WEBVTT 00:00:08.000 --> 00:00:14.000 For the rest of the term, we are going to be studying not 00:00:11.000 --> 00:00:17.000 just one differential equation at a time, but rather what are 00:00:15.000 --> 00:00:21.000 called systems of differential equations. 00:00:18.000 --> 00:00:24.000 Those are like systems of linear equations. 00:00:21.000 --> 00:00:27.000 They have to be solved simultaneously, 00:00:23.000 --> 00:00:29.000 in other words, not just one at a time. 00:00:25.000 --> 00:00:31.000 So, how does a system look when you write it down? 00:00:30.000 --> 00:00:36.000 Well, since we are going to be talking about systems of 00:00:34.000 --> 00:00:40.000 ordinary differential equations, there still will be only one 00:00:38.000 --> 00:00:44.000 independent variable, but there will be several 00:00:42.000 --> 00:00:48.000 dependent variables. I am going to call, 00:00:45.000 --> 00:00:51.000 let's say two. The dependent variables are 00:00:48.000 --> 00:00:54.000 going to be, I will call them x and y, and then the first order 00:00:53.000 --> 00:00:59.000 system, something involving just first derivatives, 00:00:57.000 --> 00:01:03.000 will look like this. On the left-hand side 00:01:02.000 --> 00:01:08.000 will be x prime, 00:01:04.000 --> 00:01:10.000 in other words. On the right-hand side will be 00:01:07.000 --> 00:01:13.000 the dependent variables and then also the independent variables. 00:01:12.000 --> 00:01:18.000 I will indicate that, I will separate it all from the 00:01:16.000 --> 00:01:22.000 others by putting a semicolon there. 00:01:18.000 --> 00:01:24.000 And the same way y prime, the derivative of y with 00:01:22.000 --> 00:01:28.000 respect to t, will be some other function of 00:01:25.000 --> 00:01:31.000 (x, y) and t. Let's write down explicitly 00:01:30.000 --> 00:01:36.000 that x and y are dependent variables. 00:01:37.000 --> 00:01:43.000 And what they depend upon is the independent variable t, 00:01:41.000 --> 00:01:47.000 time. A system like this is going to 00:01:43.000 --> 00:01:49.000 be called first order. And we are going to consider 00:01:47.000 --> 00:01:53.000 basically only first-order systems for a secret reason that 00:01:52.000 --> 00:01:58.000 I will explain at the end of the period. 00:01:56.000 --> 00:02:02.000 This is a first-order system, meaning that the only kind of 00:02:00.000 --> 00:02:06.000 derivatives that are up here are first derivatives. 00:02:04.000 --> 00:02:10.000 So x prime is dx over dt and so on. 00:02:08.000 --> 00:02:14.000 Now, there is still more terminology. 00:02:10.000 --> 00:02:16.000 Of course, practically all the equations after the term 00:02:15.000 --> 00:02:21.000 started, virtually all the equations we have been 00:02:18.000 --> 00:02:24.000 considering are linear equations, so it must be true 00:02:22.000 --> 00:02:28.000 that linear systems are the best kind. 00:02:25.000 --> 00:02:31.000 And, boy, they certainly are. When are we going to call a 00:02:31.000 --> 00:02:37.000 system linear? I think in the beginning you 00:02:34.000 --> 00:02:40.000 should learn a little terminology before we launch in 00:02:39.000 --> 00:02:45.000 and actually try to start to solve these things. 00:02:43.000 --> 00:02:49.000 Well, the x and y, the dependent variables must 00:02:46.000 --> 00:02:52.000 occur linearly. In other words, 00:02:49.000 --> 00:02:55.000 it must look like this, ax plus by. 00:02:53.000 --> 00:02:59.000 Now, the t can be a mess. And so I will throw in an extra 00:02:58.000 --> 00:03:04.000 function of t there. And y prime will be some 00:03:03.000 --> 00:03:09.000 other linear combination of x and y, plus some other messy 00:03:08.000 --> 00:03:14.000 function of t. But even the a, 00:03:10.000 --> 00:03:16.000 b, c, and d are allowed to be functions of t. 00:03:14.000 --> 00:03:20.000 They could be one over t cubed or sine t 00:03:19.000 --> 00:03:25.000 or something like that. So I have to distinguish those 00:03:23.000 --> 00:03:29.000 cases. The case where a, 00:03:25.000 --> 00:03:31.000 b, c, and d are constants, that I will call -- 00:03:30.000 --> 00:03:36.000 Well, there are different things you can call it. 00:03:34.000 --> 00:03:40.000 We will simply call it a constant coefficient system. 00:03:40.000 --> 00:03:46.000 A system with coefficients would probably be better 00:03:45.000 --> 00:03:51.000 English. On the other hand, 00:03:47.000 --> 00:03:53.000 a, b, c, and d, this system will still be 00:03:51.000 --> 00:03:57.000 called linear if these are functions of t. 00:03:56.000 --> 00:04:02.000 Can also be functions of t. 00:04:05.000 --> 00:04:11.000 So it would be a perfectly good linear system to have x prime 00:04:08.000 --> 00:04:14.000 equals tx plus sine t times y plus e to the minus t squared. 00:04:15.000 --> 00:04:21.000 You would never see something like that but it is okay. 00:04:18.000 --> 00:04:24.000 What else do you need to know? Well, what would a homogenous 00:04:22.000 --> 00:04:28.000 system be? A homogenous system is one 00:04:24.000 --> 00:04:30.000 without these extra guys. That doesn't mean there is no t 00:04:28.000 --> 00:04:34.000 in it. There could be t in the a, 00:04:32.000 --> 00:04:38.000 b, c and d, but these terms with no x and y in them must not 00:04:38.000 --> 00:04:44.000 occur. So, a linear homogenous. 00:04:47.000 --> 00:04:53.000 And that is the kind we are going to start studying first in 00:04:50.000 --> 00:04:56.000 the same way when we studied higher order equations. 00:04:53.000 --> 00:04:59.000 We studied first homogenous. You had to know how to solve 00:04:57.000 --> 00:05:03.000 those first, and then you could learn how to solve the more 00:05:00.000 --> 00:05:06.000 general kind. So linear homogenous means that 00:05:04.000 --> 00:05:10.000 r1 is zero and r2 is zero for all time. 00:05:07.000 --> 00:05:13.000 They are identically zero. They are not there. 00:05:10.000 --> 00:05:16.000 You don't see them. Have I left anything out? 00:05:13.000 --> 00:05:19.000 Yes, the initial conditions. Since that is quite general, 00:05:18.000 --> 00:05:24.000 let's talk about what would initial conditions look like? 00:05:28.000 --> 00:05:34.000 Well, in a general way, the reason you have to have 00:05:31.000 --> 00:05:37.000 initial conditions is to get values for the arbitrary 00:05:34.000 --> 00:05:40.000 constants that appear in the solution. 00:05:37.000 --> 00:05:43.000 The question is, how many arbitrary constants 00:05:40.000 --> 00:05:46.000 are going to appear in the solutions of these equations? 00:05:43.000 --> 00:05:49.000 Well, I will just give you the answer. 00:05:46.000 --> 00:05:52.000 Two. The number of arbitrary 00:05:48.000 --> 00:05:54.000 constants that appear is the total order of the system. 00:05:51.000 --> 00:05:57.000 For example, if this were a second 00:05:53.000 --> 00:05:59.000 derivative and this were a first derivative, I would expect three 00:05:58.000 --> 00:06:04.000 arbitrary constants in the system -- 00:06:02.000 --> 00:06:08.000 -- because the total, the sum of two and one makes 00:06:05.000 --> 00:06:11.000 three. So you must have as many 00:06:07.000 --> 00:06:13.000 initial conditions as you have arbitrary constants in the 00:06:11.000 --> 00:06:17.000 solution. And that, of course, 00:06:13.000 --> 00:06:19.000 explains when we studied second-order equations, 00:06:17.000 --> 00:06:23.000 we had to have two initial conditions. 00:06:19.000 --> 00:06:25.000 I had to specify the initial starting point and the initial 00:06:24.000 --> 00:06:30.000 velocity. And the reason we had to have 00:06:26.000 --> 00:06:32.000 two conditions was because the general solution had two 00:06:30.000 --> 00:06:36.000 arbitrary constants in it. The same thing happens here but 00:06:36.000 --> 00:06:42.000 the answer is it is more natural, the conditions here are 00:06:40.000 --> 00:06:46.000 more natural. I don't have to specify the 00:06:43.000 --> 00:06:49.000 velocity. Why not? 00:06:44.000 --> 00:06:50.000 Well, because an initial condition, of course, 00:06:48.000 --> 00:06:54.000 would want me to say what the starting value of x is, 00:06:52.000 --> 00:06:58.000 some number, and it will also want to know 00:06:55.000 --> 00:07:01.000 what the starting value of y is at that same point. 00:07:00.000 --> 00:07:06.000 Well, there are my two conditions. 00:07:02.000 --> 00:07:08.000 And since this is going to have two arbitrary constants in it, 00:07:07.000 --> 00:07:13.000 it is these initial conditions that will satisfy, 00:07:10.000 --> 00:07:16.000 the arbitrary constants will have to be picked so as to 00:07:14.000 --> 00:07:20.000 satisfy those initial conditions. 00:07:17.000 --> 00:07:23.000 In some sense, the giving of initial 00:07:19.000 --> 00:07:25.000 conditions for a system is a more natural activity than 00:07:23.000 --> 00:07:29.000 giving the initial conditions of a second order system. 00:07:29.000 --> 00:07:35.000 You don't have to be the least bit cleaver about it. 00:07:32.000 --> 00:07:38.000 Anybody would give these two numbers. 00:07:35.000 --> 00:07:41.000 Whereas, somebody faced with a second order system might 00:07:38.000 --> 00:07:44.000 scratch his head. And, in fact, 00:07:40.000 --> 00:07:46.000 there are other kinds of conditions. 00:07:43.000 --> 00:07:49.000 There are boundary conditions you learned a little bit about 00:07:47.000 --> 00:07:53.000 instead of initial conditions for a second order equation. 00:07:51.000 --> 00:07:57.000 I cannot think of any more general terminology, 00:07:54.000 --> 00:08:00.000 so it sounds like we are going to actually have to get to work. 00:08:00.000 --> 00:08:06.000 Okay, let's get to work. I want to set up a system and 00:08:04.000 --> 00:08:10.000 solve it. And since one of the things in 00:08:07.000 --> 00:08:13.000 this course is supposed to be simple modeling, 00:08:10.000 --> 00:08:16.000 it should be a system that models something. 00:08:13.000 --> 00:08:19.000 In general, the kinds of models we are going to use when we 00:08:18.000 --> 00:08:24.000 study systems are the same ones we used in studying just 00:08:22.000 --> 00:08:28.000 first-order equations. Mixing, radioactive decay, 00:08:25.000 --> 00:08:31.000 temperature, the motion of temperature. 00:08:30.000 --> 00:08:36.000 Heat, heat conduction, in other words. 00:08:32.000 --> 00:08:38.000 Diffusion. I have given you a diffusion 00:08:35.000 --> 00:08:41.000 problem for your first homework on this subject. 00:08:39.000 --> 00:08:45.000 What else did we do? That's all I can think of for 00:08:43.000 --> 00:08:49.000 the moment, but I am sure they will occur to me. 00:08:46.000 --> 00:08:52.000 When, out of those physical ideas, are we going to get a 00:08:50.000 --> 00:08:56.000 system? The answer is, 00:08:52.000 --> 00:08:58.000 whenever there are two of something that there was only 00:08:56.000 --> 00:09:02.000 one of before. For example, 00:08:59.000 --> 00:09:05.000 if I have mixing with two tanks where the fluid goes like that. 00:09:03.000 --> 00:09:09.000 Say you want to have a big tank and a little tank here and you 00:09:07.000 --> 00:09:13.000 want to put some stuff into the little tank so that it will get 00:09:10.000 --> 00:09:16.000 mixed in the big tank without having to climb a big ladder and 00:09:14.000 --> 00:09:20.000 stop and drop the stuff in. That will require two tanks, 00:09:17.000 --> 00:09:23.000 the concentration of the substance in each tank, 00:09:20.000 --> 00:09:26.000 therefore, that will require a system of equations rather than 00:09:24.000 --> 00:09:30.000 just one. Or, to give something closer to 00:09:28.000 --> 00:09:34.000 home, closer to this backboard, anyway, suppose you have dah, 00:09:33.000 --> 00:09:39.000 dah, dah, don't groan, at least not audibly, 00:09:36.000 --> 00:09:42.000 something that looks like that. And next to it put an EMF 00:09:40.000 --> 00:09:46.000 there. That is just a first order. 00:09:43.000 --> 00:09:49.000 That just leads to a single first order equation. 00:09:47.000 --> 00:09:53.000 But suppose it is a two loop circuit. 00:09:58.000 --> 00:10:04.000 Now I need a pair of equations. Each of these loops gives a 00:10:02.000 --> 00:10:08.000 first order differential equation, but they have to be 00:10:06.000 --> 00:10:12.000 solved simultaneously to find the current or the charges on 00:10:10.000 --> 00:10:16.000 the condensers. And if I want a system of three 00:10:14.000 --> 00:10:20.000 equations, throw in another loop. 00:10:16.000 --> 00:10:22.000 Now, suppose I put in a coil instead. 00:10:19.000 --> 00:10:25.000 What is this going to lead to? This is going to give me a 00:10:23.000 --> 00:10:29.000 system of three equations of which this will be first order, 00:10:27.000 --> 00:10:33.000 first order. And this will be second order 00:10:32.000 --> 00:10:38.000 because it has a coil. You are up to that, 00:10:35.000 --> 00:10:41.000 right? You've had coils, 00:10:37.000 --> 00:10:43.000 inductance? Good. 00:10:39.000 --> 00:10:45.000 So the whole thing is going to count as first-order, 00:10:43.000 --> 00:10:49.000 first-order, second-order. 00:10:45.000 --> 00:10:51.000 To find out how complicated it is, you have to add up the 00:10:50.000 --> 00:10:56.000 orders. That is one and one, 00:10:52.000 --> 00:10:58.000 and two. This is really fourth-order 00:10:55.000 --> 00:11:01.000 stuff that we are talking about here. 00:11:00.000 --> 00:11:06.000 We can expect it to be a little complicated. 00:11:03.000 --> 00:11:09.000 Well, now let's take a modest little problem. 00:11:06.000 --> 00:11:12.000 I am going to return to a problem we considered earlier in 00:11:10.000 --> 00:11:16.000 the problem of heat conduction. I had forgotten whether it was 00:11:14.000 --> 00:11:20.000 on the problem set or I did it in class, but I am choosing it 00:11:19.000 --> 00:11:25.000 because it leads to something we will be able to solve. 00:11:23.000 --> 00:11:29.000 And because it illustrates how to add a little sophistication 00:11:27.000 --> 00:11:33.000 to something that was unsophisticated before. 00:11:32.000 --> 00:11:38.000 A pot of water. External temperature Te of t. 00:11:35.000 --> 00:11:41.000 I am talking about the 00:11:38.000 --> 00:11:44.000 temperature of something. And what I am talking about the 00:11:43.000 --> 00:11:49.000 temperature of will be an egg that is cooking inside, 00:11:47.000 --> 00:11:53.000 but with a difference. This egg is not homogenous 00:11:51.000 --> 00:11:57.000 inside. Instead it has a white and it 00:11:54.000 --> 00:12:00.000 has a yolk in the middle. In other words, 00:11:59.000 --> 00:12:05.000 it is a real egg and not a phony egg. 00:12:01.000 --> 00:12:07.000 That is a small pot, or it is an ostrich egg. 00:12:04.000 --> 00:12:10.000 [LAUGHTER] That is the yoke. The yolk is contained in a 00:12:08.000 --> 00:12:14.000 little membrane inside. And there are little yucky 00:12:11.000 --> 00:12:17.000 things that hold it in position. And we are going to let the 00:12:15.000 --> 00:12:21.000 temperature of the yolk, if you can see in the back of 00:12:19.000 --> 00:12:25.000 the room, be T1. That is the temperature of the 00:12:22.000 --> 00:12:28.000 yolk. The temperature of the white, 00:12:25.000 --> 00:12:31.000 which we will assume is uniform, is going to be T2. 00:12:30.000 --> 00:12:36.000 Oh, that's the water bath. The temperature of the white is 00:12:34.000 --> 00:12:40.000 T2, and then the temperature of the external water bath. 00:12:39.000 --> 00:12:45.000 In other words, the reason for introducing two 00:12:42.000 --> 00:12:48.000 variables instead of just the one variable for the overall 00:12:47.000 --> 00:12:53.000 temperature of the egg we had is because egg white is liquid pure 00:12:52.000 --> 00:12:58.000 protein, more or less, and the T1, the yolk has a lot 00:12:57.000 --> 00:13:03.000 of fat and cholesterol and other stuff like that which is 00:13:01.000 --> 00:13:07.000 supposed to be bad for you. It certainly has different 00:13:06.000 --> 00:13:12.000 conducting. It is liquid, 00:13:07.000 --> 00:13:13.000 at the beginning at any rate, but it certainly has different 00:13:11.000 --> 00:13:17.000 constants of conductivity than the egg white would. 00:13:14.000 --> 00:13:20.000 And the condition of heat through the shell of the egg 00:13:17.000 --> 00:13:23.000 would be different from the conduction of heat through the 00:13:20.000 --> 00:13:26.000 membrane that keeps the yoke together. 00:13:22.000 --> 00:13:28.000 So it is quite reasonable to consider that the white and the 00:13:26.000 --> 00:13:32.000 yolk will be at different temperatures and will have 00:13:29.000 --> 00:13:35.000 different conductivity properties. 00:13:32.000 --> 00:13:38.000 I am going to use Newton's laws but with this further 00:13:36.000 --> 00:13:42.000 refinement. In other words, 00:13:38.000 --> 00:13:44.000 introducing two temperatures. Whereas, before we only had one 00:13:43.000 --> 00:13:49.000 temperature. But let's use Newton's law. 00:13:46.000 --> 00:13:52.000 Let's see. The question is how does T1, 00:13:49.000 --> 00:13:55.000 the temperature of the yolk, vary with time? 00:13:52.000 --> 00:13:58.000 Well, the yolk is getting all its heat from the white. 00:13:57.000 --> 00:14:03.000 Therefore, Newton's law of conduction will be some constant 00:14:01.000 --> 00:14:07.000 of conductivity for the yolk times T2 minus T1. 00:14:08.000 --> 00:14:14.000 The yolk does not know anything about the external temperature 00:14:12.000 --> 00:14:18.000 of the water bath. It is completely surrounded, 00:14:15.000 --> 00:14:21.000 snug and secure within itself. But how about the temperature 00:14:20.000 --> 00:14:26.000 of the egg white? That gets heat and gives heat 00:14:23.000 --> 00:14:29.000 to two sources, from the external water and 00:14:26.000 --> 00:14:32.000 also from the internal yolk inside. 00:14:30.000 --> 00:14:36.000 So you have to take into account both of those. 00:14:33.000 --> 00:14:39.000 Its conduction of the heat through that membrane, 00:14:36.000 --> 00:14:42.000 we will use the same a, which is going to be a times T1 00:14:40.000 --> 00:14:46.000 minus T2. Remember the order in which you 00:14:44.000 --> 00:14:50.000 have to write these is governed by the yolk outside to the 00:14:48.000 --> 00:14:54.000 white. Therefore, that has to come 00:14:51.000 --> 00:14:57.000 first when I write it in order that a be a positive constant. 00:14:55.000 --> 00:15:01.000 But it is also getting heat from the water bath. 00:15:00.000 --> 00:15:06.000 And, presumably, the conductivity through the 00:15:03.000 --> 00:15:09.000 shell is different from what it is through this membrane around 00:15:08.000 --> 00:15:14.000 the yolk. So I am going to call that by a 00:15:11.000 --> 00:15:17.000 different constant. This is the conductivity 00:15:14.000 --> 00:15:20.000 through the shell into the white. 00:15:17.000 --> 00:15:23.000 And that is going to be T, the external temperature minus 00:15:21.000 --> 00:15:27.000 the temperature of the egg white. 00:15:24.000 --> 00:15:30.000 Here I have a system of equations because I want to make 00:15:28.000 --> 00:15:34.000 two dependent variables by refining the original problem. 00:15:34.000 --> 00:15:40.000 Now, you always have to write a system in standard form to solve 00:15:39.000 --> 00:15:45.000 it. You will see that the left-hand 00:15:42.000 --> 00:15:48.000 side will give the dependent variables in a certain order. 00:15:47.000 --> 00:15:53.000 In this case, the temperature of the yolk and 00:15:51.000 --> 00:15:57.000 then the temperature of the white. 00:15:54.000 --> 00:16:00.000 The law is that in order not to make mistakes -- 00:16:00.000 --> 00:16:06.000 And it's a very frequent source of error so learn from the 00:16:03.000 --> 00:16:09.000 beginning not to do this. You must write the variables on 00:16:07.000 --> 00:16:13.000 the right-hand side in the same order left to right in which 00:16:11.000 --> 00:16:17.000 they occur top to bottom here. In other words, 00:16:14.000 --> 00:16:20.000 this is not a good way to leave that. 00:16:16.000 --> 00:16:22.000 This is the first attempt in writing this system, 00:16:20.000 --> 00:16:26.000 but the final version should like this. 00:16:22.000 --> 00:16:28.000 T1 prime, I won't bother writing dT / dt, 00:16:25.000 --> 00:16:31.000 is equal to -- T1 must come first, 00:16:29.000 --> 00:16:35.000 so minus a times T1 plus a times T2. 00:16:34.000 --> 00:16:40.000 And the same law for the second one. 00:16:38.000 --> 00:16:44.000 It must come in the same order. Now, the coefficient of T1, 00:16:44.000 --> 00:16:50.000 that is easy. That's a times T1. 00:16:47.000 --> 00:16:53.000 The coefficient of T2 is minus a minus b, 00:16:52.000 --> 00:16:58.000 so minus (a plus b) times T2. 00:16:56.000 --> 00:17:02.000 But I am not done yet. There is still this external 00:17:02.000 --> 00:17:08.000 temperature I must put into the equation. 00:17:06.000 --> 00:17:12.000 Now, that is not a variable. This is some given function of 00:17:11.000 --> 00:17:17.000 t. And what the function of t is, 00:17:14.000 --> 00:17:20.000 of course, depends upon what the problem is. 00:17:18.000 --> 00:17:24.000 So that, for example, what might be some 00:17:22.000 --> 00:17:28.000 possibilities, well, suppose the problem was I 00:17:26.000 --> 00:17:32.000 wanted to coddle the egg. I think there is a generation 00:17:32.000 --> 00:17:38.000 gap here. How many of you know what a 00:17:35.000 --> 00:17:41.000 coddled egg is? How many of you don't know? 00:17:38.000 --> 00:17:44.000 Well, I'm just saying my daughter didn't know. 00:17:42.000 --> 00:17:48.000 I mentioned it to her. I said I think I'm going to do 00:17:46.000 --> 00:17:52.000 a coddled egg tomorrow in class. And she said what is that? 00:17:51.000 --> 00:17:57.000 And so I said a cuddled egg? She said why would someone 00:17:55.000 --> 00:18:01.000 cuddle an egg? I said coddle. 00:17:59.000 --> 00:18:05.000 And she said, oh, you mean like a person, 00:18:02.000 --> 00:18:08.000 like what you do to somebody you like or don't like or I 00:18:07.000 --> 00:18:13.000 don't know. Whatever. 00:18:09.000 --> 00:18:15.000 I thought a while and said, yeah, more like that. 00:18:13.000 --> 00:18:19.000 [LAUGHTER] Anyway, for the enrichment of your 00:18:17.000 --> 00:18:23.000 cooking skills, to coddle an egg, 00:18:20.000 --> 00:18:26.000 it is considered to produce a better quality product than 00:18:25.000 --> 00:18:31.000 boiling an egg. That is why people do it. 00:18:30.000 --> 00:18:36.000 You heat up the water to boiling, the egg should be at 00:18:34.000 --> 00:18:40.000 room temperature, and then you carefully lower 00:18:37.000 --> 00:18:43.000 the egg into the water. And you turn off the heat so 00:18:41.000 --> 00:18:47.000 the water bath cools exponentially while the egg 00:18:45.000 --> 00:18:51.000 inside is rising in temperature. And then you wait four minutes 00:18:50.000 --> 00:18:56.000 or six minutes or whatever and take it out. 00:18:53.000 --> 00:18:59.000 You have a perfect egg. So for coddling, 00:18:56.000 --> 00:19:02.000 spelled so, what will the external temperature be? 00:19:02.000 --> 00:19:08.000 Well, it starts out at time zero at 100 degrees centigrade 00:19:06.000 --> 00:19:12.000 because the water is supposed to be boiling. 00:19:09.000 --> 00:19:15.000 The reason you have it boiling is for calibration so that you 00:19:13.000 --> 00:19:19.000 can know what temperature it is without having to use a 00:19:17.000 --> 00:19:23.000 thermometer, unless you're on Pike's Peak or some place. 00:19:20.000 --> 00:19:26.000 It starts out at 100 degrees. And after that, 00:19:24.000 --> 00:19:30.000 since the light is off, it cools exponential because 00:19:27.000 --> 00:19:33.000 that is another law. You only have to know what K is 00:19:32.000 --> 00:19:38.000 for your particular pot and you will be able to solve the 00:19:37.000 --> 00:19:43.000 coddled egg problem. In other words, 00:19:40.000 --> 00:19:46.000 you will then be able to solve these equations and know how the 00:19:45.000 --> 00:19:51.000 temperature rises. I am going to solve a different 00:19:49.000 --> 00:19:55.000 problem because I don't want to have to deal with this 00:19:54.000 --> 00:20:00.000 inhomogeneous term. Let's use, as a different 00:19:58.000 --> 00:20:04.000 problem, a person cooks an egg. Coddles the egg by the first 00:20:04.000 --> 00:20:10.000 process, decides the egg is done, let's say hardboiled, 00:20:09.000 --> 00:20:15.000 and then you are supposed to drop a hardboiled egg into cold 00:20:14.000 --> 00:20:20.000 water. Not just to cool it but also 00:20:17.000 --> 00:20:23.000 because I think it prevents that dark thing from forming that 00:20:23.000 --> 00:20:29.000 looks sort of unattractive. Let's ice bath. 00:20:28.000 --> 00:20:34.000 The only reason for dropping the egg into an ice bath is so 00:20:32.000 --> 00:20:38.000 that you could have a homogenous equation to solve. 00:20:36.000 --> 00:20:42.000 And since this a first system we are going to solve, 00:20:40.000 --> 00:20:46.000 let's make life easy for ourselves. 00:20:43.000 --> 00:20:49.000 Now, all my work in preparing this example, 00:20:47.000 --> 00:20:53.000 and it took considerably longer time than actually solving the 00:20:52.000 --> 00:20:58.000 problem, was in picking values for a and b which would make 00:20:56.000 --> 00:21:02.000 everything come out nice. It's harder than it looks. 00:21:02.000 --> 00:21:08.000 The values that we are going to use, which make no physical 00:21:07.000 --> 00:21:13.000 sense whatsoever, but a equals 2 and b 00:21:11.000 --> 00:21:17.000 equals 3. These are called nice numbers. 00:21:15.000 --> 00:21:21.000 What is the equation? What is the system? 00:21:18.000 --> 00:21:24.000 Can somebody read it off for me? 00:21:21.000 --> 00:21:27.000 It is T1 prime equals, what is it, minus 2T1 plus 2T2. 00:21:26.000 --> 00:21:32.000 That's good. 00:21:30.000 --> 00:21:36.000 Minus 2T1 plus 2T2. 00:21:40.000 --> 00:21:46.000 T2 prime is, what is it? 00:21:42.000 --> 00:21:48.000 I think this is 2T1. And the other one is minus a 00:21:48.000 --> 00:21:54.000 plus b, so minus 5. 00:21:51.000 --> 00:21:57.000 This is a system. Now, on Wednesday I will teach 00:21:57.000 --> 00:22:03.000 you a fancy way of solving this. But, to be honest, 00:22:03.000 --> 00:22:09.000 the fancy way will take roughly about as long as the way I am 00:22:07.000 --> 00:22:13.000 going to do it now. The main reason for doing it is 00:22:10.000 --> 00:22:16.000 that it introduces new vocabulary which everyone wants 00:22:14.000 --> 00:22:20.000 you to have. And also, more important 00:22:16.000 --> 00:22:22.000 reasons, it gives more insight into the solution than this 00:22:20.000 --> 00:22:26.000 method. This method just produces the 00:22:22.000 --> 00:22:28.000 answer, but you want insight, also. 00:22:24.000 --> 00:22:30.000 And that is just as important. But for now, 00:22:28.000 --> 00:22:34.000 let's use a method which always works and which in 40 years, 00:22:33.000 --> 00:22:39.000 after you have forgotten all other fancy methods, 00:22:36.000 --> 00:22:42.000 will still be available to you because it is method you can 00:22:40.000 --> 00:22:46.000 figure out yourself. You don't have to remember 00:22:43.000 --> 00:22:49.000 anything. The method is to eliminate one 00:22:46.000 --> 00:22:52.000 of the dependent variables. It is just the way you solve 00:22:50.000 --> 00:22:56.000 systems of linear equations in general if you aren't doing 00:22:54.000 --> 00:23:00.000 something fancy with determinants and matrices. 00:22:59.000 --> 00:23:05.000 If you just eliminate variables. 00:23:01.000 --> 00:23:07.000 We are going to eliminate one of these variables. 00:23:05.000 --> 00:23:11.000 Let's eliminate T2. You could also eliminate T1. 00:23:08.000 --> 00:23:14.000 The main thing is eliminate one of them so you will have just 00:23:13.000 --> 00:23:19.000 one left to work with. How do I eliminate T2? 00:23:16.000 --> 00:23:22.000 Beg your pardon? Is something wrong? 00:23:19.000 --> 00:23:25.000 If somebody thinks something is wrong raise his hand. 00:23:23.000 --> 00:23:29.000 No? 00:23:30.000 --> 00:23:36.000 Why do I want to get rid of T1? Well, I can add them. 00:23:33.000 --> 00:23:39.000 But, on the left-hand side, I will have T1 prime plus T2 00:23:36.000 --> 00:23:42.000 prime. What good is that? 00:23:39.000 --> 00:23:45.000 [LAUGHTER] 00:23:48.000 --> 00:23:54.000 I think you will want to do it my way. 00:23:49.000 --> 00:23:55.000 [APPLAUSE] 00:24:03.000 --> 00:24:09.000 Solve for T2 in terms of T1. That is going to be T1 prime 00:24:08.000 --> 00:24:14.000 plus 2T1 divided by 2. 00:24:12.000 --> 00:24:18.000 Now, take that and substitute it into the second equation. 00:24:18.000 --> 00:24:24.000 Wherever you see a T2, put that in, 00:24:21.000 --> 00:24:27.000 and what you will be left with is something just in T1. 00:24:28.000 --> 00:24:34.000 To be honest, I don't know any other good way 00:24:31.000 --> 00:24:37.000 of doing this. There is a fancy method that I 00:24:34.000 --> 00:24:40.000 think is talked about in your book, which leads to extraneous 00:24:39.000 --> 00:24:45.000 solutions and so on, but you don't want to know 00:24:43.000 --> 00:24:49.000 about that. This will work for a simple 00:24:46.000 --> 00:24:52.000 linear equation with constant coefficients, 00:24:49.000 --> 00:24:55.000 always. Substitute in. 00:24:51.000 --> 00:24:57.000 What do I do? Now, here I do not advise doing 00:24:54.000 --> 00:25:00.000 this mentally. It is just too easy to make a 00:24:57.000 --> 00:25:03.000 mistake. Here, I will do it carefully, 00:25:04.000 --> 00:25:10.000 writing everything out just as you would. 00:25:10.000 --> 00:25:16.000 T1 prime plus 2T1 over 2, prime, equals 2T1 minus 5 time 00:25:18.000 --> 00:25:24.000 T1 prime plus 2T1 over two. 00:25:27.000 --> 00:25:33.000 I took that and substituted 00:25:32.000 --> 00:25:38.000 into this equation. Now, I don't like those two's. 00:25:38.000 --> 00:25:44.000 Let's get rid of them by multiplying. 00:25:42.000 --> 00:25:48.000 This will become 4. 00:25:52.000 --> 00:25:58.000 And now write this out. What is this when you look at 00:25:57.000 --> 00:26:03.000 it? This is an equation just in T1. 00:26:00.000 --> 00:26:06.000 It has constant coefficients. And what is its order? 00:26:05.000 --> 00:26:11.000 Its order is two because T1 prime primed. 00:26:10.000 --> 00:26:16.000 In other words, I can eliminate T2 okay, 00:26:13.000 --> 00:26:19.000 but the equation I am going to get is no longer a first-order. 00:26:19.000 --> 00:26:25.000 It becomes a second-order differential equation. 00:26:24.000 --> 00:26:30.000 And that's a basic law. Even if you have a system of 00:26:30.000 --> 00:26:36.000 more equations, three or four or whatever, 00:26:33.000 --> 00:26:39.000 the law is that after you do the elimination successfully and 00:26:37.000 --> 00:26:43.000 end up with a single equation, normally the order of that 00:26:42.000 --> 00:26:48.000 equation will be the sum of the orders of the things you started 00:26:46.000 --> 00:26:52.000 with. So two first-order equations 00:26:49.000 --> 00:26:55.000 will always produce a second-order equation in just 00:26:53.000 --> 00:26:59.000 one dependent variable, three will produce a third 00:26:56.000 --> 00:27:02.000 order equation and so on. So you trade one complexity for 00:27:02.000 --> 00:27:08.000 another. You trade the complexity of 00:27:04.000 --> 00:27:10.000 having to deal with two equations simultaneously instead 00:27:09.000 --> 00:27:15.000 of just one for the complexity of having to deal with a single 00:27:13.000 --> 00:27:19.000 higher order equation which is more trouble to solve. 00:27:17.000 --> 00:27:23.000 It is like all mathematical problems. 00:27:20.000 --> 00:27:26.000 Unless you are very lucky, if you push them down one way, 00:27:24.000 --> 00:27:30.000 they are really simple now, they just pop up some place 00:27:28.000 --> 00:27:34.000 else. You say, oh, 00:27:30.000 --> 00:27:36.000 I didn't save anything after all. 00:27:32.000 --> 00:27:38.000 That is the law of conservation of mathematical difficulty. 00:27:36.000 --> 00:27:42.000 [LAUGHTER] You saw that even with the Laplace transform. 00:27:40.000 --> 00:27:46.000 In the beginning it looks great, you've got these tables, 00:27:44.000 --> 00:27:50.000 take the equation, horrible to solve. 00:27:46.000 --> 00:27:52.000 Take some transform, trivial to solve for capital Y. 00:27:50.000 --> 00:27:56.000 Now I have to find the inverse Laplace transform. 00:27:53.000 --> 00:27:59.000 And suddenly all the work is there, partial fractions, 00:27:57.000 --> 00:28:03.000 funny formulas and so on. It is very hard in mathematics 00:28:02.000 --> 00:28:08.000 to get away with something. It happens now and then and 00:28:06.000 --> 00:28:12.000 everybody cheers. Let's write this out now in the 00:28:09.000 --> 00:28:15.000 form in which it looks like an equation we can actually solve. 00:28:13.000 --> 00:28:19.000 Just be careful. Now it is all right to use the 00:28:17.000 --> 00:28:23.000 method by which you collect terms. 00:28:19.000 --> 00:28:25.000 There is only one term involving T1 double prime. 00:28:23.000 --> 00:28:29.000 It's the one that comes from here. 00:28:25.000 --> 00:28:31.000 How about the terms in T1 prime? 00:28:27.000 --> 00:28:33.000 There is a 2. Here, there is minus 5 T1 00:28:33.000 --> 00:28:39.000 prime. If I put it on the other side 00:28:37.000 --> 00:28:43.000 it makes plus 5 T1 prime plus this two makes 7 T1 prime. 00:28:44.000 --> 00:28:50.000 And how many T1's are there? Well, none on the left-hand 00:28:51.000 --> 00:28:57.000 side. On the right-hand side I have 4 00:28:55.000 --> 00:29:01.000 here minus 10. 4 minus 10 is negative 6. 00:29:02.000 --> 00:29:08.000 Negative 6 T1 put on this left-hand side the way we want 00:29:06.000 --> 00:29:12.000 to do makes plus 6 T1. 00:29:15.000 --> 00:29:21.000 There are no inhomogeneous terms, so that is equal to zero. 00:29:18.000 --> 00:29:24.000 If I had gotten a negative number for one of these 00:29:22.000 --> 00:29:28.000 coefficients, I would instantly know if I had 00:29:25.000 --> 00:29:31.000 made a mistake. Why? 00:29:26.000 --> 00:29:32.000 Why must those numbers come out to be positive? 00:29:30.000 --> 00:29:36.000 It is because the system must be, the system must be, 00:29:33.000 --> 00:29:39.000 fill in with one word, stable. 00:29:36.000 --> 00:29:42.000 And why must this system be stable? 00:29:38.000 --> 00:29:44.000 In other words, the long-term solutions must be 00:29:42.000 --> 00:29:48.000 zero, must all go to zero, whatever they are. 00:29:45.000 --> 00:29:51.000 Why is that? Well, because you are putting 00:29:48.000 --> 00:29:54.000 the egg into an ice bath. Or, because we know it was 00:29:52.000 --> 00:29:58.000 living but after being hardboiled it is dead and, 00:29:56.000 --> 00:30:02.000 therefore, dead systems are stable. 00:30:00.000 --> 00:30:06.000 That's not a good reason but it is, so to speak, 00:30:03.000 --> 00:30:09.000 the real one. It's clear anyway that all 00:30:05.000 --> 00:30:11.000 solutions must tend to zero physically. 00:30:08.000 --> 00:30:14.000 That's obvious. And, therefore, 00:30:10.000 --> 00:30:16.000 the differential equation must have the same property, 00:30:14.000 --> 00:30:20.000 and that means that its coefficients must be positive. 00:30:17.000 --> 00:30:23.000 All its coefficients must be positive. 00:30:20.000 --> 00:30:26.000 If this weren't there, I would get oscillating 00:30:23.000 --> 00:30:29.000 solutions, which wouldn't go to zero. 00:30:25.000 --> 00:30:31.000 That is physical impossible for this egg. 00:30:30.000 --> 00:30:36.000 Now the rest is just solving. The characteristic equation, 00:30:34.000 --> 00:30:40.000 if you can remember way, way back in prehistoric times 00:30:39.000 --> 00:30:45.000 when we were solving these equations, is this. 00:30:43.000 --> 00:30:49.000 And what you want to do is factor it. 00:30:46.000 --> 00:30:52.000 This is where all the work was, getting those numbers so that 00:30:51.000 --> 00:30:57.000 this would factor. So it's r plus 1 times r plus 6 00:31:04.000 --> 00:31:10.000 And so the solutions are, the roots are r equals 00:31:07.000 --> 00:31:13.000 negative 1. I am just making marks on the 00:31:10.000 --> 00:31:16.000 board, but you have done this often enough, 00:31:13.000 --> 00:31:19.000 you know what I am talking about. 00:31:15.000 --> 00:31:21.000 So the characteristic roots are those two numbers. 00:31:18.000 --> 00:31:24.000 And, therefore, the solution is, 00:31:20.000 --> 00:31:26.000 I could write down immediately with its arbitrary constant as 00:31:24.000 --> 00:31:30.000 c1 times e to the negative t plus c2 times e to the negative 00:31:28.000 --> 00:31:34.000 6t. Now, I have got to get T2. 00:31:34.000 --> 00:31:40.000 Here the first worry is T2 is going to give me two more 00:31:39.000 --> 00:31:45.000 arbitrary constants. It better not. 00:31:42.000 --> 00:31:48.000 The system is only allowed to have two arbitrary constants in 00:31:47.000 --> 00:31:53.000 its solution because that is the initial conditions we are giving 00:31:52.000 --> 00:31:58.000 it. By the way, I forgot to give 00:31:55.000 --> 00:32:01.000 initial conditions. Let's give initial conditions. 00:32:01.000 --> 00:32:07.000 Let's say the initial temperature of the yolk, 00:32:05.000 --> 00:32:11.000 when it is put in the ice bath, is 40 degrees centigrade, 00:32:10.000 --> 00:32:16.000 Celsius. And T2, let's say the white 00:32:13.000 --> 00:32:19.000 ought to be a little hotter than the yolk is always cooler than 00:32:18.000 --> 00:32:24.000 the white for a soft boiled egg, I don't know, 00:32:22.000 --> 00:32:28.000 or a hardboiled egg if it hasn't been chilled too long. 00:32:27.000 --> 00:32:33.000 Let's make this 45. Realistic numbers. 00:32:32.000 --> 00:32:38.000 Now, the thing not to do is to say, hey, I found T1. 00:32:35.000 --> 00:32:41.000 Okay, I will find T2 by the same procedure. 00:32:39.000 --> 00:32:45.000 I will go through the whole thing. 00:32:41.000 --> 00:32:47.000 I will eliminate T1 instead. Then I will end up with an 00:32:45.000 --> 00:32:51.000 equation T2 and I will solve that and get T2 equals blah, 00:32:50.000 --> 00:32:56.000 blah, blah. That is no good, 00:32:52.000 --> 00:32:58.000 A, because you are working too hard and, B, because you are 00:32:56.000 --> 00:33:02.000 going to get two more arbitrary constants unrelated to these 00:33:01.000 --> 00:33:07.000 two. And that is no good. 00:33:04.000 --> 00:33:10.000 Because the correct solution only has two constants in it. 00:33:09.000 --> 00:33:15.000 Not four. So that procedure is wrong. 00:33:12.000 --> 00:33:18.000 You must calculate T2 from the T1 that you found, 00:33:15.000 --> 00:33:21.000 and that is the equation which does it. 00:33:18.000 --> 00:33:24.000 That's the one we have to have. Where is the chalk? 00:33:22.000 --> 00:33:28.000 Yes. Maybe I can have a little thing 00:33:25.000 --> 00:33:31.000 so I can just carry this around with me. 00:33:37.000 --> 00:33:43.000 That is the relation between T2 and T1. 00:33:40.000 --> 00:33:46.000 Or, if you don't like it, either one of these equations 00:33:44.000 --> 00:33:50.000 will express T2 in terms of T1 for you. 00:33:47.000 --> 00:33:53.000 It doesn't matter. Whichever one you use, 00:33:50.000 --> 00:33:56.000 however you do it, that's the way you must 00:33:53.000 --> 00:33:59.000 calculate T2. So what is it? 00:33:56.000 --> 00:34:02.000 T2 is calculated from that pink box. 00:34:00.000 --> 00:34:06.000 It is one-half of T1 prime plus T1. 00:34:05.000 --> 00:34:11.000 Now, if I take the derivative of this, I get minus c1 times 00:34:11.000 --> 00:34:17.000 the exponential. The coefficient is minus c1, 00:34:16.000 --> 00:34:22.000 take half of that, that is minus a half c1 00:34:21.000 --> 00:34:27.000 and add it to T1. Minus one-half c1 plus c1 gives 00:34:26.000 --> 00:34:32.000 me one-half c1. 00:34:32.000 --> 00:34:38.000 And here I take the derivative, it is minus 6 c2. 00:34:38.000 --> 00:34:44.000 Take half of that, minus 3 c2 and add this c2 to 00:34:44.000 --> 00:34:50.000 it, minus 3 plus 1 makes minus 2. 00:34:48.000 --> 00:34:54.000 That is T2. And notice it uses the same 00:34:53.000 --> 00:34:59.000 arbitrary constants that T1 uses. 00:34:59.000 --> 00:35:05.000 So we end up with just two because we calculated T2 from 00:35:02.000 --> 00:35:08.000 that formula or from the equation which is equivalent to 00:35:06.000 --> 00:35:12.000 it, not from scratch. We haven't put in the initial 00:35:09.000 --> 00:35:15.000 conditions yet, but that is easy to do. 00:35:11.000 --> 00:35:17.000 Everybody, when working with exponentials, 00:35:14.000 --> 00:35:20.000 of course, you always want the initial conditions to be when T 00:35:18.000 --> 00:35:24.000 is equal to zero because that makes all the 00:35:21.000 --> 00:35:27.000 exponentials one and you don't have to worry about them. 00:35:25.000 --> 00:35:31.000 But this you know. If I put in the initial 00:35:27.000 --> 00:35:33.000 conditions, at time zero, T1 has the value 40. 00:35:32.000 --> 00:35:38.000 So 40 should be equal to c1 + c2. 00:35:38.000 --> 00:35:44.000 And the other equation will say that 45 is equal to one-half c1 00:35:45.000 --> 00:35:51.000 minus 2 c2. Now we are supposed to 00:35:52.000 --> 00:35:58.000 solve these. Well, this is called solving 00:35:57.000 --> 00:36:03.000 simultaneous linear equations. We could use Kramer's rule, 00:36:05.000 --> 00:36:11.000 inverse matrices, but why don't we just 00:36:09.000 --> 00:36:15.000 eliminate. Let me see. 00:36:12.000 --> 00:36:18.000 If I multiply by, 45, so multiply by two, 00:36:17.000 --> 00:36:23.000 you get 90 equals c1 minus 4 c2. 00:36:23.000 --> 00:36:29.000 Then subtract this guy from that guy. 00:36:27.000 --> 00:36:33.000 So, 40 taken from 90 makes 50. And c1 taken from c1, 00:36:35.000 --> 00:36:41.000 because I multiplied by two, makes zero. 00:36:40.000 --> 00:36:46.000 And c2 taken from minus 4 c2, that makes minus 5 c2, 00:36:47.000 --> 00:36:53.000 I guess. 00:36:49.000 --> 00:36:55.000 I seem to get c2 is equal to negative 10. 00:36:56.000 --> 00:37:02.000 And if c2 is negative 10, then c1 must be 50. 00:37:04.000 --> 00:37:10.000 There are two ways of checking the answer. 00:37:07.000 --> 00:37:13.000 One is to plug it into the equations, and the other is to 00:37:13.000 --> 00:37:19.000 peak. Yes, that's right. 00:37:15.000 --> 00:37:21.000 [LAUGHTER] 00:37:25.000 --> 00:37:31.000 The final answer is, in other words, 00:37:27.000 --> 00:37:33.000 you put a 50 here, 25 there, negative 10 here, 00:37:30.000 --> 00:37:36.000 and positive 20 there. That gives the answer to the 00:37:34.000 --> 00:37:40.000 problem. It tells you, 00:37:35.000 --> 00:37:41.000 in other words, how the temperature of the yolk 00:37:39.000 --> 00:37:45.000 varies with time and how the temperature of the white varies 00:37:43.000 --> 00:37:49.000 with time. As I said, we are going to 00:37:46.000 --> 00:37:52.000 learn a slick way of doing this problem, or at least a very 00:37:51.000 --> 00:37:57.000 different way of doing the same problem next time, 00:37:54.000 --> 00:38:00.000 but let's put that on ice for the moment. 00:37:57.000 --> 00:38:03.000 And instead I would like to spend the rest of the period 00:38:01.000 --> 00:38:07.000 doing for first order systems the same thing that I did for 00:38:05.000 --> 00:38:11.000 you the very first day of the term. 00:38:09.000 --> 00:38:15.000 Remember, I walked in assuming that you knew how to separate 00:38:13.000 --> 00:38:19.000 variables the first day of the term, and I did not talk to you 00:38:17.000 --> 00:38:23.000 about how to solve fancier equations by fancier methods. 00:38:21.000 --> 00:38:27.000 I instead talked to you about the geometric significance, 00:38:25.000 --> 00:38:31.000 what the geometric meaning of a single first order equation was 00:38:29.000 --> 00:38:35.000 and how that geometric meaning enabled you to solve it 00:38:33.000 --> 00:38:39.000 numerically. And we spent a little while 00:38:36.000 --> 00:38:42.000 working on such problems because nowadays with computers it is 00:38:40.000 --> 00:38:46.000 really important that you get a feeling for what these things 00:38:44.000 --> 00:38:50.000 mean as opposed to just algorithms for solving them. 00:38:47.000 --> 00:38:53.000 As I say, most differential equations, especially systems, 00:38:50.000 --> 00:38:56.000 are likely to be solved by a computer anyway. 00:38:54.000 --> 00:39:00.000 You have to be the guiding genius that interprets the 00:38:57.000 --> 00:39:03.000 answers and can see when mistakes are being made, 00:39:01.000 --> 00:39:07.000 stuff like that. The problem is, 00:39:04.000 --> 00:39:10.000 therefore, what is the meaning of this system? 00:39:15.000 --> 00:39:21.000 Well, you are not going to get anywhere interpreting it 00:39:18.000 --> 00:39:24.000 geometrically, unless you get rid of that t on 00:39:21.000 --> 00:39:27.000 the right-hand side. And the only way of getting rid 00:39:25.000 --> 00:39:31.000 of the t is to declare it is not there. 00:39:28.000 --> 00:39:34.000 So I hereby declare that I will only consider, 00:39:31.000 --> 00:39:37.000 for the rest of the period, that is only ten minutes, 00:39:34.000 --> 00:39:40.000 systems in which no t appears explicitly on the right-hand 00:39:38.000 --> 00:39:44.000 side. Because I don't know what to do 00:39:42.000 --> 00:39:48.000 if it does up here. We have a word for these. 00:39:45.000 --> 00:39:51.000 Remember what the first order word was? 00:39:48.000 --> 00:39:54.000 A first order equation where there was no t explicitly on the 00:39:53.000 --> 00:39:59.000 right-hand side, we called it, 00:39:55.000 --> 00:40:01.000 anybody remember? Just curious. 00:39:57.000 --> 00:40:03.000 Autonomous, right. 00:40:05.000 --> 00:40:11.000 This is an autonomous system. It is not a linear system 00:40:08.000 --> 00:40:14.000 because these are messy functions. 00:40:10.000 --> 00:40:16.000 This could be x times y or x squared minus 3y squared 00:40:14.000 --> 00:40:20.000 divided by sine of x plus y. 00:40:18.000 --> 00:40:24.000 It could be a mess. Definitely not linear. 00:40:21.000 --> 00:40:27.000 But autonomous means no t. t means the independent 00:40:24.000 --> 00:40:30.000 variable appears on the right-hand side. 00:40:27.000 --> 00:40:33.000 Of course, it is there. It is buried in the dx/dt and 00:40:30.000 --> 00:40:36.000 dy/dt. But it is not on the right-hand 00:40:33.000 --> 00:40:39.000 side. No t appears on the right-hand 00:40:35.000 --> 00:40:41.000 side. 00:40:41.000 --> 00:40:47.000 Because no t appears on the right-hand side, 00:40:44.000 --> 00:40:50.000 I can now draw a picture of this. 00:40:47.000 --> 00:40:53.000 But, let's see, what does a solution look like? 00:40:52.000 --> 00:40:58.000 I never even talked about what a solution was, 00:40:56.000 --> 00:41:02.000 did I? Well, pretend that immediately 00:40:59.000 --> 00:41:05.000 after I talked about that, I talked about this. 00:41:05.000 --> 00:41:11.000 What is the solution? Well, the solution, 00:41:07.000 --> 00:41:13.000 maybe you took it for granted, is a pair of functions, 00:41:10.000 --> 00:41:16.000 x of t, y of t if when you plug it in 00:41:13.000 --> 00:41:19.000 it satisfies the equation. And so what else is new? 00:41:16.000 --> 00:41:22.000 The solution is x equals x of t, 00:41:19.000 --> 00:41:25.000 y equals y of t. 00:41:27.000 --> 00:41:33.000 If I draw a picture of that what would it look like? 00:41:30.000 --> 00:41:36.000 This is where your previous knowledge of physics above all 00:41:35.000 --> 00:41:41.000 18.02, maybe 18.01 if you learned this in high school, 00:41:39.000 --> 00:41:45.000 what is x equals x of t and y equals y of t? 00:41:44.000 --> 00:41:50.000 How do you draw a picture of 00:41:47.000 --> 00:41:53.000 that? What does it represent? 00:41:49.000 --> 00:41:55.000 A curve. And what will be the title of 00:41:52.000 --> 00:41:58.000 the chapter of the calculus book in which that is discussed? 00:41:56.000 --> 00:42:02.000 Parametric equations. This is a parameterized curve. 00:42:12.000 --> 00:42:18.000 So we know what the solution looks like. 00:42:15.000 --> 00:42:21.000 Our solution is a parameterized curve. 00:42:18.000 --> 00:42:24.000 And what does a parameterized curve look like? 00:42:21.000 --> 00:42:27.000 Well, it travels, and in a certain direction. 00:42:34.000 --> 00:42:40.000 Okay. That's enough. 00:42:35.000 --> 00:42:41.000 Why do I have several of those curves? 00:42:38.000 --> 00:42:44.000 Well, because I have several solutions. 00:42:40.000 --> 00:42:46.000 In fact, given any initial starting point, 00:42:43.000 --> 00:42:49.000 there is a solution that goes through it. 00:42:46.000 --> 00:42:52.000 I will put in possible starting points. 00:42:49.000 --> 00:42:55.000 And you can do this on the computer screen with a little 00:42:53.000 --> 00:42:59.000 program you will have, one of the visuals you'll have. 00:42:56.000 --> 00:43:02.000 It's being made right now. You put down starter point, 00:43:01.000 --> 00:43:07.000 put down a click, and then it just draws the 00:43:04.000 --> 00:43:10.000 curve passing through that point. 00:43:06.000 --> 00:43:12.000 Didn't we do this early in the term? 00:43:09.000 --> 00:43:15.000 Yes. But there is a difference now 00:43:11.000 --> 00:43:17.000 which I will explain. These are various possible 00:43:14.000 --> 00:43:20.000 starting points at time zero for this solution, 00:43:17.000 --> 00:43:23.000 and then you see what happens to it afterwards. 00:43:20.000 --> 00:43:26.000 In fact, through every point in the plane will pass a solution 00:43:25.000 --> 00:43:31.000 curve, parameterized curve. Now, what is then the 00:43:29.000 --> 00:43:35.000 representation of this? Well, what is the meaning of x 00:43:32.000 --> 00:43:38.000 prime of t and y prime of t? 00:43:40.000 --> 00:43:46.000 I am not going to worry for the moment about the right-hand 00:43:44.000 --> 00:43:50.000 side. What does this mean by itself? 00:43:47.000 --> 00:43:53.000 If this is the curve, the parameterized motion, 00:43:50.000 --> 00:43:56.000 then this represents its velocity vector. 00:43:53.000 --> 00:43:59.000 It is the velocity of the solution at time t. 00:43:58.000 --> 00:44:04.000 If I think of the solution as being a parameterized motion. 00:44:03.000 --> 00:44:09.000 All I have drawn here is the trace, the path of the motion. 00:44:08.000 --> 00:44:14.000 This hasn't indicated how fast it was going. 00:44:11.000 --> 00:44:17.000 One solution might go whoosh and another one might go rah. 00:44:16.000 --> 00:44:22.000 That is a velocity, and that velocity changes from 00:44:20.000 --> 00:44:26.000 point to point. It changes direction. 00:44:23.000 --> 00:44:29.000 Well, we know its direction at each point. 00:44:27.000 --> 00:44:33.000 That's tangent. What I cannot tell is the 00:44:31.000 --> 00:44:37.000 speed. From this picture, 00:44:33.000 --> 00:44:39.000 I cannot tell what the speed was. 00:44:36.000 --> 00:44:42.000 Too bad. Now, what is then the meaning 00:44:39.000 --> 00:44:45.000 of the system? What the system does, 00:44:41.000 --> 00:44:47.000 it prescribes at each point the velocity vector. 00:44:45.000 --> 00:44:51.000 If you tell me what the point (x, y) is in the plane then 00:44:50.000 --> 00:44:56.000 these equations give you the velocity vector at that point. 00:44:54.000 --> 00:45:00.000 And, therefore, what I end up with, 00:44:57.000 --> 00:45:03.000 the system is what you call in physics and what you call in 00:45:01.000 --> 00:45:07.000 18.02 a velocity field. So at each point there is a 00:45:06.000 --> 00:45:12.000 certain vector. The vector is always tangent to 00:45:09.000 --> 00:45:15.000 the solution curve through there, but I cannot predict from 00:45:13.000 --> 00:45:19.000 just this picture what its length will be because at some 00:45:17.000 --> 00:45:23.000 points, it might be going slow. The solution might be going 00:45:21.000 --> 00:45:27.000 slowly. In other words, 00:45:22.000 --> 00:45:28.000 the plane is filled up with these guys. 00:45:33.000 --> 00:45:39.000 Stop me. Not enough here. 00:45:37.000 --> 00:45:43.000 So on and so on. We can say a system of first 00:45:44.000 --> 00:45:50.000 order equations, ODEs of first order equations, 00:45:52.000 --> 00:45:58.000 autonomous because there must be no t on the right-hand side, 00:46:03.000 --> 00:46:09.000 is equal to a velocity field. A field of velocity. 00:46:12.000 --> 00:46:18.000 The plane covered with velocity vectors. 00:46:18.000 --> 00:46:24.000 And a solution is a parameterized curve with the 00:46:25.000 --> 00:46:31.000 right velocity everywhere. 00:46:38.000 --> 00:46:44.000 Now, there obviously must be a connection between that and the 00:47:39.000 --> 00:47:45.000 direction fields we studied at the beginning of the term. 00:48:36.000 --> 00:48:42.000 And there is. It is a very important 00:49:11.000 --> 00:49:17.000 connection. It is too important to talk 00:49:49.000 --> 00:49:55.000 about in minus one minute. When we need it, 00:50:32.000 --> 00:50:38.000 I will have to spend some time talking about it then.