order 00:05:21.250 --> 00:05:21.875 equation? 00:05:21.875 --> 00:05:26.142 So, I'm going to say, not this. 00:05:26.142 --> 00:05:31.400 The standard first order form, what would that be? 00:05:31.400 --> 00:05:36.776 Well, it would be y prime equals, and everything else 00:05:36.776 --> 00:05:39.000 on the left hand side. 00:05:39.000 --> 00:05:41.304 So, it would be y prime. 00:05:41.304 --> 00:05:46.305 And now, if I turn this into the standard first order form, 00:05:46.305 --> 00:05:53.000 it would be negative p of x y plus q of x. 00:05:53.000 --> 00:05:57.750 But, of course, nobody would write negative p of x. 00:05:57.750 --> 00:06:00.384 So, now, I explicitly want to say 00:06:00.384 --> 00:06:05.777 that this is a form which I will never use for this equation, 00:06:05.777 --> 00:06:12.000 although half the books of the world do. 00:06:12.000 --> 00:06:14.400 In short, this poor little first order equation 00:06:14.400 --> 00:06:16.125 belongs to two ethnic groups. 00:06:16.125 --> 00:06:18.375 It's both a first order equation, 00:06:18.375 --> 00:06:22.125 and therefore, its standard form should be written this way, 00:06:22.125 --> 00:06:26.664 but it's also a linear equation, and therefore its standard form 00:06:26.664 --> 00:06:28.570 should be used this way. 00:06:28.570 --> 00:06:32.400 Well, it has to decide, and I have decided for it. 00:06:32.400 --> 00:06:35.875 It is, above all, a linear equation, not just 00:06:35.875 --> 00:06:37.600 a first order equation. 00:06:37.600 --> 00:06:43.000 And, in this course, this will always be the standard form. 00:06:43.000 --> 00:06:46.000 Now, well, what on earth is the difference? 00:06:46.000 --> 00:06:49.744 If you don't do it that way, the difference 00:06:49.744 --> 00:06:51.800 is entirely in the sin(p). 00:06:51.800 --> 00:06:56.816 But, if you get the sign of p wrong in the answers, 00:06:56.816 --> 00:07:01.000 it is just a disaster from that point on. 00:07:01.000 --> 00:07:05.500 A trivial little change of sign in the answer 00:07:05.500 --> 00:07:08.500 produces solutions and functions which have 00:07:08.500 --> 00:07:09.888 totally different behavior. 00:07:09.888 --> 00:07:14.452 And, you are going to be really lost in this course. 00:07:14.452 --> 00:07:17.888 So, maybe I should draw a line through it 00:07:17.888 --> 00:07:21.000 to indicate, please don't pay any attention 00:07:21.000 --> 00:07:25.856 to this whatsoever, except that we are not going to do that. 00:07:25.856 --> 00:07:29.998 Okay, well, what's so important about this equation? 00:07:29.998 --> 00:07:34.500 Well, number one, it can always be solved. 00:07:34.500 --> 00:07:39.000 That's a very, very big thing in differential equations. 00:07:39.000 --> 00:07:41.724 But, it's also the equation which 00:07:41.724 --> 00:07:44.500 arises in a variety of models. 00:07:44.500 --> 00:07:48.900 Now, I'm just going to list a few of them. 00:07:48.900 --> 00:07:54.227 All of them I think you will need either in part one or part 00:07:54.227 --> 00:08:00.500 two of problem sets over these first couple of problem sets, 00:08:00.500 --> 00:08:04.000 or second and third maybe. 00:08:04.000 --> 00:08:11.152 But, of them, I'm going to put at the very top of the list 00:08:11.152 --> 00:08:16.800 of what I'll call here, I'll give it two names: 00:08:16.800 --> 00:08:21.600 the temperature diffusion model, well, it 00:08:21.600 --> 00:08:27.570 would be better to call it temperature concentration 00:08:27.570 --> 00:08:35.000 by analogy, temperature concentration model. 00:08:35.000 --> 00:08:40.000 There's the mixing model, which is hardly less important. 00:08:40.000 --> 00:08:43.000 In other words, it's almost as important. 00:08:43.000 --> 00:08:46.000 You have that in your problem set. 00:08:46.000 --> 00:08:52.000 And then, there are other, slightly less important models. 00:08:52.000 --> 00:08:55.000 There is the model of radioactive decay. 00:08:55.000 --> 00:09:00.000 There's the model of a bank interest, bank account, 00:09:00.000 --> 00:09:04.428 various motion models, you know, Newton's Law type problems 00:09:04.428 --> 00:09:08.280 if you can figure out a way of getting 00:09:08.280 --> 00:09:15.000 rid of the second derivative, some motion problems. 00:09:15.000 --> 00:09:23.000 A classic example is the motion of a rocket being fired off, 00:09:23.000 --> 00:09:24.500 etc., etc., etc. 00:09:24.500 --> 00:09:28.625 Now, today I have to pick a model. 00:09:28.625 --> 00:09:37.200 And, the one I'm going to pick is this temperature 00:09:37.200 --> 00:09:40.000 concentration model. 00:09:40.000 --> 00:09:43.000 So, this is going to be today's model. 00:09:43.000 --> 00:09:45.500 Tomorrow's model in the recitation, 00:09:45.500 --> 00:09:50.000 I'm asking the recitations to, among other things, 00:09:50.000 --> 00:09:54.224 make sure they do a mixing problem, A) to show you 00:09:54.224 --> 00:10:00.000 how to do it, and B) because it's on the problem sets. 00:10:00.000 --> 00:10:04.000 That's not a good reason, but it's not a bad one. 00:10:04.000 --> 00:10:06.688 The others are either in part one 00:10:06.688 --> 00:10:10.452 or we will take them up later in the term. 00:10:10.452 --> 00:10:16.000 This is not going to be the only lecture on the linear equation. 00:10:16.000 --> 00:10:21.000 There will be another one next week of equal importance. 00:10:21.000 --> 00:10:24.000 But, we can't do everything today. 00:10:24.000 --> 00:10:27.500 So, let's talk about the temperature concentration 00:10:27.500 --> 00:10:32.000 model, except I'm going to change its name. 00:10:32.000 --> 00:10:39.000 I'm going to change its name to the conduction diffusion model. 00:10:39.000 --> 00:10:44.000 I'll put conduction over there, and diffusion over here, 00:10:44.000 --> 00:10:48.428 let's say, since, as you will see, the similarities, 00:10:48.428 --> 00:10:52.428 they are practically the same model. 00:10:52.428 --> 00:10:56.250 All that's changed from one to the other 00:10:56.250 --> 00:11:00.000 is the name of the ideas. 00:11:00.000 --> 00:11:03.000 In one case, you call it temperature, 00:11:03.000 --> 00:11:06.800 and the other, you should call it concentration. 00:11:06.800 --> 00:11:11.332 But, the actual mathematics isn't identical. 00:11:11.332 --> 00:11:14.500 So, let's begin with conduction. 00:11:14.500 --> 00:11:19.330 All right, so, I need a simple physical situation 00:11:19.330 --> 00:11:21.000 that I'm modeling. 00:11:21.000 --> 00:11:24.000 So, imagine a tank of some liquid. 00:11:24.000 --> 00:11:27.000 Water will do as well as anything. 00:11:27.000 --> 00:11:30.500 And, in the inside is a suspended, 00:11:30.500 --> 00:11:33.250 somehow, is a chamber. 00:11:33.250 --> 00:11:38.665 A metal cube will do, and let's suppose 00:11:38.665 --> 00:11:43.332 that its walls are partly insulated, not so much 00:11:43.332 --> 00:11:46.000 that no heat can get through. 00:11:46.000 --> 00:11:50.440 There is no such thing as perfect insulation 00:11:50.440 --> 00:11:56.000 anyway, except maybe an absolute perfect vacuum. 00:11:56.000 --> 00:11:59.500 Now, inside, so here on the outside is liquid. 00:11:59.500 --> 00:12:04.400 Okay, on the inside is, what I'm interested 00:12:04.400 --> 00:12:10.000 in is the temperature of this thing. 00:12:10.000 --> 00:12:13.666 I'll call that T. Now, that's different from the temperature 00:12:13.666 --> 00:12:17.000 of the external water bath. 00:12:17.000 --> 00:12:21.000 So, I'll call that T sub e, T for temperature 00:12:21.000 --> 00:12:25.500 measured in Celsius, let's say, for the sake of definiteness. 00:12:25.500 --> 00:12:28.625 But, this is the external temperature. 00:12:28.625 --> 00:12:33.000 So, I'll indicate it with an e. 00:12:33.000 --> 00:12:35.775 Now, what is the model? 00:12:35.775 --> 00:12:43.328 Well, in other words, how do I set up a differential equation 00:12:43.328 --> 00:12:46.000 to model the situation? 00:12:46.000 --> 00:12:53.000 Well, it's based on a physical law, which I think you know, 00:12:53.000 --> 00:12:57.362 you've had simple examples like this, 00:12:57.362 --> 00:13:03.000 the so called Newton's Law of cooling, -- -- 00:13:03.000 --> 00:13:06.545 which says that the rate of change, 00:13:06.545 --> 00:13:13.000 the temperature of the heat goes from the outside to the inside 00:13:13.000 --> 00:13:14.500 by conduction only. 00:13:14.500 --> 00:13:18.140 Heat, of course, can travel in various ways, 00:13:18.140 --> 00:13:23.000 by convection, by conduction, as here, or by radiation, 00:13:23.000 --> 00:13:25.270 are the three most common. 00:13:25.270 --> 00:13:31.332 Of these, I only want one, namely transmission of heat 00:13:31.332 --> 00:13:33.000 by conduction. 00:13:33.000 --> 00:13:36.267 And, that's the way it's probably a little better 00:13:36.267 --> 00:13:41.000 to call it the conduction model, rather than the temperature 00:13:41.000 --> 00:13:45.086 model, which might involve other ways for the heat 00:13:45.086 --> 00:13:46.600 to be traveling. 00:13:46.600 --> 00:13:49.333 So, dt, the independent variable, 00:13:49.333 --> 00:13:53.000 is not going to be x, as it was over there. 00:13:53.000 --> 00:13:55.688 It's going to be t for time. 00:13:55.688 --> 00:13:59.000 So, maybe I should write that down. t equals time. 00:13:59.000 --> 00:14:05.000 Capital T equals temperature in degrees Celsius. 00:14:05.000 --> 00:14:08.000 So, you can put in the degrees Celsius if you want. 00:14:08.000 --> 00:14:12.375 So, it's proportional to the temperature difference 00:14:12.375 --> 00:14:13.750 between these two. 00:14:13.750 --> 00:14:16.428 Now, how shall I write the difference? 00:14:16.428 --> 00:14:21.000 Write it this way because if you don't you will be in trouble. 00:14:21.000 --> 00:14:24.000 Now, why do I write it that way? 00:14:24.000 --> 00:14:26.664 Well, I write it that way because I 00:14:26.664 --> 00:14:32.000 want this constant to be positive, a positive constant. 00:14:32.000 --> 00:14:35.800 In general, any constant, so, parameters which are physical, 00:14:35.800 --> 00:14:39.726 have some physical significance, one always 00:14:39.726 --> 00:14:44.142 wants to arrange the equation so that they are positive numbers, 00:14:44.142 --> 00:14:47.855 the way people normally think of these things. 00:14:47.855 --> 00:14:49.600 This is called the conductivity. 00:14:49.600 --> 00:14:52.000 The conductivity of what? 00:14:52.000 --> 00:14:54.800 Well, I don't know, of the system 00:14:54.800 --> 00:14:58.080 of the situation, the conductivity of the wall, 00:14:58.080 --> 00:15:01.999 or the wall if the metal were just by itself. 00:15:01.999 --> 00:15:04.332 At any rate, it's a constant. 00:15:04.332 --> 00:15:07.000 It's thought of as a constant. 00:15:07.000 --> 00:15:11.142 And, why positive, well, because if the external temperature is 00:15:11.142 --> 00:15:14.000 bigger than the internal temperature, 00:15:14.000 --> 00:15:18.666 I expect T to rise, the internal temperature to rise. 00:15:18.666 --> 00:15:24.220 That means dT / dt, its slope, should be positive. 00:15:24.220 --> 00:15:27.500 So, in other words, if Te is bigger than T, 00:15:27.500 --> 00:15:29.375 I expect this number to be positive. 00:15:29.375 --> 00:15:33.452 And, that tells you that k must be a positive constant. 00:15:33.452 --> 00:15:37.000 If I had turned it the other way, expressed 00:15:37.000 --> 00:15:40.000 the difference in the reverse order, 00:15:40.000 --> 00:15:44.000 K would then be negative, have to be negative in order 00:15:44.000 --> 00:15:47.000 that this turn out to be positive in that situation I 00:15:47.000 --> 00:15:47.600 described. 00:15:47.600 --> 00:15:51.200 And, since nobody wants negative values of k, 00:15:51.200 --> 00:15:53.999 you have to write the equation in this form 00:15:53.999 --> 00:15:56.000 rather than the other way around. 00:15:56.000 --> 00:15:59.000 So, there's our differential equation. 00:15:59.000 --> 00:16:02.000 It will probably have an initial condition. 00:16:02.000 --> 00:16:05.663 So, it could be the temperature at the starting time should 00:16:05.663 --> 00:16:10.000 be some given number, T zero. 00:16:10.000 --> 00:16:13.000 But, the condition could be given in other ways. 00:16:13.000 --> 00:16:15.000 One can ask, what's the temperature as time 00:16:15.000 --> 00:16:17.000 goes to infinity, for example? 00:16:17.000 --> 00:16:21.000 There are different ways of getting that initial condition. 00:16:21.000 --> 00:16:23.000 Okay, that's the conduction model. 00:16:23.000 --> 00:16:25.000 What would the diffusion model be? 00:16:25.000 --> 00:16:28.142 The diffusion model, mathematically, would be, 00:16:28.142 --> 00:16:31.000 word for word, the same. 00:16:31.000 --> 00:16:33.912 The only difference is that now, what 00:16:33.912 --> 00:16:38.270 I imagine is I'll draw the picture the same way, 00:16:38.270 --> 00:16:43.541 except now I'm going to put, label the inside not with a T 00:16:43.541 --> 00:16:46.665 but with a C, C for concentration. 00:16:46.665 --> 00:16:50.856 It's in an external water bath, let's say. 00:16:50.856 --> 00:16:53.571 So, there is an external concentration. 00:16:53.571 --> 00:16:57.888 And, what I'm talking about is some chemical, 00:16:57.888 --> 00:17:02.250 let's say salt will do as well as anything. 00:17:02.250 --> 00:17:07.500 So, C is equal to salt concentration inside, 00:17:07.500 --> 00:17:13.714 and Ce would be the salt concentration outside, 00:17:13.714 --> 00:17:18.000 outside in the water bath. 00:17:18.000 --> 00:17:26.000 Now, I imagine some mechanism, so this is a salt solution. 00:17:26.000 --> 00:17:28.664 That's a salt solution. 00:17:28.664 --> 00:17:34.178 And, I imagine some mechanism by which the salt can diffuse, 00:17:34.178 --> 00:17:36.999 it's a diffusion model now, diffuse from here 00:17:36.999 --> 00:17:40.000 into the air or possibly out the other way. 00:17:40.000 --> 00:17:43.500 And that's usually done by vaguely referring 00:17:43.500 --> 00:17:47.140 to the outside as a semi permeable membrane, 00:17:47.140 --> 00:17:50.080 semi permeable, so that the salt will 00:17:50.080 --> 00:17:53.750 have a little hard time getting through but permeable, 00:17:53.750 --> 00:17:56.800 so that it won't be blocked completely. 00:17:56.800 --> 00:18:00.000 So, there's a membrane. 00:18:00.000 --> 00:18:06.000 You write the semi permeable membrane outside, 00:18:06.000 --> 00:18:07.713 outside the inside. 00:18:07.713 --> 00:18:10.000 Well, I give up. 00:18:10.000 --> 00:18:14.000 You know, membrane somewhere. 00:18:14.000 --> 00:18:17.000 Sorry, membrane wall. 00:18:17.000 --> 00:18:18.332 How's that? 00:18:18.332 --> 00:18:21.000 Now, what's the equation? 00:18:21.000 --> 00:18:26.600 Well, the equation is the same, except it's 00:18:26.600 --> 00:18:29.500 called the diffusion equation. 00:18:29.500 --> 00:18:35.284 I don't think Newton got his name on this. 00:18:35.284 --> 00:18:40.178 The diffusion equation says that the rate at which the salt 00:18:40.178 --> 00:18:42.384 diffuses across the membrane, which 00:18:42.384 --> 00:18:47.000 is the same up to a constant as the rate at which 00:18:47.000 --> 00:18:51.000 the concentration inside changes, is some constant, 00:18:51.000 --> 00:18:54.000 usually called k still, okay. 00:18:54.000 --> 00:18:55.332 Do I contradict? 00:18:55.332 --> 00:18:58.000 Okay, let's keep calling it k1. 00:18:58.000 --> 00:19:01.000 Now it's different, times Ce minus C. 00:19:01.000 --> 00:19:03.800 And, for the same reason as before, 00:19:03.800 --> 00:19:07.142 if the external concentration is bigger 00:19:07.142 --> 00:19:12.724 than the internal concentration, we expect salt to flow in. 00:19:12.724 --> 00:19:15.000 That will make C rise. 00:19:15.000 --> 00:19:18.307 It will make this positive, and therefore, we 00:19:18.307 --> 00:19:21.070 want k to be positive, just k1 to be 00:19:21.070 --> 00:19:25.000 positive for the same reason it had to be positive before. 00:19:25.000 --> 00:19:28.600 So, in each case, the model that I'm talking about 00:19:28.600 --> 00:19:31.000 is the differential equation. 00:19:31.000 --> 00:19:34.500 So, maybe I should, let's put that, make that clear. 00:19:34.500 --> 00:19:39.713 Or, I would say that this first order differential equation 00:19:39.713 --> 00:19:43.332 models this physical situation, and the same thing 00:19:43.332 --> 00:19:46.000 is true on the other side over here. 00:19:46.000 --> 00:19:48.800 This is the diffusion equation, and this 00:19:48.800 --> 00:19:50.375 is the conduction equation. 00:19:50.375 --> 00:19:55.220 Now, if you are in any doubt about the power of differential 00:19:55.220 --> 00:19:58.842 equations, the point is, when I talk about this thing, 00:19:58.842 --> 00:20:02.875 I don't have to say which of these I'm following. 00:20:02.875 --> 00:20:07.284 I'll use neutral variables like Y and X 00:20:07.284 --> 00:20:09.000 to solve these equations. 00:20:09.000 --> 00:20:12.600 But, with a single stroke, I will be handling 00:20:12.600 --> 00:20:13.750 those situations together. 00:20:13.750 --> 00:20:16.500 And, that's the power of the method. 00:20:16.500 --> 00:20:20.200 Now, you obviously must be wondering, look, 00:20:20.200 --> 00:20:22.666 these look very, very special. 00:20:22.666 --> 00:20:27.140 He said he was going to talk about the first, general first 00:20:27.140 --> 00:20:28.000 order equation. 00:20:28.000 --> 00:20:31.000 But, these look rather special to me. 00:20:31.000 --> 00:20:33.220 Well, not too special. 00:20:33.220 --> 00:20:36.000 How should we write it? 00:20:36.000 --> 00:20:41.332 Suppose I write, let's take the temperature equation just 00:20:41.332 --> 00:20:44.000 to have something definite. 00:20:44.000 --> 00:20:48.000 Notice that it's in a form corresponding to Newton's Law. 00:20:48.000 --> 00:20:52.000 But it is not in the standard linear form. 00:20:52.000 --> 00:20:54.688 Let's put it in standard linear form, 00:20:54.688 --> 00:20:59.080 so at least you could see that it's a linear equation. 00:20:59.080 --> 00:21:02.384 So, if I put it in standard form, 00:21:02.384 --> 00:21:06.224 it's going to look like DTDTD little t plus KT 00:21:06.224 --> 00:21:09.220 is equal to K times TE. 00:21:09.220 --> 00:21:13.200 Now, compare that with the general, the way 00:21:13.200 --> 00:21:16.000 the general equation is supposed to look, 00:21:16.000 --> 00:21:20.000 the yellow box over there, the standard linear form. 00:21:20.000 --> 00:21:21.998 How are they going to compare? 00:21:21.998 --> 00:21:24.500 Well, this is a pretty general function. 00:21:24.500 --> 00:21:26.000 This is general. 00:21:26.000 --> 00:21:29.840 This is a general function of T because I can 00:21:29.840 --> 00:21:31.428 make the external temperature. 00:21:31.428 --> 00:21:37.000 I could suppose it behaves in anyway I like, steadily rising, 00:21:37.000 --> 00:21:39.000 decaying exponentially, maybe oscillating 00:21:39.000 --> 00:21:42.180 back and forth for some reason. 00:21:42.180 --> 00:21:46.500 The only way in which it's not general 00:21:46.500 --> 00:21:50.000 is that this K is a constant. 00:21:50.000 --> 00:21:53.000 So, I will ask you to be generous. 00:21:53.000 --> 00:21:57.000 Let's imagine the conductivity is changing over time. 00:21:57.000 --> 00:22:00.815 So, this is usually constant, but there's 00:22:00.815 --> 00:22:05.220 no law which says it has to be. 00:22:05.220 --> 00:22:09.142 How could a conductivity change over time? 00:22:09.142 --> 00:22:12.888 Well, we could suppose that this wall 00:22:12.888 --> 00:22:17.142 was made of slowly congealing Jell O, for instance. 00:22:17.142 --> 00:22:23.000 It starts out as liquid, and then it gets solid. 00:22:23.000 --> 00:22:26.444 And, Jell O doesn't transmit heat, 00:22:26.444 --> 00:22:32.180 I believe, quite as well as liquid does, as a liquid would. 00:22:32.180 --> 00:22:36.000 Is Jell O a solid or liquid? 00:22:36.000 --> 00:22:37.713 I don't know. 00:22:37.713 --> 00:22:40.000 Let's forget about that. 00:22:40.000 --> 00:22:46.000 So, with this understanding, so let's say not necessarily 00:22:46.000 --> 00:22:51.500 here, but not necessarily, I can think of this, therefore, 00:22:51.500 --> 00:22:55.000 by allowing K to vary with time. 00:22:55.000 --> 00:23:00.000 And the external temperature to vary with time. 00:23:00.000 --> 00:23:07.000 I can think of it as a general, linear equation. 00:23:07.000 --> 00:23:09.000 So, these models are not special. 00:23:09.000 --> 00:23:10.452 They are fairly general. 00:23:10.452 --> 00:23:14.284 Well, I did promise you I would solve an equation, 00:23:14.284 --> 00:23:18.000 and that this lecture, I still have not solved any equations. 00:23:18.000 --> 00:23:21.000 OK, time to stop temporizing and solve. 00:23:21.000 --> 00:23:24.000 So, I'm going to, in order not to play favorites 00:23:24.000 --> 00:23:27.000 with these two models, I'll go back to, 00:23:27.000 --> 00:23:31.000 and to get you used to thinking of the variables all the time, 00:23:31.000 --> 00:23:35.444 that is, you know, be eclectic switching from one variable 00:23:35.444 --> 00:23:38.552 to another according to which particular lecture 00:23:38.552 --> 00:23:42.635 you happened to be sitting in. 00:23:42.635 --> 00:23:52.712 So, let's take our equation in the form, Y prime plus P of XY, 00:23:52.712 --> 00:23:58.000 the general form using the old variables 00:23:58.000 --> 00:24:04.000 equals Q of X. Solve me. 00:24:04.000 --> 00:24:06.928 Well, there are different ways of describing the solution 00:24:06.928 --> 00:24:07.428 process. 00:24:07.428 --> 00:24:10.614 No matter how you do it, it amounts 00:24:10.614 --> 00:24:13.684 to the same amount of work and there is always 00:24:13.684 --> 00:24:16.331 a trick involved at each one of them 00:24:16.331 --> 00:24:19.125 since you can't suppress a trick by doing 00:24:19.125 --> 00:24:21.000 the problem some other way. 00:24:21.000 --> 00:24:24.000 The way I'm going to do it, I think, is the best. 00:24:24.000 --> 00:24:26.000 That's why I'm giving it to you. 00:24:26.000 --> 00:24:27.815 It's the easiest to remember. 00:24:27.815 --> 00:24:31.000 It leads to the least work, but I 00:24:31.000 --> 00:24:36.000 have colleagues who would fight with me about that point. 00:24:36.000 --> 00:24:39.330 So, since they are not here to fight with me 00:24:39.330 --> 00:24:42.332 I am free to do whatever I like. 00:24:42.332 --> 00:24:45.400 One of the main reasons for doing 00:24:45.400 --> 00:24:49.400 it the way I'm going to do is because I 00:24:49.400 --> 00:24:55.555 want you to get what our word into your consciousness, two 00:24:55.555 --> 00:24:57.220 words, integrating factor. 00:24:57.220 --> 00:25:02.500 I'm going to solve this equation by finding and integrating 00:25:02.500 --> 00:25:09.000 factor of the form U of X. What's an integrating factor? 00:25:09.000 --> 00:25:13.000 Well, I'll show you not by writing an elaborate definition 00:25:13.000 --> 00:25:16.666 on the board, but showing you what its function is. 00:25:16.666 --> 00:25:19.332 It's a certain function, U of X, I 00:25:19.332 --> 00:25:23.200 don't know what it is, but here's what I wanted to do. 00:25:23.200 --> 00:25:26.763 I want to multiply, I'm going to drop the X's a just so 00:25:26.763 --> 00:25:29.110 that the thing looks less complicated. 00:25:29.110 --> 00:25:34.125 So, what I want to do is multiply this equation 00:25:34.125 --> 00:25:36.000 through by U of X. 00:25:36.000 --> 00:25:40.360 That's why it's called a factor because you're 00:25:40.360 --> 00:25:43.816 going to multiply everything through by it. 00:25:43.816 --> 00:25:49.220 So, it's going to look like UY prime plus PUY equals QU, 00:25:49.220 --> 00:25:52.776 and now, so far, it's just a factor. 00:25:52.776 --> 00:25:55.375 What makes it an integrating factor 00:25:55.375 --> 00:26:00.766 is that this, after I do that, I want this to turn out 00:26:00.766 --> 00:26:04.500 to be the derivative of something with respect 00:26:04.500 --> 00:26:08.250 to X. You see the motivation for that. 00:26:08.250 --> 00:26:11.125 If this turns out to be the derivative of something, 00:26:11.125 --> 00:26:13.363 because I've chosen U so cleverly, 00:26:13.363 --> 00:26:17.000 then I will be able to solve the equation immediately 00:26:17.000 --> 00:26:19.000 just by integrating this with respect to X, 00:26:19.000 --> 00:26:21.000 and integrating that with respect to X. 00:26:21.000 --> 00:26:24.750 You just, then, integrate both sides with respect to X, 00:26:24.750 --> 00:26:26.000 and the equation is solved. 00:26:26.000 --> 00:26:31.000 Now, the only question is, what should I choose for U? 00:26:31.000 --> 00:26:34.000 Well, if you think of the product formula, 00:26:34.000 --> 00:26:37.000 there might be many things to try here. 00:26:37.000 --> 00:26:40.000 But there's only one reasonable thing to try. 00:26:40.000 --> 00:26:45.000 Try to pick U so that it's the derivative of U times Y. 00:26:45.000 --> 00:26:46.665 See how reasonable that is? 00:26:46.665 --> 00:26:49.333 If I use the product rule on this, 00:26:49.333 --> 00:26:52.000 the first term is U times Y prime. 00:26:52.000 --> 00:26:56.000 The second term would be U prime times Y. 00:26:56.000 --> 00:26:59.000 Well, I've got the Y there. 00:26:59.000 --> 00:27:01.000 So, this will work. 00:27:01.000 --> 00:27:05.888 It works if, what's the condition that you must satisfy 00:27:05.888 --> 00:27:09.000 in order for that to be true? 00:27:09.000 --> 00:27:15.000 Well, it must be that after it to the differentiation, 00:27:15.000 --> 00:27:19.000 U prime turns out to be P times U. 00:27:19.000 --> 00:27:20.816 So, is it clear? 00:27:20.816 --> 00:27:26.625 This is something we want to be equal to, and the thing I will 00:27:26.625 --> 00:27:32.331 try to do it is by choosing U in such a way 00:27:32.331 --> 00:27:37.000 that this equality will take place. 00:27:37.000 --> 00:27:40.000 And then I will be able to solve the equation. 00:27:40.000 --> 00:27:43.000 And so, here's what my U prime must satisfy. 00:27:43.000 --> 00:27:44.815 Hey, we can solve that. 00:27:44.815 --> 00:27:47.222 But please don't forget that P is 00:27:47.222 --> 00:27:49.000 P of X. It's a function of X. 00:27:49.000 --> 00:27:53.000 So, if you separate variables, I'm going to do this. 00:27:53.000 --> 00:27:56.600 So, what is it, DU over U equals P of X times DX. 00:27:56.600 --> 00:28:00.500 If I integrate that, so, separate variables, 00:28:00.500 --> 00:28:04.665 integrate, and you're going to get DU over U integrates 00:28:04.665 --> 00:28:08.428 to the be the log of U, and the other side 00:28:08.428 --> 00:28:12.000 integrates to be the integral of P of X DX. 00:28:12.000 --> 00:28:16.000 Now, you can put an arbitrary constant there, 00:28:16.000 --> 00:28:18.448 or you can think of it as already implied 00:28:18.448 --> 00:28:20.142 by the indefinite integral. 00:28:20.142 --> 00:28:24.500 Well, that doesn't tell us, yet, what U is. 00:28:24.500 --> 00:28:26.000 What should U be? 00:28:26.000 --> 00:28:30.250 Notice, I don't have to find every possible U, which works. 00:28:30.250 --> 00:28:34.000 All I'm looking for is one. 00:28:34.000 --> 00:28:38.000 All I want is a single view which satisfies that equation. 00:28:38.000 --> 00:28:42.000 Well, U equals the integral, E to the integral of PDX. 00:28:42.000 --> 00:28:44.500 That's not too beautiful looking, 00:28:44.500 --> 00:28:46.726 but by differential equations, things 00:28:46.726 --> 00:28:50.454 can get so complicated that in a week or two, 00:28:50.454 --> 00:28:55.000 you will think of this as an extremely simple formula. 00:28:55.000 --> 00:29:00.000 So, there is a formula for our integrating factor. 00:29:00.000 --> 00:29:01.500 We found it. 00:29:01.500 --> 00:29:07.250 We will always be able to write an integrating factor. 00:29:07.250 --> 00:29:14.750 Don't worry about the arbitrary constant because you only need 00:29:14.750 --> 00:29:17.000 one such U. 00:29:17.000 --> 00:29:23.000 So: no arbitrary constant since only one U needed. 00:29:23.000 --> 00:29:26.600 And, that's the solution, the way 00:29:26.600 --> 00:29:30.200 we solve the linear equation. 00:29:30.200 --> 00:29:37.544 OK, let's take over, and actually do it. 00:29:37.544 --> 00:29:43.714 I think it would be better to summarize it 00:29:43.714 --> 00:29:48.000 as a clear cut method. 00:29:48.000 --> 00:29:51.000 So, let's do that. 00:29:51.000 --> 00:29:54.000 So, what's our method? 00:29:54.000 --> 00:30:04.000 It's the method for solving Y prime plus PY equals Q. 00:30:04.000 --> 00:30:07.800 Well, the first place, make sure it's in standard linear form. 00:30:07.800 --> 00:30:11.331 If it isn't, you must put it in that form. 00:30:11.331 --> 00:30:14.712 Notice, the formula for the integrating factor, the formula 00:30:14.712 --> 00:30:16.856 for the integrating factor involves 00:30:16.856 --> 00:30:19.000 P, the integral of PDX. 00:30:19.000 --> 00:30:21.000 So, you'd better get the right P. 00:30:21.000 --> 00:30:22.600 Otherwise, you are sunk. 00:30:22.600 --> 00:30:25.750 OK, so put it in standard linear form. 00:30:25.750 --> 00:30:29.332 That way, you will have the right P. Notice 00:30:29.332 --> 00:30:32.544 that if you wrote it in that form, 00:30:32.544 --> 00:30:35.375 and all you remembered was E to the integral PDX, 00:30:35.375 --> 00:30:38.000 the P would have the wrong sign. 00:30:38.000 --> 00:30:41.070 If you're going to write, that P should have a negative sign 00:30:41.070 --> 00:30:41.570 there. 00:30:41.570 --> 00:30:44.500 So, do it this way, and no other way. 00:30:44.500 --> 00:30:47.665 Otherwise, you will get confused and get wrong signs. 00:30:47.665 --> 00:30:51.331 And, as I say, that will produce wrong answers, and not 00:30:51.331 --> 00:30:54.000 just slightly wrong answers, but disastrously 00:30:54.000 --> 00:30:57.664 wrong answers from the point of view of the modeling 00:30:57.664 --> 00:31:02.108 if you really want answers to physical problems. 00:31:02.108 --> 00:31:06.600 So, here's a standard linear form. 00:31:06.600 --> 00:31:09.666 Then, find the integrating factor. 00:31:09.666 --> 00:31:17.000 So, calculate E to the integral, PDX, the integrating factor, 00:31:17.000 --> 00:31:22.000 and that multiply both, I'm putting this as both, 00:31:22.000 --> 00:31:31.000 underlined that as many times as you have room in your notes. 00:31:31.000 --> 00:31:38.362 Multiply both sides by this integrating factor by E 00:31:38.362 --> 00:31:42.400 to the integral PDX. 00:31:42.400 --> 00:31:46.000 And then, integrate. 00:31:46.000 --> 00:31:51.000 OK, let's take a simple example. 00:31:51.000 --> 00:31:57.400 Suppose we started with the equation XY prime 00:31:57.400 --> 00:32:06.000 minus Y equals, I had X2, X3, something like that, 00:32:06.000 --> 00:32:12.000 X3, I think, yeah, X2. 00:32:12.000 --> 00:32:16.000 OK, what's the first thing to do? 00:32:16.000 --> 00:32:18.915 Put it in standard form. 00:32:18.915 --> 00:32:25.915 So, step zero will be to write it as Y prime minus 00:32:25.915 --> 00:32:30.000 one over X times Y equals X2. 00:32:30.000 --> 00:32:34.833 Let's do the work first, and then I'll talk about mistakes. 00:32:34.833 --> 00:32:39.888 Well, we now calculate the integrating factor. 00:32:39.888 --> 00:32:43.000 So, I would do it in steps. 00:32:43.000 --> 00:32:48.000 You can integrate negative one over X, right? 00:32:48.000 --> 00:32:51.500 That integrates to minus log X. So, 00:32:51.500 --> 00:32:57.000 the integrating factor is E to the integral of this, DX. 00:32:57.000 --> 00:33:02.000 So, it's E to the negative log X. 00:33:02.000 --> 00:33:06.500 Now, in real life, that's not the way to leave that. 00:33:06.500 --> 00:33:10.333 What is E to the negative log X? 00:33:10.333 --> 00:33:16.000 Well, think of it as E to the log X to the minus one. 00:33:16.000 --> 00:33:21.570 Or, in other words, it is E to the log X is X. So, 00:33:21.570 --> 00:33:23.000 it's one over X. 00:33:23.000 --> 00:33:27.000 So, the integrating factor is one over X. 00:33:27.000 --> 00:33:34.000 OK, multiply both sides by the integrating factor. 00:33:34.000 --> 00:33:35.500 Both sides of what? 00:33:35.500 --> 00:33:41.000 Both sides of this: the equation written in standard form, 00:33:41.000 --> 00:33:42.200 and both sides. 00:33:42.200 --> 00:33:47.800 So, it's going to be one over XY prime minus one over X2 Y 00:33:47.800 --> 00:33:52.815 is equal to X2 times one over X, which is simply X. Now, 00:33:52.815 --> 00:33:55.545 if you have done the work correctly, 00:33:55.545 --> 00:34:01.000 you should be able, now, to integrate the left hand 00:34:01.000 --> 00:34:02.110 side directly. 00:34:02.110 --> 00:34:06.428 So, I'm going to write it this way. 00:34:06.428 --> 00:34:10.815 I always recommend that you put it as extra step, well, 00:34:10.815 --> 00:34:14.142 put it as an extra step the reason 00:34:14.142 --> 00:34:17.000 for using that integrating factor, 00:34:17.000 --> 00:34:21.200 in other words, that the left hand side is supposed to be, 00:34:21.200 --> 00:34:24.000 now, one over X times Y prime. 00:34:24.000 --> 00:34:27.328 I always put it that because there's always 00:34:27.328 --> 00:34:31.775 a chance you made a mistake or forgot something. 00:34:31.775 --> 00:34:34.555 Look at it, mentally differentiated 00:34:34.555 --> 00:34:39.855 using the product rule just to check that, in fact, it 00:34:39.855 --> 00:34:43.000 turns out to be the same as the left hand side. 00:34:43.000 --> 00:34:44.535 So, what do we get? 00:34:44.535 --> 00:34:48.500 One over X times Y prime plus Y times the derivative 00:34:48.500 --> 00:34:53.800 of one over X, which indeed is negative one over X2. 00:34:53.800 --> 00:34:58.000 And now, finally, that's 3A, continue, do the integration. 00:34:58.000 --> 00:35:00.149 So, you're going to get, let's see 00:35:00.149 --> 00:35:04.904 if we can do it all on one board, one over X times Y 00:35:04.904 --> 00:35:07.904 is equal to X plus a constant, X, sorry, X2 00:35:07.904 --> 00:35:09.857 over two plus a constant. 00:35:09.857 --> 00:35:16.332 And, the final step will be, therefore, now 00:35:16.332 --> 00:35:21.000 I want to isolate Y by itself. 00:35:21.000 --> 00:35:26.000 So, Y will be equal to multiply through by X. 00:35:26.000 --> 00:35:34.000 X3 over two plus C times X. And, that's the solution. 00:35:34.000 --> 00:35:40.000 OK, let's do one a little slightly more complicated. 00:35:40.000 --> 00:35:41.816 Let's try this one. 00:35:41.816 --> 00:35:45.545 Now, my equation is going to be one, 00:35:45.545 --> 00:35:51.000 I'll still keep two, Y and X, as the variables. 00:35:51.000 --> 00:35:57.000 I'll use T and F for a minute or two. 00:35:57.000 --> 00:36:01.360 One plus cosine X, so, I'm not going 00:36:01.360 --> 00:36:07.000 to give you this one in standard form either. 00:36:07.000 --> 00:36:09.332 It's a trick question. 00:36:09.332 --> 00:36:17.332 Y prime minus sine X times Y is equal to anything reasonable, 00:36:17.332 --> 00:36:19.000 I guess. 00:36:19.000 --> 00:36:24.000 I think X, 2X, make it more exciting. 00:36:24.000 --> 00:36:28.920 OK, now, I think I should warn you 00:36:28.920 --> 00:36:38.000 where the mistakes are just so that you can make all of them. 00:36:38.000 --> 00:36:41.000 So, this is mistake number one. 00:36:41.000 --> 00:36:44.750 You don't put it in standard form. 00:36:44.750 --> 00:36:51.000 Mistake number two: generally people can do step one fine. 00:36:51.000 --> 00:36:57.000 Mistake number two is, this is my most common mistake, 00:36:57.000 --> 00:37:00.000 so I'm very sensitive to it. 00:37:00.000 --> 00:37:03.750 But that doesn't mean if you make it, 00:37:03.750 --> 00:37:06.000 you'll get any sympathy from me. 00:37:06.000 --> 00:37:08.000 I don't give sympathy to myself. 00:37:08.000 --> 00:37:10.400 You are so intense, so happy at having 00:37:10.400 --> 00:37:12.332 found the integrating factor, you 00:37:12.332 --> 00:37:16.332 forget to multiply Q by the integrating factor also. 00:37:16.332 --> 00:37:20.332 You just handle the left hand side of the equation, 00:37:20.332 --> 00:37:22.999 if you forget about the right hand side. 00:37:22.999 --> 00:37:27.220 So, the emphasis on the both here is the right hand, 00:37:27.220 --> 00:37:29.000 please include the Q. 00:37:29.000 --> 00:37:33.000 Please include the right hand side. 00:37:33.000 --> 00:37:34.332 Any other mistakes? 00:37:34.332 --> 00:37:37.250 Well, nothing that I can think of. 00:37:37.250 --> 00:37:39.664 Well, maybe only, anyway, we are not 00:37:39.664 --> 00:37:43.570 going to make any mistakes the rest of this lecture. 00:37:43.570 --> 00:37:45.000 So, what do we do? 00:37:45.000 --> 00:37:47.304 We write this in standard form. 00:37:47.304 --> 00:37:51.600 So, it's going to look like Y prime minus sine X, 00:37:51.600 --> 00:37:55.816 sine X divided by one plus cosine X times Y 00:37:55.816 --> 00:37:59.000 equals, my heart sinks because I know 00:37:59.000 --> 00:38:04.000 I'm supposed to integrate something like this. 00:38:04.000 --> 00:38:07.000 And, boy, that's going to give me problems. 00:38:07.000 --> 00:38:08.713 Well, not yet. 00:38:08.713 --> 00:38:11.000 With the integrating factor? 00:38:11.000 --> 00:38:14.000 The integrating factor is, well, we 00:38:14.000 --> 00:38:18.220 want to calculate the integral of negative sine X 00:38:18.220 --> 00:38:20.000 over one plus cosine. 00:38:20.000 --> 00:38:21.875 That's the integral of PDX. 00:38:21.875 --> 00:38:25.500 And, after that, we have to exponentiate it. 00:38:25.500 --> 00:38:28.000 Well, can you do this? 00:38:28.000 --> 00:38:31.000 Yeah, but if you stare at it a little while, 00:38:31.000 --> 00:38:38.000 you can see that the top is the derivative of the bottom. 00:38:38.000 --> 00:38:39.332 That is great. 00:38:39.332 --> 00:38:43.600 That means it integrates to be the log of one 00:38:43.600 --> 00:38:47.362 plus cosine X. Is that right, one over one 00:38:47.362 --> 00:38:51.625 plus cosine X times the derivative of this, which 00:38:51.625 --> 00:38:56.375 is negative cosine X. Therefore, the integrating factor 00:38:56.375 --> 00:38:57.875 is E to that. 00:38:57.875 --> 00:39:02.270 In other words, it is one plus cosine X. 00:39:02.270 --> 00:39:05.500 Therefore, so this was step zero. 00:39:05.500 --> 00:39:09.332 Step one, we found the integrating factor. 00:39:09.332 --> 00:39:17.000 And now, step two, we multiply through the integrating factor. 00:39:17.000 --> 00:39:19.220 And what do we get? 00:39:19.220 --> 00:39:23.142 We multiply through the standard for equation 00:39:23.142 --> 00:39:29.000 by the integrating factor, if you do that, what you get is, 00:39:29.000 --> 00:39:35.000 well, Y prime gets the coefficient one plus cosine X, 00:39:35.000 --> 00:39:38.885 Y prime minus sign X equals 2X. 00:39:38.885 --> 00:39:40.000 Oh, dear. 00:39:40.000 --> 00:39:45.000 Well, I hope somebody would giggle at this point. 00:39:45.000 --> 00:39:47.270 What's giggle able about it? 00:39:47.270 --> 00:39:51.000 Well, that all this was totally wasted work. 00:39:51.000 --> 00:39:53.500 It's called spinning your wheels. 00:39:53.500 --> 00:39:56.500 No, it's not spinning your wheels. 00:39:56.500 --> 00:40:00.000 It's doing what you're supposed to do, 00:40:00.000 --> 00:40:05.000 and finding out that you wasted the entire time doing 00:40:05.000 --> 00:40:08.000 what you were supposed to do. 00:40:08.000 --> 00:40:12.140 Well, in other words, that net effect of this 00:40:12.140 --> 00:40:16.856 is to end up with the same equation we started with. 00:40:16.856 --> 00:40:19.000 But, what is the point? 00:40:19.000 --> 00:40:22.178 The point of having done all this 00:40:22.178 --> 00:40:26.775 was because now the left hand side is exactly 00:40:26.775 --> 00:40:32.270 the derivative of something, and the left hand side should 00:40:32.270 --> 00:40:35.000 be the derivative of what? 00:40:35.000 --> 00:40:37.904 Well, it should be the derivative of one 00:40:37.904 --> 00:40:40.600 plus cosine X times Y, all prime. 00:40:40.600 --> 00:40:44.776 Now, you can check that that's in fact the case. 00:40:44.776 --> 00:40:48.815 It's one plus cosine X, Y prime, plus minus sine 00:40:48.815 --> 00:40:51.856 X, the derivative of this side times Y. 00:40:51.856 --> 00:40:56.220 So, if you had thought, in looking at the equation, 00:40:56.220 --> 00:41:00.724 to say to yourself, this is a derivative of that, 00:41:00.724 --> 00:41:04.248 maybe I'll just check right away to see 00:41:04.248 --> 00:41:08.571 if it's the derivative of one plus cosine X sine. 00:41:08.571 --> 00:41:12.000 You would have saved that work. 00:41:12.000 --> 00:41:16.000 Well, you don't have to be brilliant or clever, 00:41:16.000 --> 00:41:17.332 or anything like that. 00:41:17.332 --> 00:41:20.200 You can follow your nose, and it's just, 00:41:20.200 --> 00:41:24.875 I want to give you a positive experience in solving 00:41:24.875 --> 00:41:28.000 linear equations, not too negative. 00:41:28.000 --> 00:41:31.000 Anyway, so we got to this point. 00:41:31.000 --> 00:41:37.000 So, now this is 2X, and now we are ready to solve 00:41:37.000 --> 00:41:43.500 the equation, which is the solution now will be one plus 00:41:43.500 --> 00:41:49.000 cosine X times Y is equal to X2 plus a constant, 00:41:49.000 --> 00:41:55.524 and so Y is equal to X2 divided by X2 plus a constant divided 00:41:55.524 --> 00:42:01.400 by one plus cosine X. Suppose I have given you an initial 00:42:01.400 --> 00:42:03.000 condition, which I didn't. 00:42:03.000 --> 00:42:07.000 But, suppose the initial condition said that Y of zero 00:42:07.000 --> 00:42:08.332 were one, for instance. 00:42:08.332 --> 00:42:11.665 Then, the solution would be, so, this is an if, 00:42:11.665 --> 00:42:15.498 I'm throwing in at the end just to make it a little bit more 00:42:15.498 --> 00:42:17.400 of a problem, how would I put, then 00:42:17.400 --> 00:42:21.400 I could evaluate the constant by using the initial condition. 00:42:21.400 --> 00:42:23.000 What would it be? 00:42:23.000 --> 00:42:25.454 This would be, on the left hand side, one, 00:42:25.454 --> 00:42:30.000 on the right hand side would be C over two. 00:42:30.000 --> 00:42:34.000 So, I would get one equals C over two. 00:42:34.000 --> 00:42:35.875 Is that correct? 00:42:35.875 --> 00:42:42.180 Cosine of zero is one, so that's two down below. 00:42:42.180 --> 00:42:48.856 Therefore, C is equal to two, and that would then 00:42:48.856 --> 00:42:51.000 complete the solution. 00:42:51.000 --> 00:42:57.000 We would be X2 plus two over one plus cosine X. 00:42:57.000 --> 00:43:03.000 Now, you can do this in general, of course, 00:43:03.000 --> 00:43:06.180 and get a general formula. 00:43:06.180 --> 00:43:11.712 And, we will have occasion to use that next week. 00:43:11.712 --> 00:43:15.220 But for now, why don't we concentrate 00:43:15.220 --> 00:43:18.555 on the most interesting case, namely 00:43:18.555 --> 00:43:21.885 that of the most linear equation, 00:43:21.885 --> 00:43:24.332 with constant coefficient, that is, 00:43:24.332 --> 00:43:27.714 so let's look at the linear equation 00:43:27.714 --> 00:43:31.284 with constant coefficient, because that's 00:43:31.284 --> 00:43:37.000 the one that most closely models the conduction and diffusion 00:43:37.000 --> 00:43:37.666 equations. 00:43:37.666 --> 00:43:43.000 So, what I'm interested in, is since this is the, of them all, 00:43:43.000 --> 00:43:45.400 probably it's the most important case 00:43:45.400 --> 00:43:48.500 is the one where P is a constant because 00:43:48.500 --> 00:43:50.000 of its application to that. 00:43:50.000 --> 00:43:53.500 And, many of the other, the bank account, for example, 00:43:53.500 --> 00:43:55.666 all of those will use a constant coefficient. 00:43:55.666 --> 00:43:58.428 So, how is the thing going to look? 00:43:58.428 --> 00:44:01.000 Well, I will use the cooling. 00:44:01.000 --> 00:44:05.000 Let's use the temperature model, for example. 00:44:05.000 --> 00:44:07.220 The temperature model, the equation 00:44:07.220 --> 00:44:10.600 will be DTDT plus KT is equal to. 00:44:10.600 --> 00:44:14.800 Now, notice on the right hand side, this is a common error. 00:44:14.800 --> 00:44:16.000 You don't put TE. 00:44:16.000 --> 00:44:21.000 You have to put KTE because that's what the equation says. 00:44:21.000 --> 00:44:25.000 If you think units, you won't have any trouble. 00:44:25.000 --> 00:44:30.000 Units have to be compatible on both sides of a differential 00:44:30.000 --> 00:44:30.666 equation. 00:44:30.666 --> 00:44:34.331 And therefore, whatever the units were for capital KT, 00:44:34.331 --> 00:44:38.541 I'd have to have the same units on the right hand side, 00:44:38.541 --> 00:44:42.664 which indicates I cannot have KT on the left of the differential 00:44:42.664 --> 00:44:44.998 equation, and just T on the right, 00:44:44.998 --> 00:44:47.500 and expect the units to be compatible. 00:44:47.500 --> 00:44:49.000 That's not possible. 00:44:49.000 --> 00:44:51.100 So, that's a good way of remembering 00:44:51.100 --> 00:44:54.000 that if you're modeling temperature or concentration, 00:44:54.000 --> 00:44:57.000 you have to have the K on both sides. 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 00:45:02.080 --> 00:45:05.000 now because this is really too easy. 00:45:05.000 --> 00:45:07.220 What's the integrating factor? 00:45:07.220 --> 00:45:13.227 Well, the integrating factor is going to be the integral of K, 00:45:13.227 --> 00:45:16.000 the coefficient now is just K. 00:45:16.000 --> 00:45:21.284 P is a constant, K, and if I integrate KDT, I get KT, 00:45:21.284 --> 00:45:23.000 and I exponentiate that. 00:45:23.000 --> 00:45:28.000 So, the integrating factor is E to the KT. 00:45:28.000 --> 00:45:34.000 I multiply through both sides, multiply by E to the KT, 00:45:34.000 --> 00:45:38.000 and what's the resulting equation? 00:45:38.000 --> 00:45:44.333 Well, it's going to be , I'll write it in the compact form. 00:45:44.333 --> 00:45:50.000 It's going to be E to the KT times T, all prime. 00:45:50.000 --> 00:45:55.284 The differentiation is now, of course, with respect 00:45:55.284 --> 00:45:57.000 to the time. 00:45:57.000 --> 00:46:00.750 And, that's equal to KTE, whatever 00:46:00.750 --> 00:46:04.725 that is, times E to the KT. 00:46:04.725 --> 00:46:09.000 This is a function of T, of course, 00:46:09.000 --> 00:46:13.500 the function of little time, sorry, little T time. 00:46:13.500 --> 00:46:18.000 OK, and now, finally, we are going to integrate. 00:46:18.000 --> 00:46:19.332 What's the answer? 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 00:46:26.224 --> 00:46:31.750 T is, sorry, K little t, K times time times the temperature 00:46:31.750 --> 00:46:37.000 is equal to the integral of KTE. 00:46:37.000 --> 00:46:40.840 I'll put the fact that it's a function of T 00:46:40.840 --> 00:46:43.998 inside just to remind you, E to the KT, 00:46:43.998 --> 00:46:46.500 and now I'll put the arbitrary constant. 00:46:46.500 --> 00:46:50.142 Let's put in the arbitrary constant explicitly. 00:46:50.142 --> 00:46:53.000 So, what will T be? 00:46:53.000 --> 00:46:56.000 OK, T will look like this, finally. 00:46:56.000 --> 00:46:59.000 It will be E to the negative KT. 00:46:59.000 --> 00:47:01.500 That's on the outside. 00:47:01.500 --> 00:47:04.000 Then, you will integrate. 00:47:04.000 --> 00:47:07.552 Of course, the difficulty of doing this integral 00:47:07.552 --> 00:47:12.000 depends entirely upon how this external temperature varies. 00:47:12.000 --> 00:47:16.726 But anyways, it's going to be K times that function, which 00:47:16.726 --> 00:47:20.000 I haven't specified, E to the KT plus C 00:47:20.000 --> 00:47:22.664 times E to the negative KT. 00:47:22.664 --> 00:47:27.000 Now, some people, many, in fact, that almost always, 00:47:27.000 --> 00:47:30.750 in the engineering literature, almost never 00:47:30.750 --> 00:47:36.285 write indefinite integrals because an indefinite integral 00:47:36.285 --> 00:47:38.000 is indefinite. 00:47:38.000 --> 00:47:40.400 In other words, this covers not just one function, 00:47:40.400 --> 00:47:42.800 but a whole multitude of functions 00:47:42.800 --> 00:47:46.000 which differ from each other by an arbitrary constant. 00:47:46.000 --> 00:47:49.000 So, in a formula like this, there's a certain vagueness, 00:47:49.000 --> 00:47:51.541 and it's further compounded by the fact 00:47:51.541 --> 00:47:55.000 that I don't know whether the arbitrary constant is here. 00:47:55.000 --> 00:47:58.571 I seem to have put it explicitly on the outside the way 00:47:58.571 --> 00:48:02.000 you're used to doing from calculus. 00:48:02.000 --> 00:48:04.570 Many people, therefore, prefer, and I 00:48:04.570 --> 00:48:06.565 think you should learn this, to do 00:48:06.565 --> 00:48:11.885 what is done in the very first section of the notes called 00:48:11.885 --> 00:48:13.500 definite integral solutions. 00:48:13.500 --> 00:48:16.400 If there's an initial condition saying 00:48:16.400 --> 00:48:19.200 that the internal temperature at time zero 00:48:19.200 --> 00:48:22.912 is some given value, what they like to do 00:48:22.912 --> 00:48:26.248 is make this thing definite by integrating here 00:48:26.248 --> 00:48:30.625 from zero to T, and making this a dummy variable. 00:48:30.625 --> 00:48:36.272 You see, what that does is it gives you 00:48:36.272 --> 00:48:39.452 a particular function, whereas, I'm 00:48:39.452 --> 00:48:44.766 sorry I didn't put in the DT one minus two. 00:48:44.766 --> 00:48:49.554 What it does is that when time is zero, 00:48:49.554 --> 00:48:52.662 all this automatically disappears, 00:48:52.662 --> 00:49:00.000 and the arbitrary constant will then be, it's T. 00:49:00.000 --> 00:49:03.125 So, in other words, C times this, which is one, 00:49:03.125 --> 00:49:05.000 is that equal to [T?]. 00:49:05.000 --> 00:49:07.000 In other words, if I make this zero, 00:49:07.000 --> 00:49:12.000 that I can write C as equal to this arbitrary starting value. 00:49:12.000 --> 00:49:15.000 Now, when you do this, the essential thing, 00:49:15.000 --> 00:49:18.000 and we're going to come back to this next week, 00:49:18.000 --> 00:49:21.000 but right away, because K is positive, 00:49:21.000 --> 00:49:25.400 I want to emphasize that so much at the beginning of the period, 00:49:25.400 --> 00:49:30.428 I want to conclude by showing you what its significance is. 00:49:30.428 --> 00:49:35.332 This part disappears because K is positive. 00:49:35.332 --> 00:49:38.000 The conductivity is positive. 00:49:38.000 --> 00:49:41.000 This part disappears as T goes to zero. 00:49:41.000 --> 00:49:45.000 This goes to zero as T goes to infinity. 00:49:45.000 --> 00:49:48.000 So, this is a solution that remains. 00:49:48.000 --> 00:49:52.625 This, therefore, is called the steady state solution, 00:49:52.625 --> 00:49:57.000 the thing which the temperature behaves like, 00:49:57.000 --> 00:49:59.000 as T goes to infinity. 00:49:59.000 --> 00:50:01.000 This is called the transient. 00:50:01.000 --> 00:50:07.000 because it disappears as T goes to infinity. 00:50:07.000 --> 00:50:09.664 It depends on the initial condition, 00:50:09.664 --> 00:50:12.200 but it disappears, which shows you, 00:50:12.200 --> 00:50:16.665 then, in the long run for this type of problem 00:50:16.665 --> 00:50:20.000 the initial condition makes no difference. 00:50:20.000 --> 00:50:24.550 The function behaves always the same way as T goes to infinity.