WEBVTT 00:00:07.000 --> 00:00:13.000 We are going to start today in a serious way on the 00:00:12.000 --> 00:00:18.000 inhomogenous equation, second-order linear 00:00:17.000 --> 00:00:23.000 differential, I'll simply write it out 00:00:22.000 --> 00:00:28.000 instead of writing out all the words which go with it. 00:00:28.000 --> 00:00:34.000 So, such an equation looks like, the second-order equation 00:00:35.000 --> 00:00:41.000 is going to look like y double prime plus p of x, 00:00:40.000 --> 00:00:46.000 t, x plus q of x times y. Now, up to now the right-hand 00:00:47.000 --> 00:00:53.000 side has been zero. So, now we are going to make it 00:00:53.000 --> 00:00:59.000 not be zero. So, this is going to be f of x. 00:00:57.000 --> 00:01:03.000 In the most frequent 00:01:00.000 --> 00:01:06.000 applications, x is time. 00:01:02.000 --> 00:01:08.000 x is usually time, often, but not always. 00:01:08.000 --> 00:01:14.000 So, maybe just for today, I will use X in talking about 00:01:13.000 --> 00:01:19.000 the general theory. And, from now on, 00:01:16.000 --> 00:01:22.000 I'll probably make X equal time because that's what is most of 00:01:22.000 --> 00:01:28.000 the time in the applications. So, this is the part we've been 00:01:27.000 --> 00:01:33.000 studying up until now. It has a lot of names. 00:01:31.000 --> 00:01:37.000 It's input, signal, commas between those, 00:01:35.000 --> 00:01:41.000 a driving term, or sometimes it's called the 00:01:39.000 --> 00:01:45.000 forcing term. You'll see all of these in the 00:01:44.000 --> 00:01:50.000 literature, and it pretty much depends upon what course you're 00:01:48.000 --> 00:01:54.000 sitting at, what the professor habitually calls it. 00:01:52.000 --> 00:01:58.000 I will try to use all these terms now and then, 00:01:55.000 --> 00:02:01.000 probably most often I will lapse into input as the most 00:01:59.000 --> 00:02:05.000 generic term, suggesting nothing in 00:02:01.000 --> 00:02:07.000 particular, and therefore, equally acceptable or 00:02:05.000 --> 00:02:11.000 unacceptable to everybody. The response, 00:02:09.000 --> 00:02:15.000 the solution, then, the solution as you know 00:02:15.000 --> 00:02:21.000 is then called the response. The response, 00:02:20.000 --> 00:02:26.000 sometimes it's called the output. 00:02:24.000 --> 00:02:30.000 I think I'll stick pretty much with response. 00:02:31.000 --> 00:02:37.000 So, I'm using pretty much the same terminology we use for 00:02:35.000 --> 00:02:41.000 studying first-order equations. Now, as you will see, 00:02:40.000 --> 00:02:46.000 the reason we had to study the homogeneous case first was 00:02:45.000 --> 00:02:51.000 because you cannot solve this without knowing the homogeneous 00:02:50.000 --> 00:02:56.000 solutions. So, that's the inhomogeneous 00:02:53.000 --> 00:02:59.000 case. But the homogeneous one, 00:02:56.000 --> 00:03:02.000 the corresponding homogeneous thing, y double prime plus p of 00:03:01.000 --> 00:03:07.000 x y prime plus q of x times y equals zero 00:03:06.000 --> 00:03:12.000 is an essential part of the solution to this 00:03:11.000 --> 00:03:17.000 equation. That's called, 00:03:15.000 --> 00:03:21.000 therefore, it has names. Now, unfortunately, 00:03:19.000 --> 00:03:25.000 it doesn't have a single name. I don't know what to call it, 00:03:24.000 --> 00:03:30.000 but I think I'll probably call it the associated homogeneous 00:03:30.000 --> 00:03:36.000 equation, or ODE, the associated homogeneous 00:03:34.000 --> 00:03:40.000 equation, the one associated to the guy on the left. 00:03:40.000 --> 00:03:46.000 It's also called the reduced equation by some people. 00:03:44.000 --> 00:03:50.000 There is some other term for it, which escapes me totally, 00:03:48.000 --> 00:03:54.000 but what the heck. Now, its solution has a name. 00:03:52.000 --> 00:03:58.000 So, its solution, of course, doesn't depend on 00:03:55.000 --> 00:04:01.000 anything in particular, the general solution, 00:03:58.000 --> 00:04:04.000 because the right-hand side is always zero. 00:04:03.000 --> 00:04:09.000 So, its solution, we know can be written as y 00:04:06.000 --> 00:04:12.000 equals in the form c1 y1 plus c2 y2, 00:04:10.000 --> 00:04:16.000 where y1 and y2 are any two independent solutions of that, 00:04:14.000 --> 00:04:20.000 and then c1's and c2's are arbitrary constants. 00:04:17.000 --> 00:04:23.000 Now, what you are looking at this equation, 00:04:20.000 --> 00:04:26.000 you're going to need this also. And therefore, 00:04:24.000 --> 00:04:30.000 it has a name. It has various names. 00:04:26.000 --> 00:04:32.000 Sometimes there is a subscript, c, there. 00:04:31.000 --> 00:04:37.000 Sometimes there's a subscript, h. 00:04:34.000 --> 00:04:40.000 Sometimes there's no subscript at all, which is the most 00:04:39.000 --> 00:04:45.000 confusing of all. But, anyway, 00:04:42.000 --> 00:04:48.000 what's the name given to it? Well, there is no name. 00:04:47.000 --> 00:04:53.000 Many books call it the solution to the associated homogeneous 00:04:53.000 --> 00:04:59.000 equation. That's maximally long. 00:04:56.000 --> 00:05:02.000 Your book calls it the complementary solution. 00:05:02.000 --> 00:05:08.000 Many people call it that, and many will look at you with 00:05:06.000 --> 00:05:12.000 a blank, who know differential equations very well, 00:05:10.000 --> 00:05:16.000 and will not have the faintest idea what you're talking about. 00:05:14.000 --> 00:05:20.000 If you call it (y)h, then you are thinking of it as 00:05:18.000 --> 00:05:24.000 the solution; the h is for homogeneous to 00:05:21.000 --> 00:05:27.000 indicate it's the solution. So, it's the solution to the, 00:05:26.000 --> 00:05:32.000 I'm not going to write that. You put it in her books if you 00:05:30.000 --> 00:05:36.000 like writing. Write solution to the 00:05:33.000 --> 00:05:39.000 associated homogeneous equation, y(h). 00:05:36.000 --> 00:05:42.000 But, it's all the same thing. Now, or the solution to the 00:05:41.000 --> 00:05:47.000 reduced equation, I see I have in my notes. 00:05:45.000 --> 00:05:51.000 Okay, good, the solution to the reduced equation, 00:05:48.000 --> 00:05:54.000 too. Okay, now, the examples, 00:05:50.000 --> 00:05:56.000 there are, of course, two classical examples, 00:05:53.000 --> 00:05:59.000 of which you know one. But, use them as the model for 00:05:57.000 --> 00:06:03.000 what solutions of these things should look like and how they 00:06:02.000 --> 00:06:08.000 should behave. So, the model you know already 00:06:06.000 --> 00:06:12.000 is the one, I won't make the leading coefficient one because 00:06:11.000 --> 00:06:17.000 it usually isn't, is the one, m x double prime, 00:06:15.000 --> 00:06:21.000 so t is the independent variable, 00:06:18.000 --> 00:06:24.000 plus b x prime plus k x equals f of t. 00:06:23.000 --> 00:06:29.000 That's the spring-mass system, 00:06:26.000 --> 00:06:32.000 the spring-mass-dashpot system. Mass, the damping constant and 00:06:32.000 --> 00:06:38.000 the spring constant, except up to now, 00:06:35.000 --> 00:06:41.000 it's always been zero here. What does this f of t 00:06:40.000 --> 00:06:46.000 represent? Well, if you think of the way 00:06:43.000 --> 00:06:49.000 in which I derived the equation, the mx, that was the Newton's 00:06:49.000 --> 00:06:55.000 law. That's the acceleration. 00:06:51.000 --> 00:06:57.000 So, it's the acceleration, the mass times the 00:06:55.000 --> 00:07:01.000 acceleration. By Newton's law, 00:06:57.000 --> 00:07:03.000 this is equal to the imposed force on the little mass truck. 00:07:04.000 --> 00:07:10.000 Okay, you got that truck, there. 00:07:06.000 --> 00:07:12.000 I'm not going to draw the truck for the nth time. 00:07:09.000 --> 00:07:15.000 You'll have to imagine it. So, here's our truck. 00:07:13.000 --> 00:07:19.000 Okay, forces are acting on it. Remember, the forces were 00:07:17.000 --> 00:07:23.000 minus kx. That came from the spring. 00:07:20.000 --> 00:07:26.000 There was a force, minus b x prime. 00:07:23.000 --> 00:07:29.000 That came from the dashpot, the damping force. 00:07:26.000 --> 00:07:32.000 So, this other guy is f of t. 00:07:30.000 --> 00:07:36.000 What's this? This is the external force, 00:07:32.000 --> 00:07:38.000 which is acting out. In other words, 00:07:34.000 --> 00:07:40.000 instead of the little truck going back and forth and doing 00:07:38.000 --> 00:07:44.000 its own thing all by itself, here's someone with an 00:07:41.000 --> 00:07:47.000 electromagnet, and the mass it's carrying is a 00:07:44.000 --> 00:07:50.000 big pile of iron ore. You're turning it on and off, 00:07:47.000 --> 00:07:53.000 and pulling that thing from afar where nobody can see it. 00:07:51.000 --> 00:07:57.000 So, this is the external force. Now, think, that is the model 00:07:55.000 --> 00:08:01.000 you must have in your mind of how these equations are treated. 00:08:00.000 --> 00:08:06.000 In other words, when f of t is zero, 00:08:03.000 --> 00:08:09.000 the system is passive. There is no external force on 00:08:07.000 --> 00:08:13.000 it when this is zero. The system is sitting, 00:08:10.000 --> 00:08:16.000 and just doing what it wants to do, all by itself. 00:08:14.000 --> 00:08:20.000 You wanted up by giving it an initial push, 00:08:17.000 --> 00:08:23.000 and putting its initial position somewhere. 00:08:20.000 --> 00:08:26.000 But after that, you lay your hands off. 00:08:23.000 --> 00:08:29.000 The system then just passively responds to its initial 00:08:27.000 --> 00:08:33.000 conditions and does what it wants. 00:08:31.000 --> 00:08:37.000 The other model is that you don't let it respond the way it 00:08:35.000 --> 00:08:41.000 wants to. You force it from the outside 00:08:37.000 --> 00:08:43.000 by pushing it with an external force. 00:08:40.000 --> 00:08:46.000 Now, those are clearly two entirely different problems: 00:08:44.000 --> 00:08:50.000 what it does by itself, or what it does when it's acted 00:08:48.000 --> 00:08:54.000 on from outside. And, when I explained to you 00:08:51.000 --> 00:08:57.000 how the thing is to be solved, you have to keep in mind those 00:08:55.000 --> 00:09:01.000 two models. So, this is the forced system. 00:09:00.000 --> 00:09:06.000 I'll just use the word, forced system, 00:09:02.000 --> 00:09:08.000 that's where f of t is not zero, versus the passive 00:09:07.000 --> 00:09:13.000 system where there is no external applied force. 00:09:11.000 --> 00:09:17.000 The passive system, the forced system, 00:09:14.000 --> 00:09:20.000 now, you have to both, even if you wanted to solve the 00:09:18.000 --> 00:09:24.000 forced system, the way the system would behave 00:09:22.000 --> 00:09:28.000 if nothing would be done to it from the outside is nonetheless 00:09:27.000 --> 00:09:33.000 going to be an important part of the solution. 00:09:32.000 --> 00:09:38.000 And, I won't be able to give you that solution without 00:09:35.000 --> 00:09:41.000 knowing this also. Now, I'd like to give you the 00:09:38.000 --> 00:09:44.000 other model very rapidly because it's in your book. 00:09:42.000 --> 00:09:48.000 It's in the problems I have to give you. 00:09:45.000 --> 00:09:51.000 You know, it's part of everybody's culture, 00:09:48.000 --> 00:09:54.000 whether they like it or not. So, that's example number one. 00:09:52.000 --> 00:09:58.000 Example number two, which follows the differential 00:09:55.000 --> 00:10:01.000 equation just as perfectly as the spring-mass-dashpot system 00:09:59.000 --> 00:10:05.000 is the simple electric circuit. The inductance, 00:10:04.000 --> 00:10:10.000 you don't know yet what an inductance is, 00:10:08.000 --> 00:10:14.000 officially, but you will, a resistance, 00:10:11.000 --> 00:10:17.000 sorry, that's okay, put the capacitance up there, 00:10:15.000 --> 00:10:21.000 resistance, and then maybe a thing. 00:10:18.000 --> 00:10:24.000 So, this is a resistance. I think you know these symbols. 00:10:24.000 --> 00:10:30.000 By now, you certainly know the system for capacitance. 00:10:28.000 --> 00:10:34.000 What I mean when I say C is the capacitance, you may not know 00:10:34.000 --> 00:10:40.000 yet what L is. That's called the inductance. 00:10:39.000 --> 00:10:45.000 So, this is something called a coil because it looks like one. 00:10:43.000 --> 00:10:49.000 L is what's called its inductance. 00:10:45.000 --> 00:10:51.000 And, the differential equation, there are two differential 00:10:50.000 --> 00:10:56.000 equations which can be used in this. 00:10:52.000 --> 00:10:58.000 They are essentially the same. One is simply the derivative of 00:10:56.000 --> 00:11:02.000 the other. Both differential equations 00:10:59.000 --> 00:11:05.000 come from Kirchhoff's voltage law, that the sum of the voltage 00:11:04.000 --> 00:11:10.000 drops as you move around the circuit -- 00:11:08.000 --> 00:11:14.000 -- has to be zero because otherwise, I don't have to, 00:11:11.000 --> 00:11:17.000 that's because of somebody's law, Kirchhoff, 00:11:14.000 --> 00:11:20.000 with two h's. The sum of the voltage drops to 00:11:17.000 --> 00:11:23.000 zero, and now you know the voltage drop across this, 00:11:21.000 --> 00:11:27.000 and you know the voltage drop across that because you learned 00:11:25.000 --> 00:11:31.000 in 8.02. You will, one day, 00:11:27.000 --> 00:11:33.000 learn the voltage drop across this. 00:11:29.000 --> 00:11:35.000 But, I already know it. It's Li. 00:11:32.000 --> 00:11:38.000 So, i is the current. I'll write this thing in its 00:11:37.000 --> 00:11:43.000 primitive form first. So, i is the current that's 00:11:42.000 --> 00:11:48.000 flowing in the circuit. q is the charge on the 00:11:47.000 --> 00:11:53.000 capacitance. So, the voltage drop across the 00:11:51.000 --> 00:11:57.000 coil is L times i. The voltage shop across the, 00:11:57.000 --> 00:12:03.000 Li prime, the voltage drop across the 00:12:02.000 --> 00:12:08.000 resistance is, well, you know that. 00:12:07.000 --> 00:12:13.000 And, the voltage shop across the capacitance is q 00:12:10.000 --> 00:12:16.000 divided by C. And so, that's equal to, 00:12:13.000 --> 00:12:19.000 well, it's equal to zero, except if there's a battery 00:12:16.000 --> 00:12:22.000 here or something generating a voltage drop, 00:12:19.000 --> 00:12:25.000 so, let's call that E is a generic word. 00:12:21.000 --> 00:12:27.000 E could be a battery. It could be a source of 00:12:24.000 --> 00:12:30.000 alternating current, something like that. 00:12:27.000 --> 00:12:33.000 But, there's a voltage drop across it, and I'm giving E the 00:12:31.000 --> 00:12:37.000 name of the voltage drop. So, and then there's the 00:12:35.000 --> 00:12:41.000 question of the signs, which I know I'll never 00:12:38.000 --> 00:12:44.000 understand. But, let's assume you've chosen 00:12:41.000 --> 00:12:47.000 the sign convention so that this comes out nicely on the 00:12:44.000 --> 00:12:50.000 right-hand side. So, this might be varying 00:12:47.000 --> 00:12:53.000 sinusoidally, in which case you'd have source 00:12:50.000 --> 00:12:56.000 of alternating current. Or, it might be constant, 00:12:53.000 --> 00:12:59.000 in which that would be a battery, a little dry cell 00:12:56.000 --> 00:13:02.000 giving you direct current of a constant voltage, 00:12:59.000 --> 00:13:05.000 stuff like that. So, you could make this minus 00:13:03.000 --> 00:13:09.000 if you want, but everything will have the wrong signs, 00:13:07.000 --> 00:13:13.000 so don't do it. Now, this doesn't look like 00:13:09.000 --> 00:13:15.000 what it's supposed to look like because it's got q and i. 00:13:13.000 --> 00:13:19.000 So, the final thing you have to know is that q prime is 00:13:17.000 --> 00:13:23.000 equal to i. The rate at which that charge 00:13:19.000 --> 00:13:25.000 leaves the condenser and hurries around the circuit to find its 00:13:23.000 --> 00:13:29.000 little soul mate on the other side is the current that's 00:13:27.000 --> 00:13:33.000 flowing in the circuit. That's why current flows, 00:13:30.000 --> 00:13:36.000 except nothing really happens. Electrons just push on each 00:13:35.000 --> 00:13:41.000 other, and they stay where they are. 00:13:37.000 --> 00:13:43.000 I don't understand this at all. So, if I differentiate this, 00:13:42.000 --> 00:13:48.000 you can do two things. Either you could integrate i, 00:13:46.000 --> 00:13:52.000 and expressed the thing entirely in terms of q, 00:13:50.000 --> 00:13:56.000 or you can differentiate it, and express everything in terms 00:13:54.000 --> 00:14:00.000 of i. Your book does nicely both, 00:13:57.000 --> 00:14:03.000 does not take sides. So, let's differentiate it, 00:14:02.000 --> 00:14:08.000 and then it will look like L i double prime plus R i prime plus 00:14:06.000 --> 00:14:12.000 i divided by C equals, 00:14:09.000 --> 00:14:15.000 and now, watch out, you have now not the 00:14:12.000 --> 00:14:18.000 electromotive force, but its derivative. 00:14:15.000 --> 00:14:21.000 So, if you were so unfortunate as to put a little dry cell 00:14:19.000 --> 00:14:25.000 there, now you've got nothing, and you've got the homogeneous 00:14:24.000 --> 00:14:30.000 case. That's okay. 00:14:25.000 --> 00:14:31.000 Where are the erasers? One eraser? 00:14:28.000 --> 00:14:34.000 I don't believe this. 00:14:43.000 --> 00:14:49.000 So, there's the equation. There are our two equations. 00:14:47.000 --> 00:14:53.000 Why don't we put them up in colored chalk. 00:14:50.000 --> 00:14:56.000 There's the spring equation. And, here's the equation that 00:14:54.000 --> 00:15:00.000 governs the current, for how the current flows in 00:14:58.000 --> 00:15:04.000 that circuit. And now, you can see, 00:15:01.000 --> 00:15:07.000 again, what does it mean? If this is zero, 00:15:04.000 --> 00:15:10.000 for example, if I have a dry cell there, 00:15:07.000 --> 00:15:13.000 or if I have nothing at all in the circuit, then this 00:15:11.000 --> 00:15:17.000 represents the passive circuit. It's just sitting there. 00:15:16.000 --> 00:15:22.000 It wouldn't do anything at all, except that you've put a charge 00:15:20.000 --> 00:15:26.000 on the capacitor, and waited, and of course, 00:15:22.000 --> 00:15:28.000 when you put a charge on there, it's got a discharge, 00:15:26.000 --> 00:15:32.000 and discharges through the circuit, and swings back and 00:15:29.000 --> 00:15:35.000 forth a little bit if it's under-damped until finally 00:15:32.000 --> 00:15:38.000 towards the end the current dies away to zero. 00:15:36.000 --> 00:15:42.000 But, what usually happens is that you drive this passive 00:15:39.000 --> 00:15:45.000 circuit by putting an effective E in it, and then you want to 00:15:44.000 --> 00:15:50.000 know how the current behaves. So, those are the two problems, 00:15:48.000 --> 00:15:54.000 the passive circuit without an applied electromotive force, 00:15:52.000 --> 00:15:58.000 or plugging it into the wall, and wanting it to do things. 00:15:56.000 --> 00:16:02.000 That's the normal state of affairs. 00:16:00.000 --> 00:16:06.000 People don't want passive circuits, they want circuits 00:16:03.000 --> 00:16:09.000 which do things because, okay, that's why they want to 00:16:07.000 --> 00:16:13.000 solve inhomogeneous equations instead of homogeneous 00:16:11.000 --> 00:16:17.000 equations. But as I said, 00:16:13.000 --> 00:16:19.000 you have to do the homogeneous case first. 00:16:16.000 --> 00:16:22.000 Okay, you are now officially responsible for this, 00:16:20.000 --> 00:16:26.000 and I don't care that you haven't had it in physics yet. 00:16:24.000 --> 00:16:30.000 You will before the next exam. So, I don't even feel guilty. 00:16:30.000 --> 00:16:36.000 But, you're going to start using it on the problem set 00:16:34.000 --> 00:16:40.000 right away. So, it's never too soon to 00:16:37.000 --> 00:16:43.000 start learning it. Okay, now, the main theorem, 00:16:41.000 --> 00:16:47.000 I now want to go, so that was just examples to 00:16:44.000 --> 00:16:50.000 give you some physical feeling for the sorts of differential 00:16:49.000 --> 00:16:55.000 equations we'll be talking about. 00:16:52.000 --> 00:16:58.000 I now want to tell you briefly about the key theorem about 00:16:56.000 --> 00:17:02.000 solving the homogeneous equation. 00:16:59.000 --> 00:17:05.000 So, the main theorem about solving the homogeneous equation 00:17:04.000 --> 00:17:10.000 is, the inhomogeneous equation. So, I'm going to write the 00:17:09.000 --> 00:17:15.000 inhomogeneous equation out. I'm going to make the left-hand 00:17:14.000 --> 00:17:20.000 side a linear operator, and am going to write the 00:17:18.000 --> 00:17:24.000 equation as Ly equals f of x. 00:17:21.000 --> 00:17:27.000 That's the inhomogeneous equation. 00:17:23.000 --> 00:17:29.000 So, L is the linear operator, second order because I'm only 00:17:28.000 --> 00:17:34.000 talking about second-order equations. 00:17:32.000 --> 00:17:38.000 L is a linear operator, and then this is the 00:17:37.000 --> 00:17:43.000 differential equation. So, here's our differential 00:17:43.000 --> 00:17:49.000 equation. It's inhomogeneous because it's 00:17:48.000 --> 00:17:54.000 go the f of x on the right hand side. 00:17:53.000 --> 00:17:59.000 And, what the theorem says is that the solution has the 00:18:00.000 --> 00:18:06.000 following form, y sub p, I'll explain what that 00:18:05.000 --> 00:18:11.000 is in just a moment, plus y sub c. 00:18:13.000 --> 00:18:19.000 So, the hypothesis is we've got the linear equation, 00:18:17.000 --> 00:18:23.000 and the conclusion is that that's what its solution looks 00:18:21.000 --> 00:18:27.000 like. Now, you already know what y 00:18:24.000 --> 00:18:30.000 sub c looks like. In other words, 00:18:27.000 --> 00:18:33.000 if I write this out in more detail, it would be i.e., 00:18:31.000 --> 00:18:37.000 department of fuller explanation, -- 00:18:35.000 --> 00:18:41.000 -- the general solution looks like y equals yp, 00:18:38.000 --> 00:18:44.000 and then this thing is going to look like an arbitrary constant 00:18:43.000 --> 00:18:49.000 times y1 plus an arbitrary constant times y2, 00:18:47.000 --> 00:18:53.000 where these are solutions of the homogeneous equation. 00:18:51.000 --> 00:18:57.000 So, Yc looks like this part, and the yp, what's yp? 00:18:55.000 --> 00:19:01.000 p stands for particular, the most confusing word in this 00:19:00.000 --> 00:19:06.000 subject. But, you've got at least four 00:19:04.000 --> 00:19:10.000 weeks to learn what it means. Okay, yp is a particular 00:19:10.000 --> 00:19:16.000 solution to Ly equals f of x. 00:19:14.000 --> 00:19:20.000 Now, I'm not going to explain what particular means. 00:19:19.000 --> 00:19:25.000 First, I'll chat as if you knew what it meant, 00:19:24.000 --> 00:19:30.000 and then we'll see if you have picked it up. 00:19:30.000 --> 00:19:36.000 In other words, the procedure for solving this 00:19:33.000 --> 00:19:39.000 equation is composed of two steps. 00:19:36.000 --> 00:19:42.000 First, to find this part. In other words, 00:19:40.000 --> 00:19:46.000 to find the complementary solution, in other words, 00:19:44.000 --> 00:19:50.000 to do what we've been doing for the last week, 00:19:48.000 --> 00:19:54.000 solve not the equation you are given, but the reduced equation. 00:19:53.000 --> 00:19:59.000 So, the first step is to find this. 00:19:56.000 --> 00:20:02.000 The second step is to find yp. Now, what's yp? 00:20:01.000 --> 00:20:07.000 yp is a particular solution to the whole equation. 00:20:05.000 --> 00:20:11.000 Yeah, but which one? Well, if it's any one, 00:20:09.000 --> 00:20:15.000 then it's not a particular solution, yeah. 00:20:12.000 --> 00:20:18.000 I say, unfortunately the word particular here is not being 00:20:17.000 --> 00:20:23.000 used in exactly the same sense in which most people use it in 00:20:22.000 --> 00:20:28.000 ordinary English. It's a perfectly valid way to 00:20:26.000 --> 00:20:32.000 use it. It's just confusing, 00:20:29.000 --> 00:20:35.000 and no one has ever come up with a better word. 00:20:33.000 --> 00:20:39.000 So, particular means any one solution. 00:20:38.000 --> 00:20:44.000 Any one will do. Okay, even these have slightly 00:20:45.000 --> 00:20:51.000 different meanings. Any questions about this? 00:20:51.000 --> 00:20:57.000 I refuse to answer them. [LAUGHTER] 00:20:58.000 --> 00:21:04.000 Now, well, examples of course will make it all clear. 00:21:03.000 --> 00:21:09.000 But I'd like, first, to prove the theorem, 00:21:08.000 --> 00:21:14.000 to show you how simple it is. It's extremely simple if you 00:21:15.000 --> 00:21:21.000 just use the fact that L is a linear operator. 00:21:20.000 --> 00:21:26.000 We've got two things to prove. What have we got to prove? 00:21:26.000 --> 00:21:32.000 Well, I have to prove two statements, first of all, 00:21:32.000 --> 00:21:38.000 that all the yp plus c1 y1 plus c2 y2 are 00:21:39.000 --> 00:21:45.000 solutions. How are we going to prove that? 00:21:43.000 --> 00:21:49.000 Well, how do you know if something is a solution? 00:21:46.000 --> 00:21:52.000 Well, you plug it into the equation, and you see if it 00:21:49.000 --> 00:21:55.000 satisfies the equation. Good, let's do it, 00:21:52.000 --> 00:21:58.000 proof. L, I'm going to plug it into 00:21:54.000 --> 00:22:00.000 the equation. That means I calculate L of yp 00:21:56.000 --> 00:22:02.000 plus c1 y1 plus c2 y2. 00:22:00.000 --> 00:22:06.000 Now, what's the answer? Because this is a linear 00:22:04.000 --> 00:22:10.000 operator, and notice, the argument doesn't use the 00:22:08.000 --> 00:22:14.000 fact that the equation is second order. 00:22:12.000 --> 00:22:18.000 It immediately generalizes to a linear equation of any order, 00:22:17.000 --> 00:22:23.000 whatever-- 47. Okay, this is L of yp plus L of 00:22:21.000 --> 00:22:27.000 c1 y1 plus c2 y2. 00:22:25.000 --> 00:22:31.000 Well, what's that? What's L of the complementary 00:22:29.000 --> 00:22:35.000 solution? What does it mean to be the 00:22:34.000 --> 00:22:40.000 complementary solution? It means when you apply the 00:22:37.000 --> 00:22:43.000 operator L to it, you get zero because this 00:22:40.000 --> 00:22:46.000 satisfies the homogeneous equation. 00:22:43.000 --> 00:22:49.000 So, this is zero. What's L of yp? 00:22:46.000 --> 00:22:52.000 Well, it was a particular solution to the equation. 00:22:50.000 --> 00:22:56.000 Therefore, when I plugged it into the equation, 00:22:53.000 --> 00:22:59.000 I must have gotten out on the right-hand side, 00:22:57.000 --> 00:23:03.000 f of x. So, this is since yp is a 00:23:00.000 --> 00:23:06.000 solution to the whole equation. So, what's the conclusion? 00:23:06.000 --> 00:23:12.000 That, if I take any one of these guys, no matter what c1 00:23:10.000 --> 00:23:16.000 and c2 are, apply the linear operator, L to it, 00:23:13.000 --> 00:23:19.000 the answer comes out to be f of. 00:23:17.000 --> 00:23:23.000 Therefore, this proves that this shows that these are all 00:23:21.000 --> 00:23:27.000 solutions because that's what it means. 00:23:24.000 --> 00:23:30.000 Therefore, they satisfy L of y equals f of x. 00:23:30.000 --> 00:23:36.000 They satisfy the whole inhomogeneous differential 00:23:34.000 --> 00:23:40.000 equation, and that's it. Well, that's only half the 00:23:38.000 --> 00:23:44.000 story. The other half of the story is 00:23:41.000 --> 00:23:47.000 to show that there are no other solutions. 00:23:45.000 --> 00:23:51.000 Okay, so we got our little u of x coming up again, 00:23:50.000 --> 00:23:56.000 and he thinks he's a solution. Okay, so, to prove there are no 00:23:55.000 --> 00:24:01.000 other solutions, it almost sounds biblical, 00:23:59.000 --> 00:24:05.000 thou shalt have no other solutions before me, 00:24:03.000 --> 00:24:09.000 okay. There are no other solutions 00:24:07.000 --> 00:24:13.000 accept these guys for different values of c1 and c2. 00:24:10.000 --> 00:24:16.000 Okay, so, u of x is a solution. 00:24:13.000 --> 00:24:19.000 I have to show that u of x is one of these guys. 00:24:16.000 --> 00:24:22.000 How am I going to do that? Easy. 00:24:19.000 --> 00:24:25.000 If it's a solution that, L of u, 00:24:21.000 --> 00:24:27.000 okay, I'm going to drop the x, okay, just to make the, 00:24:25.000 --> 00:24:31.000 like I dropped the x over there. 00:24:29.000 --> 00:24:35.000 If it's a solution to the whole inhomogeneous equation, 00:24:33.000 --> 00:24:39.000 then this must come out to be f of x. 00:24:37.000 --> 00:24:43.000 Now, what's L of yp? That's f of x too, 00:24:41.000 --> 00:24:47.000 by secret little particular solution I've got in my pocket. 00:24:47.000 --> 00:24:53.000 Okay, I pull it out, ah-ha, L of yp, 00:24:50.000 --> 00:24:56.000 that's f of x, too. 00:24:51.000 --> 00:24:57.000 Now, I'm going to not add them. I'm going to subtract them. 00:24:56.000 --> 00:25:02.000 What is L of u minus yp? 00:25:00.000 --> 00:25:06.000 Well, it's zero. It's zero because this is a 00:25:05.000 --> 00:25:11.000 linear operator. This would be L of u minus L of 00:25:08.000 --> 00:25:14.000 yp. I guess the answer is zero on 00:25:12.000 --> 00:25:18.000 the right-hand side. And therefore, 00:25:14.000 --> 00:25:20.000 what is the conclusion? If that's zero, 00:25:17.000 --> 00:25:23.000 it must be a solution to the homogeneous equation. 00:25:21.000 --> 00:25:27.000 Therefore, u minus yp is equal to, 00:25:24.000 --> 00:25:30.000 there must be c1 and c2. I won't give them the generic 00:25:28.000 --> 00:25:34.000 names. I'll give them a name, 00:25:30.000 --> 00:25:36.000 a particular one. I'll put a tilde to indicate 00:25:34.000 --> 00:25:40.000 it's a particular one. c1 plus c2 y2 tilde, 00:25:37.000 --> 00:25:43.000 so, in other words, for some choice of these 00:25:41.000 --> 00:25:47.000 constants, and I'll call those particular choices c1 tilde and 00:25:45.000 --> 00:25:51.000 c2 tilde, it must be that these are equal. 00:25:48.000 --> 00:25:54.000 Well, what does that say? It says that u is equal to yp 00:25:52.000 --> 00:25:58.000 plus c1 tilde, blah, blah, blah, 00:25:54.000 --> 00:26:00.000 blah, plus c2 tilde, blah, blah, blah, 00:25:57.000 --> 00:26:03.000 blah, and therefore chose that u wasn't a new solution. 00:26:02.000 --> 00:26:08.000 It was one of these. So, u isn't new. 00:26:08.000 --> 00:26:14.000 So, I should write it down. Otherwise some of you will have 00:26:18.000 --> 00:26:24.000 missed the punch line. Okay, therefore, 00:26:25.000 --> 00:26:31.000 u is equal to yp plus c1 tilde y1 plus c2 tilde y2. 00:26:36.000 --> 00:26:42.000 And, it shows. This guy who thought he was new 00:26:39.000 --> 00:26:45.000 was not new at all. It was just one of the other 00:26:43.000 --> 00:26:49.000 solutions. Okay, well, now, 00:26:45.000 --> 00:26:51.000 since the coefficient's a constant, apparently we've done 00:26:49.000 --> 00:26:55.000 half the work. We know what the complementary 00:26:52.000 --> 00:26:58.000 solution is because you know how to do those in terms of 00:26:56.000 --> 00:27:02.000 exponentials and complex exponentials, 00:26:59.000 --> 00:27:05.000 signs and cosines, and so on. 00:27:03.000 --> 00:27:09.000 So, what's left to do? All we have to do is find to 00:27:07.000 --> 00:27:13.000 solve equations, which are inhomogeneous. 00:27:10.000 --> 00:27:16.000 All we have to do is find a particular solution, 00:27:14.000 --> 00:27:20.000 find one solution. It doesn't matter which one, 00:27:18.000 --> 00:27:24.000 any one. Just find one, 00:27:20.000 --> 00:27:26.000 okay? Now, we're going to spend the 00:27:23.000 --> 00:27:29.000 next two weeks trying to do this. 00:27:26.000 --> 00:27:32.000 I'll give you various methods. I'll give you a general method 00:27:32.000 --> 00:27:38.000 involving Fourier series because it's a good excuse for learning 00:27:37.000 --> 00:27:43.000 what Fourier series are. But, the answer is that in 00:27:41.000 --> 00:27:47.000 general, for a few standard functions, it's known how to do 00:27:45.000 --> 00:27:51.000 this. You will learn those methods 00:27:48.000 --> 00:27:54.000 for finding those using operators. 00:27:50.000 --> 00:27:56.000 For all the others, it's done by a series, 00:27:54.000 --> 00:28:00.000 or a method involving approximation. 00:27:58.000 --> 00:28:04.000 Or, the worse comes to worst, you throw it on a computer and 00:28:02.000 --> 00:28:08.000 just take a graph and the numerical output of answers as 00:28:07.000 --> 00:28:13.000 the particular solution. Okay, now before, 00:28:10.000 --> 00:28:16.000 we are going to start that work, not today. 00:28:14.000 --> 00:28:20.000 We'll start it next Monday, and it will last, 00:28:18.000 --> 00:28:24.000 as I say the next two weeks. And, we will be up to spring 00:28:22.000 --> 00:28:28.000 break. But, before we do that, 00:28:25.000 --> 00:28:31.000 I'd like to relate this to what we did for first order equations 00:28:30.000 --> 00:28:36.000 because there is something to be learned from that. 00:28:36.000 --> 00:28:42.000 Think back to the linear first-order equation, 00:28:38.000 --> 00:28:44.000 and I'm going to, since from now on for the rest 00:28:41.000 --> 00:28:47.000 of the period, I'm going to be considering the 00:28:44.000 --> 00:28:50.000 case for constant coefficients. In other words, 00:28:47.000 --> 00:28:53.000 this case of springs or circuits or simple systems which 00:28:51.000 --> 00:28:57.000 behave like those and have constant coefficients. 00:28:54.000 --> 00:29:00.000 So, for the linear, first-order equation, 00:28:56.000 --> 00:29:02.000 there, too, I'm going to think of constant coefficients. 00:29:01.000 --> 00:29:07.000 We talked quite a bit about this equation. 00:29:04.000 --> 00:29:10.000 What did I call the right-hand side? 00:29:06.000 --> 00:29:12.000 I think we usually called it q of t, 00:29:09.000 --> 00:29:15.000 right? This is in ancient history. 00:29:12.000 --> 00:29:18.000 The definition of ancient history was before the first 00:29:16.000 --> 00:29:22.000 exam. Okay, now how does that fit 00:29:18.000 --> 00:29:24.000 into this theorem that I've given you? 00:29:21.000 --> 00:29:27.000 Remember what the solution looked like. 00:29:25.000 --> 00:29:31.000 The solution looked like, remember, you took the 00:29:28.000 --> 00:29:34.000 integrating factor was e to the kt, 00:29:32.000 --> 00:29:38.000 and then after you integrated both sides, multiplied through, 00:29:37.000 --> 00:29:43.000 and then the final answer looked like this, 00:29:41.000 --> 00:29:47.000 y equaled, it was e to the negative kt times 00:29:45.000 --> 00:29:51.000 either an indefinite integral, or a definite integral 00:29:50.000 --> 00:29:56.000 depending on your preference, q of t, 00:29:54.000 --> 00:30:00.000 so, x is metamorphosed into t. I gather you've got that, 00:29:58.000 --> 00:30:04.000 e to the kt plus, what was the other term? 00:30:02.000 --> 00:30:08.000 A constant times e to the negative kt 00:30:08.000 --> 00:30:14.000 How does this fit into the paradigm I've given you over 00:30:12.000 --> 00:30:18.000 there for solving the second order equation? 00:30:15.000 --> 00:30:21.000 Which term is which? Well, this has the arbitrary 00:30:19.000 --> 00:30:25.000 constant in it. So, this must be the 00:30:22.000 --> 00:30:28.000 complementary solution. Is it? 00:30:24.000 --> 00:30:30.000 Is this the solution to the associated homogeneous equation? 00:30:30.000 --> 00:30:36.000 What's the associated homogeneous equation? 00:30:32.000 --> 00:30:38.000 Put zero here. Okay, if you put zero there, 00:30:35.000 --> 00:30:41.000 what's the solution? Now, this you ought to know. 00:30:38.000 --> 00:30:44.000 y prime equals negative ky. 00:30:40.000 --> 00:30:46.000 What's the solution? e to the negative kt. 00:30:43.000 --> 00:30:49.000 You are supposed to come into 00:30:46.000 --> 00:30:52.000 this course knowing that, except there's an arbitrary 00:30:49.000 --> 00:30:55.000 constant in front. So, right, this is exactly the 00:30:52.000 --> 00:30:58.000 solution to the associated homogeneous equation, 00:30:55.000 --> 00:31:01.000 where there is zero here. Then, what's this thing? 00:31:00.000 --> 00:31:06.000 This is a particular solution. This is my yp. 00:31:02.000 --> 00:31:08.000 But that's not a particular solution because this indefinite 00:31:06.000 --> 00:31:12.000 integral, you know, has an arbitrary constant in 00:31:09.000 --> 00:31:15.000 it. In fact, it's just that 00:31:10.000 --> 00:31:16.000 arbitrary constant. So, it's totally confusing. 00:31:13.000 --> 00:31:19.000 But, this symbol, you know when you actually 00:31:16.000 --> 00:31:22.000 solve the equation this way, all you did was you found one 00:31:20.000 --> 00:31:26.000 function here. You didn't throw in the 00:31:22.000 --> 00:31:28.000 arbitrary constant right away. All you needed to do was find 00:31:26.000 --> 00:31:32.000 one function. And, even if you really are 00:31:29.000 --> 00:31:35.000 bothered by the fact that this is so indefinite, 00:31:32.000 --> 00:31:38.000 and therefore, make it a particular solution 00:31:35.000 --> 00:31:41.000 by making this zero, make it a definite integral, 00:31:38.000 --> 00:31:44.000 zero, here, t there, and then change those t's to 00:31:42.000 --> 00:31:48.000 dummy t's, t1's or t tildes, or something like that. 00:31:45.000 --> 00:31:51.000 So, this fits into that thing. In other words, 00:31:48.000 --> 00:31:54.000 I could have done it at that time, but I didn't the point 00:31:52.000 --> 00:31:58.000 because this can be solved directly, whereas, 00:31:55.000 --> 00:32:01.000 of course, the general second order equation in homogeneous 00:31:58.000 --> 00:32:04.000 cannot be solved directly, and therefore you have to be 00:32:02.000 --> 00:32:08.000 willing to talk about what its solutions look like in advance. 00:32:08.000 --> 00:32:14.000 Now, remember I said, we talked, I said there was two 00:32:12.000 --> 00:32:18.000 different cases, although both of them had the 00:32:15.000 --> 00:32:21.000 identical looking solution. Their meaning in the physical 00:32:19.000 --> 00:32:25.000 world was so different that they really should be considered as 00:32:24.000 --> 00:32:30.000 solving the same equation. And, one of these was the case. 00:32:30.000 --> 00:32:36.000 Of the two, perhaps the more important was the case when k 00:32:34.000 --> 00:32:40.000 was positive, and of course the other is when 00:32:38.000 --> 00:32:44.000 k is negative. When k is positive, 00:32:41.000 --> 00:32:47.000 that had the effect of separating that solution into 00:32:45.000 --> 00:32:51.000 this part, which was a transient, and the other part, 00:32:49.000 --> 00:32:55.000 which was a steady state. The steady state solution, 00:32:53.000 --> 00:32:59.000 that was the yp part of it in that terminology. 00:32:57.000 --> 00:33:03.000 And, the transient part, it was trangent because it went 00:33:02.000 --> 00:33:08.000 to zero. If k is positive, 00:33:05.000 --> 00:33:11.000 the exponential dies regardless of what c is. 00:33:09.000 --> 00:33:15.000 So, the transient, that's the yc part. 00:33:12.000 --> 00:33:18.000 It goes to zero as Ttgoes to infinity. 00:33:16.000 --> 00:33:22.000 The transient depends on, uses, the initial condition, 00:33:21.000 --> 00:33:27.000 whatever it is, because that's what determines 00:33:25.000 --> 00:33:31.000 the value of c. On the other hand, 00:33:28.000 --> 00:33:34.000 this initial condition makes no difference as t goes towards 00:33:33.000 --> 00:33:39.000 infinity. All that's left is this steady 00:33:38.000 --> 00:33:44.000 state solution. And, all solutions tend to the 00:33:42.000 --> 00:33:48.000 steady state solution. So, if k is positive, 00:33:46.000 --> 00:33:52.000 one gets this analysis of the solutions into the sum of one 00:33:52.000 --> 00:33:58.000 basic solution, and the others, 00:33:55.000 --> 00:34:01.000 which just die away, have no influence on this, 00:33:59.000 --> 00:34:05.000 less and less influence as time goes to infinity. 00:34:05.000 --> 00:34:11.000 For k less than zero, this analysis does not work 00:34:09.000 --> 00:34:15.000 because this term, if k is less than zero, 00:34:12.000 --> 00:34:18.000 this term goes to infinity or negative infinity, 00:34:16.000 --> 00:34:22.000 and typically tends to dominate that. 00:34:19.000 --> 00:34:25.000 So, it's the start that the important one. 00:34:23.000 --> 00:34:29.000 It depends on the initial conditions, and the analysis is 00:34:28.000 --> 00:34:34.000 meaningless. So, the above is meaningless. 00:34:33.000 --> 00:34:39.000 And now, what I'd like to do is try to see what the analog of 00:34:39.000 --> 00:34:45.000 that is for second order equations, and higher order 00:34:45.000 --> 00:34:51.000 equations. If you understand second-order, 00:34:49.000 --> 00:34:55.000 that's good enough. Higher order goes exactly the 00:34:54.000 --> 00:35:00.000 same way. So, the question is, 00:34:58.000 --> 00:35:04.000 for second-order, let's make it with constant 00:35:02.000 --> 00:35:08.000 coefficients plus, I could call it b and k, 00:35:07.000 --> 00:35:13.000 oh, no, b k, or p. 00:35:11.000 --> 00:35:17.000 The trouble is, that wouldn't take care of the 00:35:14.000 --> 00:35:20.000 electrical circuits. So, I just want to use neutral 00:35:17.000 --> 00:35:23.000 letters, which suggest nothing. And, you can make them turn it 00:35:22.000 --> 00:35:28.000 into a circuit, so springs, or yet other 00:35:25.000 --> 00:35:31.000 examples undreamt of. But these are constants. 00:35:28.000 --> 00:35:34.000 And I'm going to think of it as time. 00:35:32.000 --> 00:35:38.000 I think I'll switch back to time, let x be the time. 00:35:37.000 --> 00:35:43.000 So, B y equals f of t. So, there is our equation. 00:35:43.000 --> 00:35:49.000 A and B are constants. And, the question is, 00:35:47.000 --> 00:35:53.000 the question I'm asking, can think of either of these 00:35:53.000 --> 00:35:59.000 two models or others, the question I'm asking is, 00:35:58.000 --> 00:36:04.000 under what circumstances can I make that same type of analysis 00:36:04.000 --> 00:36:10.000 into steady-state and transient? Well, what does the solution 00:36:11.000 --> 00:36:17.000 look like? The solution looks like y 00:36:15.000 --> 00:36:21.000 equals a particular solution plus c1 y1 plus c2 y2. 00:36:20.000 --> 00:36:26.000 Therefore, to make that look 00:36:24.000 --> 00:36:30.000 like this, the c1 and c2 contain the initial conditions. 00:36:31.000 --> 00:36:37.000 This part does not. Therefore, if I want to say 00:36:35.000 --> 00:36:41.000 that the solutions look like a steady state solution plus 00:36:41.000 --> 00:36:47.000 something that dies away, which becomes less and less 00:36:47.000 --> 00:36:53.000 important as time goes on, what I'm really asking is, 00:36:52.000 --> 00:36:58.000 under what circumstances is this part guaranteed to go to 00:36:58.000 --> 00:37:04.000 zero? So, the question is, 00:37:01.000 --> 00:37:07.000 when, in other words, under what conditions on the 00:37:06.000 --> 00:37:12.000 equation A and B, in effect, is what we are 00:37:10.000 --> 00:37:16.000 asking. When does c1 y1 plus c2 y2 go 00:37:14.000 --> 00:37:20.000 to zero as t goes to infinity, 00:37:19.000 --> 00:37:25.000 regardless of what c1 and c2 are for all c1 c2. 00:37:25.000 --> 00:37:31.000 Now, here there was no difficulty. 00:37:30.000 --> 00:37:36.000 We had the thing very explicitly, and you could see k 00:37:34.000 --> 00:37:40.000 is positive: this goes to zero. And if k is negative, 00:37:38.000 --> 00:37:44.000 it doesn't go to zero. It goes to infinity. 00:37:41.000 --> 00:37:47.000 Here, I want to make the same kind of analysis, 00:37:44.000 --> 00:37:50.000 except it's just going to take, it's a little more trouble. 00:37:49.000 --> 00:37:55.000 But the answer, when it finally comes out is 00:37:52.000 --> 00:37:58.000 very beautiful. So, when are all these guys 00:37:55.000 --> 00:38:01.000 going to go to zero? First of all, 00:37:58.000 --> 00:38:04.000 you might as well just have the definition. 00:38:01.000 --> 00:38:07.000 So, all the good things that this is going to imply, 00:38:05.000 --> 00:38:11.000 if this is so, in other words, 00:38:07.000 --> 00:38:13.000 if they all go to zero, everything in the complementary 00:38:12.000 --> 00:38:18.000 solution, then the ODE is called stable. 00:38:17.000 --> 00:38:23.000 Some people call it asymptotically stable. 00:38:21.000 --> 00:38:27.000 I don't know what to call it. I can make the analysis, 00:38:28.000 --> 00:38:34.000 and then I use the identical terminology, c1 y1 plus c2 y2. 00:38:36.000 --> 00:38:42.000 This is called the transient because it goes to zero. 00:38:40.000 --> 00:38:46.000 This is called the particular solution now that we labored so 00:38:45.000 --> 00:38:51.000 hard to get for the next two weeks. 00:38:47.000 --> 00:38:53.000 It's the important part. It's the steady-state part. 00:38:52.000 --> 00:38:58.000 It's what lasts out to infinity after the other stuff has 00:38:56.000 --> 00:39:02.000 disappeared. So, this is the steady-state 00:38:59.000 --> 00:39:05.000 solution, steady-state solution, okay? 00:39:04.000 --> 00:39:10.000 And, the differential equation is called stable. 00:39:07.000 --> 00:39:13.000 Now, it's of the highest interest to know when a 00:39:10.000 --> 00:39:16.000 differential equation is stable, linear differential equation is 00:39:14.000 --> 00:39:20.000 stable in this sense because you have a control. 00:39:17.000 --> 00:39:23.000 You know what its solutions look like. 00:39:20.000 --> 00:39:26.000 You have some feeling for how it's behaving in the long term. 00:39:24.000 --> 00:39:30.000 If this is not so, each equation is a law unto 00:39:27.000 --> 00:39:33.000 itself if you don't know. So, let's do the work. 00:39:31.000 --> 00:39:37.000 For the rest of the period, what I'd like to do is to find 00:39:36.000 --> 00:39:42.000 out what the conditions are, which make this true. 00:39:40.000 --> 00:39:46.000 Those were the equations which we will have a right to call 00:39:45.000 --> 00:39:51.000 stable. So, when does this happen, 00:39:48.000 --> 00:39:54.000 and where is it going to happen? 00:39:50.000 --> 00:39:56.000 I don't know. I guess, here. 00:40:05.000 --> 00:40:11.000 Now, I think the first step is fairly easy, and it will give 00:40:10.000 --> 00:40:16.000 you a good review of what we've been doing up until now. 00:40:14.000 --> 00:40:20.000 So, I'm simply going to make a case-by-case analysis. 00:40:19.000 --> 00:40:25.000 Don't worry, it won't take very long. 00:40:22.000 --> 00:40:28.000 What are the cases we've been studying? 00:40:25.000 --> 00:40:31.000 Well, what do the characteristic roots look like? 00:40:29.000 --> 00:40:35.000 The roots of the characteristic equation, in other words, 00:40:34.000 --> 00:40:40.000 remember, there are cases. The first case is they are real 00:40:41.000 --> 00:40:47.000 and distinct, r1 not equal to r2, 00:40:45.000 --> 00:40:51.000 real and distinct. What are the other cases? 00:40:50.000 --> 00:40:56.000 Well, r1 equals r2. And then, there's the case 00:40:56.000 --> 00:41:02.000 where there are complex. So, I will write it r equals a 00:41:02.000 --> 00:41:08.000 plus or minus b i. What do the solutions look 00:41:07.000 --> 00:41:13.000 like? So, my ham-handed approach to 00:41:09.000 --> 00:41:15.000 this problem is going to be, in each case, 00:41:12.000 --> 00:41:18.000 I'll look at the solutions, and first get the condition on 00:41:16.000 --> 00:41:22.000 the roots. So, in other words, 00:41:18.000 --> 00:41:24.000 I'm not going to worry right away about the a and the b. 00:41:21.000 --> 00:41:27.000 I'm going, instead, to worry about expressing this 00:41:24.000 --> 00:41:30.000 condition of stability in terms of the characteristic roots. 00:41:28.000 --> 00:41:34.000 In fact, that's the only way in which many people know the 00:41:32.000 --> 00:41:38.000 conditions. You're going to be smarter. 00:41:35.000 --> 00:41:41.000 Okay, what do the solutions look like? 00:41:38.000 --> 00:41:44.000 Well, the general solution looks like e to the r1 t plus c2 00:41:42.000 --> 00:41:48.000 e to the r2 t. 00:41:45.000 --> 00:41:51.000 Okay, so, what's the stability condition? 00:41:48.000 --> 00:41:54.000 In other words, if equation happened to have 00:41:51.000 --> 00:41:57.000 its characteristic roots, real and distinct, 00:41:54.000 --> 00:42:00.000 under what circumstances would it be stable? 00:41:57.000 --> 00:42:03.000 Would it, in other words, all its solutions go to zero? 00:42:01.000 --> 00:42:07.000 So, I'm talking about the homogeneous equation, 00:42:04.000 --> 00:42:10.000 the reduced equation, the associated homogeneous 00:42:07.000 --> 00:42:13.000 equation. Why? 00:42:10.000 --> 00:42:16.000 Because that's all that's involved in this. 00:42:13.000 --> 00:42:19.000 In other words, when I write that, 00:42:15.000 --> 00:42:21.000 I am no longer interested in the whole equation. 00:42:19.000 --> 00:42:25.000 All I'm interested in is the reduced equation, 00:42:22.000 --> 00:42:28.000 the equation where you turn the f of t on the 00:42:26.000 --> 00:42:32.000 right-hand side into zero. So, what's the stability 00:42:31.000 --> 00:42:37.000 condition? Well, let's write it out. 00:42:35.000 --> 00:42:41.000 Under what circumstances will all these guys go to zero? 00:42:41.000 --> 00:42:47.000 If r1 and r2 should be negative, can they be zero? 00:42:46.000 --> 00:42:52.000 No, because then it will be a constant and it will go to zero. 00:42:52.000 --> 00:42:58.000 How about this one? Well, in this one, 00:42:56.000 --> 00:43:02.000 it's (c1 plus c2 times t) multiplied by e to the r1 t. 00:43:01.000 --> 00:43:07.000 Of course, both of these are 00:43:08.000 --> 00:43:14.000 the same. I'll just arbitrarily pick one 00:43:11.000 --> 00:43:17.000 of them. What happens to this as things 00:43:14.000 --> 00:43:20.000 go to zero? Well, this part is rising, 00:43:16.000 --> 00:43:22.000 at least if c2 is positive. This part is either helping or 00:43:21.000 --> 00:43:27.000 it's hindering. But, I hope you know what these 00:43:24.000 --> 00:43:30.000 functions look like, and you know which of them go 00:43:28.000 --> 00:43:34.000 to zero. They go to zero if r1 is 00:43:31.000 --> 00:43:37.000 negative. It might rise in the beginning, 00:43:35.000 --> 00:43:41.000 but after a while they lose the energy. 00:43:39.000 --> 00:43:45.000 Of course, if r1 is equal to zero, what do these guys do? 00:43:44.000 --> 00:43:50.000 Linear, go to infinity. Well, we are doing okay. 00:43:49.000 --> 00:43:55.000 How about here? Well, here, it's a little more 00:43:53.000 --> 00:43:59.000 complicated. The solutions look like e to 00:43:57.000 --> 00:44:03.000 the at times (c1 cosine bt plus c2 sine bt). 00:44:03.000 --> 00:44:09.000 Now, this part is a pure 00:44:07.000 --> 00:44:13.000 oscillation. You know that. 00:44:09.000 --> 00:44:15.000 It might have a big amplitude, but whatever it does, 00:44:13.000 --> 00:44:19.000 it does the same thing all the time. 00:44:16.000 --> 00:44:22.000 So, whether this goes to zero depends entirely upon what that 00:44:20.000 --> 00:44:26.000 exponential is doing. And, that exponential goes to 00:44:24.000 --> 00:44:30.000 zero if a is negative. So here, the condition is 00:44:28.000 --> 00:44:34.000 negative. And now, the only thing left to 00:44:33.000 --> 00:44:39.000 do is to say it nicely. I've got three cases, 00:44:38.000 --> 00:44:44.000 and I want to say them all in one breath. 00:44:43.000 --> 00:44:49.000 So, the stability condition is, the ODE is stable. 00:44:49.000 --> 00:44:55.000 So, this is, or f of t. 00:44:53.000 --> 00:44:59.000 It doesn't matter. But, psychologically, 00:44:57.000 --> 00:45:03.000 you can put this as zero there, is stable if what? 00:45:05.000 --> 00:45:11.000 In case one, this is true. 00:45:07.000 --> 00:45:13.000 In case two, that's true. 00:45:10.000 --> 00:45:16.000 In case three, that's true. 00:45:13.000 --> 00:45:19.000 But that's ugly. Make it beautiful. 00:45:17.000 --> 00:45:23.000 The beautiful way of saying it is if all the characteristic 00:45:23.000 --> 00:45:29.000 roots have negative real parts. If the characteristic roots, 00:45:31.000 --> 00:45:37.000 the r's or the a plus or minus b i, have negative real part. 00:45:38.000 --> 00:45:44.000 That's the form in which the electrical engineers will nod 00:45:44.000 --> 00:45:50.000 their head, tell you, yeah, that's right, 00:45:49.000 --> 00:45:55.000 negative real part, sorry. 00:45:52.000 --> 00:45:58.000 Isn't it right? Is that right here? 00:45:56.000 --> 00:46:02.000 Yeah. What's the real part of these 00:45:59.000 --> 00:46:05.000 guys? They themselves, 00:46:03.000 --> 00:46:09.000 because they are real. What's the real part of this? 00:46:09.000 --> 00:46:15.000 Yeah. The only case in which I really 00:46:13.000 --> 00:46:19.000 had to use real part is when I talk about the complex case 00:46:20.000 --> 00:46:26.000 because a is just the real part of a complex number. 00:46:26.000 --> 00:46:32.000 It's not the whole thing.