WEBVTT 00:00:07.000 --> 00:00:13.000 The real topic is how to solve inhomogeneous systems, 00:00:11.000 --> 00:00:17.000 but the subtext is what I wrote on the board. 00:00:14.000 --> 00:00:20.000 I think you will see that really thinking in terms of 00:00:18.000 --> 00:00:24.000 matrices makes certain things a lot easier than they would be 00:00:23.000 --> 00:00:29.000 otherwise. And I hope to give you a couple 00:00:26.000 --> 00:00:32.000 of examples of that today in connection with solving systems 00:00:30.000 --> 00:00:36.000 of inhomogeneous equations. Now, there is a little problem. 00:00:36.000 --> 00:00:42.000 We have to have a little bit of theory ahead of time before 00:00:41.000 --> 00:00:47.000 that, which I thought rather than interrupt the presentation 00:00:47.000 --> 00:00:53.000 as I try to talk about the inhomogeneous systems it would 00:00:51.000 --> 00:00:57.000 be better to put a little theory in the beginning. 00:00:56.000 --> 00:01:02.000 I think you will find it harmless. 00:01:00.000 --> 00:01:06.000 And about half of it you know already. 00:01:03.000 --> 00:01:09.000 The theory I am talking about is, in general, 00:01:07.000 --> 00:01:13.000 the theory of the systems x prime equal a x. 00:01:11.000 --> 00:01:17.000 I will just state it when n is equal to two. 00:01:15.000 --> 00:01:21.000 A two-by-two system likely you have had up until now. 00:01:19.000 --> 00:01:25.000 It is also true for end-by-end. It is just a little more 00:01:24.000 --> 00:01:30.000 tedious to write out and to give the definitions. 00:01:30.000 --> 00:01:36.000 Here is a little two-by-two system. 00:01:33.000 --> 00:01:39.000 It is homogeneous. There are no zeros. 00:01:37.000 --> 00:01:43.000 And it is not necessary to assume this, but since the 00:01:42.000 --> 00:01:48.000 matrix is going to be constant until the end of the term let's 00:01:48.000 --> 00:01:54.000 assume it in and not go for a spurious generality. 00:01:53.000 --> 00:01:59.000 So constant matrices like you will have on your homework. 00:02:00.000 --> 00:02:06.000 Now, there are two theorems, or maybe three that I want you 00:02:05.000 --> 00:02:11.000 to know, that you need to know in order to understand what is 00:02:10.000 --> 00:02:16.000 going on. The first one, 00:02:13.000 --> 00:02:19.000 fortunately, is already in your bloodstream, 00:02:16.000 --> 00:02:22.000 I hope. Let's call it theorem A. 00:02:19.000 --> 00:02:25.000 It is simply the one that says that the general solution to the 00:02:25.000 --> 00:02:31.000 system, that system I wrote on the board, the two-by-two system 00:02:31.000 --> 00:02:37.000 is what you know it to be. Namely, from all the examples 00:02:36.000 --> 00:02:42.000 that you have calculated. It is a linear combination with 00:02:40.000 --> 00:02:46.000 arbitrary constants for the coefficients of two solutions. 00:02:44.000 --> 00:02:50.000 In other words, to solve it, 00:02:46.000 --> 00:02:52.000 to find the general solution you put all your energy into 00:02:49.000 --> 00:02:55.000 finding two independent solutions. 00:02:52.000 --> 00:02:58.000 And then, as soon as you found them, the general one is gotten 00:02:56.000 --> 00:03:02.000 by combining those with arbitrary constants. 00:03:00.000 --> 00:03:06.000 The only thing to specify is what the x1 and the x2 are. 00:03:04.000 --> 00:03:10.000 "Where," I guess, would be the right word to use. 00:03:08.000 --> 00:03:14.000 Where x1 and x2 are two solutions, but neither must be a 00:03:12.000 --> 00:03:18.000 constant multiple of the other. That is the only thing I want 00:03:17.000 --> 00:03:23.000 to stress, they have to be independent. 00:03:20.000 --> 00:03:26.000 Or, as it is better to say, linearly independent. 00:03:24.000 --> 00:03:30.000 Are two linearly independent solutions. 00:03:34.000 --> 00:03:40.000 And department of fuller explanation, i.e., 00:03:37.000 --> 00:03:43.000 neither is a constant multiple of the other. 00:03:50.000 --> 00:03:56.000 That is what it means to be linearly independent. 00:03:52.000 --> 00:03:58.000 Now, this theorem I am not going to prove. 00:03:55.000 --> 00:04:01.000 I am just going to say that the proof is a lot like the one for 00:03:59.000 --> 00:04:05.000 second order equations. It has an easy part and a hard 00:04:03.000 --> 00:04:09.000 part. The easy part is to show that 00:04:05.000 --> 00:04:11.000 all of these guys are solutions. And, in fact, 00:04:08.000 --> 00:04:14.000 that is almost self-evident by looking at the equation. 00:04:12.000 --> 00:04:18.000 For example, if x1 and x2, 00:04:14.000 --> 00:04:20.000 each of those solve that equation so does their sum 00:04:17.000 --> 00:04:23.000 because, when you plug it in, you differentiate the sum by 00:04:21.000 --> 00:04:27.000 differentiating each term and adding. 00:04:24.000 --> 00:04:30.000 And here A times x1 plus x2 is Ax1 plus Ax2. 00:04:28.000 --> 00:04:34.000 In other words, 00:04:31.000 --> 00:04:37.000 you are using the linearity and the superposition principle. 00:04:36.000 --> 00:04:42.000 It is easy to show that all of these, well, maybe I should 00:04:42.000 --> 00:04:48.000 actually write something down instead of just talking. 00:04:47.000 --> 00:04:53.000 Easy that all these are solutions. 00:04:50.000 --> 00:04:56.000 Every one of those guys, regardless of what c1 and c2 00:04:55.000 --> 00:05:01.000 is, is a solution. That is linearity, 00:04:59.000 --> 00:05:05.000 if I use that buzz word, plus the superposition 00:05:02.000 --> 00:05:08.000 principle, that the sum of two solutions is a solution. 00:05:12.000 --> 00:05:18.000 The hard thing is not to show that these are solutions but to 00:05:16.000 --> 00:05:22.000 show that these are all the solutions, that there are no 00:05:21.000 --> 00:05:27.000 other solutions. No matter how you do that it is 00:05:24.000 --> 00:05:30.000 hard. The hard thing is that there 00:05:27.000 --> 00:05:33.000 are no other solutions. These are all. 00:05:31.000 --> 00:05:37.000 Now, you could sort of say, well, it has two arbitrary 00:05:35.000 --> 00:05:41.000 constants in it. That is sort of a rough and 00:05:38.000 --> 00:05:44.000 ready reason, but it is not considered 00:05:41.000 --> 00:05:47.000 adequate by mathematicians. And, in fact, 00:05:44.000 --> 00:05:50.000 I could go into a song and dance as to just why it is 00:05:48.000 --> 00:05:54.000 inadequate. But we have other things to do, 00:05:52.000 --> 00:05:58.000 bigger fish to fry, as they say. 00:05:54.000 --> 00:06:00.000 Let's fry a fish. No, we have another theorem 00:05:58.000 --> 00:06:04.000 first. This one it is mostly the words 00:06:04.000 --> 00:06:10.000 that I am interested in. Once again, we have our old 00:06:11.000 --> 00:06:17.000 friend the Wronskian back. The Wronskian of what? 00:06:18.000 --> 00:06:24.000 Of two solutions. It is the Wronskian of the 00:06:25.000 --> 00:06:31.000 solution x1 and x2. They don't, by the way, 00:06:30.000 --> 00:06:36.000 have to be independent. Just two solutions to the 00:06:34.000 --> 00:06:40.000 system. And what is it? 00:06:36.000 --> 00:06:42.000 Hey, didn't we already have a Wronskian? 00:06:39.000 --> 00:06:45.000 Yeah. Forget about that one for the 00:06:42.000 --> 00:06:48.000 moment. Postpone it for a minute. 00:06:45.000 --> 00:06:51.000 This is a determinant, just like the old one way. 00:07:00.000 --> 00:07:06.000 This is going to be a great lecture. 00:07:03.000 --> 00:07:09.000 (x1, x2). Now what is this? 00:07:06.000 --> 00:07:12.000 x1 is a column vector, right? 00:07:08.000 --> 00:07:14.000 x2 is a column vector. Two things in it. 00:07:12.000 --> 00:07:18.000 Two things in it. Together they make a square 00:07:17.000 --> 00:07:23.000 matrix. And this means it is 00:07:19.000 --> 00:07:25.000 determinant. It is the determinant of this. 00:07:24.000 --> 00:07:30.000 It is a determinant, in other words, 00:07:27.000 --> 00:07:33.000 of a square matrix. And that is what it is. 00:07:32.000 --> 00:07:38.000 I will change this equality. To indicate it is a definition, 00:07:37.000 --> 00:07:43.000 I will put the colon there, which is what you add, 00:07:41.000 --> 00:07:47.000 to indicate this is only equal because I say so. 00:07:44.000 --> 00:07:50.000 It is a definition, in other words. 00:07:47.000 --> 00:07:53.000 Now, there is a connection between this and the earlier 00:07:51.000 --> 00:07:57.000 Wronskian which I, unfortunately, 00:07:54.000 --> 00:08:00.000 cannot explain to you because you are going to explain it to 00:07:58.000 --> 00:08:04.000 me. I gave it to you as part one of 00:08:02.000 --> 00:08:08.000 your homework problem. Make sure you do it. 00:08:05.000 --> 00:08:11.000 And, if you cannot remember what the old Wronskian is, 00:08:08.000 --> 00:08:14.000 please look it up in the book. Don't look it up in the 00:08:12.000 --> 00:08:18.000 solution to the problem. If you do that you will learn 00:08:16.000 --> 00:08:22.000 something. Then you will see how, 00:08:18.000 --> 00:08:24.000 in a certain sense, this is a more general 00:08:21.000 --> 00:08:27.000 definition than I gave you before. 00:08:23.000 --> 00:08:29.000 The one I gave you before is, in a certain sense, 00:08:27.000 --> 00:08:33.000 a special case of it. Now that is just the 00:08:31.000 --> 00:08:37.000 definition. There is a theorem. 00:08:34.000 --> 00:08:40.000 And the theorem is going to look just like the one we had 00:08:39.000 --> 00:08:45.000 for second order equations, if you can remember back that 00:08:43.000 --> 00:08:49.000 far. The theorem is that if these 00:08:46.000 --> 00:08:52.000 are two solutions there are only two possibilities for the 00:08:51.000 --> 00:08:57.000 Wronskian. So either or. 00:08:53.000 --> 00:08:59.000 Either the Wronskian is -- Now, the Wronskian, 00:08:58.000 --> 00:09:04.000 these are functions, the column vectors are the 00:09:03.000 --> 00:09:09.000 solutions, so those are functions of the variable t, 00:09:08.000 --> 00:09:14.000 so are these. The Wronskian as a whole is a 00:09:12.000 --> 00:09:18.000 function of the independent variable t after you have 00:09:18.000 --> 00:09:24.000 calculated out that determinant. I will write it now this way to 00:09:24.000 --> 00:09:30.000 indicate that it s a function of t. 00:09:28.000 --> 00:09:34.000 Either the Wronskian is -- One possibility is identically 00:09:35.000 --> 00:09:41.000 zero. That is zero for all values of 00:09:39.000 --> 00:09:45.000 t, in other words. And this happens if x1 and x2 00:09:45.000 --> 00:09:51.000 are not linearly independent. Usually people just say 00:09:51.000 --> 00:09:57.000 dependent and hope they are interpreted correctly. 00:09:57.000 --> 00:10:03.000 Are dependent. But since I did not explain 00:10:03.000 --> 00:10:09.000 what dependent means, I will say it. 00:10:07.000 --> 00:10:13.000 Not linearly independent. I know that is horrible, 00:10:12.000 --> 00:10:18.000 but nobody has figured out another way to say it. 00:10:18.000 --> 00:10:24.000 That is one possibility, or the opposite of this is 00:10:23.000 --> 00:10:29.000 never zero for any t value. I mean a normal function is 00:10:29.000 --> 00:10:35.000 zero here and there, and the rest of the time not 00:10:33.000 --> 00:10:39.000 zero. Well, not this Wronskian. 00:10:35.000 --> 00:10:41.000 You only have two choices. Either it is zero all the time 00:10:38.000 --> 00:10:44.000 or it is never zero. It is like the function e to 00:10:42.000 --> 00:10:48.000 the t. In other words, 00:10:43.000 --> 00:10:49.000 an exponential which is never zero, always positive and never 00:10:48.000 --> 00:10:54.000 zero. Or, it could be a constant. 00:10:50.000 --> 00:10:56.000 Anyway, it has to be a function which is never zero. 00:10:53.000 --> 00:10:59.000 And this happens in the other case, so this is -- 00:10:58.000 --> 00:11:04.000 There is no place to write it. This is the case if x1 and x2 00:11:05.000 --> 00:11:11.000 are independent, by which I mean linearly 00:11:11.000 --> 00:11:17.000 independent. It is just I didn't have room 00:11:16.000 --> 00:11:22.000 to write it. That is pretty much the end of 00:11:22.000 --> 00:11:28.000 the theory. And now, let's start in on the 00:11:27.000 --> 00:11:33.000 matrices. The basic new matrix we are 00:11:31.000 --> 00:11:37.000 going to be talking about this period and next one on Monday 00:11:35.000 --> 00:11:41.000 also is the way that most people who work with systems actually 00:11:40.000 --> 00:11:46.000 look at the solutions to systems, so it is important you 00:11:43.000 --> 00:11:49.000 learn this word and this way of looking at it. 00:11:47.000 --> 00:11:53.000 What they do is look not at each solution separately, 00:11:50.000 --> 00:11:56.000 as we have been doing up until now. 00:11:52.000 --> 00:11:58.000 They put them all together in a single matrix. 00:11:57.000 --> 00:12:03.000 And it is the properties of that matrix that they study and 00:12:02.000 --> 00:12:08.000 try to do the calculations using. 00:12:04.000 --> 00:12:10.000 And that matrix is called the fundamental matrix for the 00:12:09.000 --> 00:12:15.000 system. 00:12:17.000 --> 00:12:23.000 Sometimes people don't bother writing in the whole system. 00:12:20.000 --> 00:12:26.000 They just say it is a fundamental matrix for A 00:12:23.000 --> 00:12:29.000 because, after all, A is the only thing that is 00:12:25.000 --> 00:12:31.000 varying there. Once you know A, 00:12:27.000 --> 00:12:33.000 you know what the system is. So what is this guy? 00:12:31.000 --> 00:12:37.000 Well, it is a two-by-two matrix. 00:12:33.000 --> 00:12:39.000 And it is the most harmless thing. 00:12:35.000 --> 00:12:41.000 It is the precursor of the Wronskian. 00:12:38.000 --> 00:12:44.000 It is what the Wronskian was before the determinant was 00:12:42.000 --> 00:12:48.000 taken. In other words, 00:12:44.000 --> 00:12:50.000 it is the matrix whose two columns are those two solutions. 00:12:48.000 --> 00:12:54.000 The other question is what we are going to call it. 00:12:52.000 --> 00:12:58.000 I kept trying everything and settled on calling it capital X 00:12:56.000 --> 00:13:02.000 because I think that is the one that guides you in the 00:13:00.000 --> 00:13:06.000 calculations the best. This is definition two, 00:13:06.000 --> 00:13:12.000 so colon equality. Notice I am not using vertical 00:13:10.000 --> 00:13:16.000 lines now because that would mean a determinant. 00:13:15.000 --> 00:13:21.000 It is the matrix whose columns are two independent solutions. 00:13:34.000 --> 00:13:40.000 Is that all? Yeah. 00:13:35.000 --> 00:13:41.000 You just put them side-by-side. Why? 00:13:38.000 --> 00:13:44.000 That will come out. Why should one do this? 00:13:41.000 --> 00:13:47.000 Well, first of all, in order not to interrupt the 00:13:45.000 --> 00:13:51.000 basic calculation that I want to make with this during the 00:13:50.000 --> 00:13:56.000 period, it has two basic properties that we are going to 00:13:54.000 --> 00:14:00.000 need during this period. These are the properties. 00:14:00.000 --> 00:14:06.000 Just two. And one is obvious and the 00:14:02.000 --> 00:14:08.000 other you will think, I hope, is a little less 00:14:06.000 --> 00:14:12.000 familiar. I think you will see there is 00:14:09.000 --> 00:14:15.000 nothing to it. It is just a way of talking, 00:14:13.000 --> 00:14:19.000 really. The first is the one that is 00:14:16.000 --> 00:14:22.000 already embedded in the theorem, namely that the determinant of 00:14:21.000 --> 00:14:27.000 the fundamental matrix is not zero for any t. 00:14:24.000 --> 00:14:30.000 Why? Well, I just told you it 00:14:27.000 --> 00:14:33.000 wasn't. This is the Wronskian. 00:14:31.000 --> 00:14:37.000 The Wronskian is never zero? Why is it never zero? 00:14:35.000 --> 00:14:41.000 Well, because I said these columns had to be independent 00:14:40.000 --> 00:14:46.000 solutions. So this is not just not zero, 00:14:44.000 --> 00:14:50.000 it is never zero. It is not zero for any value of 00:14:48.000 --> 00:14:54.000 t. That is good. 00:14:50.000 --> 00:14:56.000 As you will see, we are going to need that 00:14:54.000 --> 00:15:00.000 property. But the other one is a little 00:14:57.000 --> 00:15:03.000 stranger. The only thing I can say is, 00:15:02.000 --> 00:15:08.000 get used to it. Namely that X prime equals AX. 00:15:06.000 --> 00:15:12.000 Now, why is that strange? That is not the same as this. 00:15:11.000 --> 00:15:17.000 This is a column vector. That is a square matrix and 00:15:15.000 --> 00:15:21.000 this is a column vector. This is not a column vector. 00:15:20.000 --> 00:15:26.000 This is a square matrix. This is what is called a matrix 00:15:26.000 --> 00:15:32.000 differential equation where the variable is not a single x or a 00:15:32.000 --> 00:15:38.000 column vector of a set of x's like the x and the y. 00:15:37.000 --> 00:15:43.000 It is a whole matrix. 00:15:50.000 --> 00:15:56.000 Well, first of all, I should say what is it saying? 00:15:53.000 --> 00:15:59.000 This is a two-by-two matrix. When I multiply them I get a 00:15:58.000 --> 00:16:04.000 two-by-two matrix. What is this? 00:16:00.000 --> 00:16:06.000 This is a two-by-two matrix, every entry of which has been 00:16:04.000 --> 00:16:10.000 differentiated. That is what it means to put 00:16:08.000 --> 00:16:14.000 that prime there. To differentiate a matrix means 00:16:11.000 --> 00:16:17.000 nothing fancy. It just means differentiate 00:16:14.000 --> 00:16:20.000 every entry. It is just like to 00:16:16.000 --> 00:16:22.000 differentiate a vector (x, y), to make a velocity vector 00:16:20.000 --> 00:16:26.000 you differentiate the x and the y. 00:16:22.000 --> 00:16:28.000 Well, a column vector is a special kind of matrix. 00:16:26.000 --> 00:16:32.000 The definition applies to any matrix. 00:16:30.000 --> 00:16:36.000 Well, why is that so? I state it as a property, 00:16:33.000 --> 00:16:39.000 but I will continue it by giving you, so to speak, 00:16:37.000 --> 00:16:43.000 the proof of it. In fact, there is nothing in 00:16:40.000 --> 00:16:46.000 this. It is nothing more than a 00:16:42.000 --> 00:16:48.000 little matrix calculation of the most primitive kind. 00:16:46.000 --> 00:16:52.000 Namely, what does this mean? Let's try to undo that. 00:16:50.000 --> 00:16:56.000 What does the left-hand side really mean? 00:16:53.000 --> 00:16:59.000 Well, if that is what x means, the left-hand side must mean 00:16:58.000 --> 00:17:04.000 the derivative of the first column. 00:17:02.000 --> 00:17:08.000 That is its first column. And the derivative of the 00:17:05.000 --> 00:17:11.000 second column. That is what it means to 00:17:07.000 --> 00:17:13.000 differentiate the matrix X. You differentiate each column 00:17:11.000 --> 00:17:17.000 separately. And to differentiate the column 00:17:14.000 --> 00:17:20.000 you need to differentiate every function in it. 00:17:17.000 --> 00:17:23.000 Well, what does the right-hand side mean? 00:17:19.000 --> 00:17:25.000 Well, I am supposed to take A and multiply that 00:17:23.000 --> 00:17:29.000 by [x1,x2]. Now, I don't know how to prove 00:17:26.000 --> 00:17:32.000 this, except ask you to think about it. 00:17:30.000 --> 00:17:36.000 Or, I could write it all out here. 00:17:34.000 --> 00:17:40.000 But think of this as a bing, bing, bing, bing. 00:17:39.000 --> 00:17:45.000 And this is a bing, bing. 00:17:42.000 --> 00:17:48.000 And this is a bong, bong. 00:17:45.000 --> 00:17:51.000 How do I do the multiplication? In other words, 00:17:50.000 --> 00:17:56.000 what is in the first column of the matrix? 00:17:55.000 --> 00:18:01.000 Well, it is dah, dah, and the lower thing is 00:18:01.000 --> 00:18:07.000 dah, dah. In other words, 00:18:03.000 --> 00:18:09.000 it is A times x1. 00:18:15.000 --> 00:18:21.000 Shut your eyes and visualize it. 00:18:17.000 --> 00:18:23.000 Got it? Dah, dah is the top entry, 00:18:19.000 --> 00:18:25.000 and dah, dah is the bottom entry. 00:18:22.000 --> 00:18:28.000 It is what you get by multiplying A by the column 00:18:25.000 --> 00:18:31.000 vector x1. And the same way the other guy 00:18:28.000 --> 00:18:34.000 is -- -- what you get by multiplying 00:18:33.000 --> 00:18:39.000 A by the column vector x2. This is just matrix 00:18:37.000 --> 00:18:43.000 multiplication. That is the law of matrix 00:18:41.000 --> 00:18:47.000 multiplication. That is how you multiply 00:18:45.000 --> 00:18:51.000 matrices. Well, good, but where does this 00:18:49.000 --> 00:18:55.000 get us? What does it mean for those two 00:18:53.000 --> 00:18:59.000 guys to be equal? That is going to happen, 00:18:57.000 --> 00:19:03.000 if and only if x1 prime is equal to A x1. 00:19:02.000 --> 00:19:08.000 This guy equals that guy. And similarly for the x2's. 00:19:14.000 --> 00:19:20.000 The end result is that this matrix, saying that the 00:19:18.000 --> 00:19:24.000 fundamental matrix satisfies this matrix differential 00:19:22.000 --> 00:19:28.000 equation is only a way of saying, in one breath, 00:19:26.000 --> 00:19:32.000 that its two columns are both solutions to the original 00:19:30.000 --> 00:19:36.000 system. It is, so to speak, 00:19:33.000 --> 00:19:39.000 an efficient way of turning these two equations into a 00:19:38.000 --> 00:19:44.000 single equation by making a matrix. 00:19:41.000 --> 00:19:47.000 I guess it is time, finally, to come to the topic 00:19:46.000 --> 00:19:52.000 of the lecture. I said the thing the matrices 00:19:50.000 --> 00:19:56.000 were going to be used for is solving inhomogeneous systems, 00:19:55.000 --> 00:20:01.000 so let's take a look at those. I thought I would give you an 00:20:00.000 --> 00:20:06.000 example. Inhomogeneous systems. 00:20:10.000 --> 00:20:16.000 Well, what is one going to look like? 00:20:12.000 --> 00:20:18.000 So far what we have done is, up until now has been solving, 00:20:17.000 --> 00:20:23.000 we spent essentially two weeks solving and plotting the 00:20:21.000 --> 00:20:27.000 solutions to homogeneous systems. 00:20:23.000 --> 00:20:29.000 There was nothing over there. And homogeneous systems, 00:20:27.000 --> 00:20:33.000 in fact, with constant coefficients. 00:20:31.000 --> 00:20:37.000 Stuff that looked like that that we abbreviated with 00:20:34.000 --> 00:20:40.000 matrices. Now, to make the system 00:20:36.000 --> 00:20:42.000 inhomogeneous what I do is add the extra term on the right-hand 00:20:41.000 --> 00:20:47.000 side, which is some function of t. 00:20:43.000 --> 00:20:49.000 Except, I will have to have two functions of t because I have 00:20:47.000 --> 00:20:53.000 two equations. Now it is inhomogeneous. 00:20:50.000 --> 00:20:56.000 And what makes it inhomogeneous is the fact that these are not 00:20:54.000 --> 00:21:00.000 zero anymore. There is something there. 00:20:57.000 --> 00:21:03.000 Functions of t are there. These are given functions of t 00:21:02.000 --> 00:21:08.000 like exponentials, polynomials, 00:21:04.000 --> 00:21:10.000 the usual stuff you have on the right-hand side of the 00:21:08.000 --> 00:21:14.000 differential equation. What is confusing here is that 00:21:11.000 --> 00:21:17.000 when we studied second order equations it was homogeneous if 00:21:15.000 --> 00:21:21.000 the right-hand side was zero, and if there was something else 00:21:20.000 --> 00:21:26.000 there it was inhomogeneous. Unfortunately, 00:21:23.000 --> 00:21:29.000 I have stuck this stuff on the right-hand side so it is not 00:21:27.000 --> 00:21:33.000 quite so clear anymore. It has got to look like that, 00:21:32.000 --> 00:21:38.000 in other words. How would the matrix 00:21:34.000 --> 00:21:40.000 abbreviation look? Well, the left-hand side is x 00:21:37.000 --> 00:21:43.000 prime. The homogenous part is ax, 00:21:40.000 --> 00:21:46.000 just as it has always been. The only extra part is those 00:21:43.000 --> 00:21:49.000 functions r. And this is a column vector, 00:21:46.000 --> 00:21:52.000 after the multiplication this is a column vector, 00:21:50.000 --> 00:21:56.000 what is left is column vector. Now, explicitly it is a 00:21:53.000 --> 00:21:59.000 function of t, given by explicit functions of 00:21:56.000 --> 00:22:02.000 t, again, like exponentials. Or, they could be fancy 00:22:02.000 --> 00:22:08.000 functions. That is the thing we are trying 00:22:06.000 --> 00:22:12.000 to solve. Why don't I put it up in green? 00:22:10.000 --> 00:22:16.000 Our new and better and improved system. 00:22:13.000 --> 00:22:19.000 Think back to what we did when we studied inhomogeneous 00:22:18.000 --> 00:22:24.000 equations. We are not talking about 00:22:22.000 --> 00:22:28.000 systems but just a single equation. 00:22:25.000 --> 00:22:31.000 What we did was the main theorem -- 00:22:30.000 --> 00:22:36.000 I guess there are going to be three theorems today, 00:22:36.000 --> 00:22:42.000 not just two. Theorem C. 00:22:39.000 --> 00:22:45.000 Is that right? Yes, A, B. 00:22:42.000 --> 00:22:48.000 We are up to C. Theorem C says that the general 00:22:48.000 --> 00:22:54.000 solution, that is, the general solution to the 00:22:54.000 --> 00:23:00.000 system, is equal to the complimentary function, 00:23:00.000 --> 00:23:06.000 which is the general solution to x prime equals Ax, 00:23:06.000 --> 00:23:12.000 -- -- the homogeneous equation, 00:23:11.000 --> 00:23:17.000 in other words, plus, what am I going to call 00:23:15.000 --> 00:23:21.000 it? (x)p, right you are, 00:23:17.000 --> 00:23:23.000 a particular solution. But the principle is the same 00:23:22.000 --> 00:23:28.000 and is proved exactly the same way. 00:23:25.000 --> 00:23:31.000 It is just linearity and superposition. 00:23:35.000 --> 00:23:41.000 The linearity of the original system and the superposition 00:23:40.000 --> 00:23:46.000 principle. The essence is that to solve 00:23:43.000 --> 00:23:49.000 this inhomogeneous system, what we have to do is find a 00:23:48.000 --> 00:23:54.000 particular solution. This part I already know how to 00:23:52.000 --> 00:23:58.000 do. We have been doing that for two 00:23:55.000 --> 00:24:01.000 weeks. The new thing is to find this. 00:24:05.000 --> 00:24:11.000 Now, if you remember back before spring break, 00:24:08.000 --> 00:24:14.000 most of the work in solving the second order equation was in 00:24:12.000 --> 00:24:18.000 finding that particular solution. 00:24:15.000 --> 00:24:21.000 You quickly enough learned how to solve the homogeneous 00:24:19.000 --> 00:24:25.000 equation, but there was no real general method for finding this. 00:24:24.000 --> 00:24:30.000 We had an exponential input theorem with some modifications 00:24:28.000 --> 00:24:34.000 to it. We took a week's detour in 00:24:32.000 --> 00:24:38.000 Fourier series to see how to do it for periodic functions or 00:24:37.000 --> 00:24:43.000 functions defined on finite intervals. 00:24:40.000 --> 00:24:46.000 There were other techniques which I did not get around to 00:24:45.000 --> 00:24:51.000 showing you, techniques involving the so-called method 00:24:50.000 --> 00:24:56.000 of undetermined coefficients. Although, some of you peaked in 00:24:55.000 --> 00:25:01.000 your book and learned it from there. 00:25:00.000 --> 00:25:06.000 But the work is in finding (x)p. 00:25:03.000 --> 00:25:09.000 The miracle that occurs here, by contrast, 00:25:07.000 --> 00:25:13.000 is that it turns out to be easy to find (x)p. 00:25:11.000 --> 00:25:17.000 And easy in this further sense that I do not have to restrict 00:25:17.000 --> 00:25:23.000 the kind of function I use. For example, 00:25:21.000 --> 00:25:27.000 the second homework problem I have given you, 00:25:26.000 --> 00:25:32.000 the second part two homework problem. 00:25:36.000 --> 00:25:42.000 You will see how to use systems. 00:25:38.000 --> 00:25:44.000 For example, to solve this simple equation, 00:25:42.000 --> 00:25:48.000 I will write it out for you, consider that equation, 00:25:46.000 --> 00:25:52.000 tangent t. What technique will you apply 00:25:50.000 --> 00:25:56.000 to solve that? In other words, 00:25:52.000 --> 00:25:58.000 suppose you wanted to find a particular solution to that. 00:25:57.000 --> 00:26:03.000 The right-hand side is not an exponential. 00:26:00.000 --> 00:26:06.000 It is not a polynomial. It is not like sine or cosine 00:26:06.000 --> 00:26:12.000 of bt. I could use the Laplace 00:26:10.000 --> 00:26:16.000 transform. No, because you don't know how 00:26:14.000 --> 00:26:20.000 to take the Laplace transform of tangent t. 00:26:19.000 --> 00:26:25.000 Neither, for that matter, do I. 00:26:21.000 --> 00:26:27.000 Fourier series. Not a good choice for a 00:26:25.000 --> 00:26:31.000 function that goes to infinity at pi over two. 00:26:35.000 --> 00:26:41.000 So you cannot do this until you do your homework. 00:26:39.000 --> 00:26:45.000 Now you will be able to do it. In other words, 00:26:44.000 --> 00:26:50.000 one of the big things is not only will I give you a formula 00:26:50.000 --> 00:26:56.000 for the Xp but that formula will work even for tangent t, 00:26:56.000 --> 00:27:02.000 any function at all. Well, I thought I would try to 00:27:00.000 --> 00:27:06.000 put a little meat on the bones of the inhomogeneous systems by 00:27:04.000 --> 00:27:10.000 actually giving you a physical problem so we would actually be 00:27:08.000 --> 00:27:14.000 able to solve a physical problem instead of just demonstrate a 00:27:12.000 --> 00:27:18.000 solution method. Here is a mixing problem. 00:27:23.000 --> 00:27:29.000 Just to illustrate what makes a system of equations 00:27:27.000 --> 00:27:33.000 inhomogeneous, here at two ugly tanks. 00:27:31.000 --> 00:27:37.000 I am not going to draw these carefully, but they are both 1 00:27:37.000 --> 00:27:43.000 liter. And they are connected by 00:27:40.000 --> 00:27:46.000 pipes. And I won't bother opening 00:27:43.000 --> 00:27:49.000 holes in them. There is a pipe with fluids 00:27:47.000 --> 00:27:53.000 flowing back there and this direction it is flowing this 00:27:52.000 --> 00:27:58.000 way, but that is not the end. The end is there is stuff 00:28:00.000 --> 00:28:06.000 coming in to both of them. And I think I will just make it 00:28:08.000 --> 00:28:14.000 coming out of this one. There is something realistic. 00:28:15.000 --> 00:28:21.000 The numbers 2, 3, 2. 00:28:18.000 --> 00:28:24.000 Let's start there and see what the others have to be. 00:28:25.000 --> 00:28:31.000 So these are flow rates. One liter tanks. 00:28:31.000 --> 00:28:37.000 The flow rates are in, let's say, liters per hour. 00:28:38.000 --> 00:28:44.000 And I have some dissolved substance in, 00:28:42.000 --> 00:28:48.000 so here is going to be x salt in there and the same chemical 00:28:50.000 --> 00:28:56.000 in there, whatever it is. x is the amount of salt, 00:28:56.000 --> 00:29:02.000 let's say, in tank one. And y, the same thing in tank 00:29:02.000 --> 00:29:08.000 two. Now, if you have stuff flowing 00:29:06.000 --> 00:29:12.000 unequally this way, you must have balance. 00:29:09.000 --> 00:29:15.000 You have to make sure that neither tank is getting emptied 00:29:15.000 --> 00:29:21.000 or bursting and exploding. What is flowing in? 00:29:19.000 --> 00:29:25.000 What is x? Three is going out, 00:29:22.000 --> 00:29:28.000 two is coming in, so this has to be one in order 00:29:27.000 --> 00:29:33.000 that tank x stay full and not explode. 00:29:32.000 --> 00:29:38.000 And how about y? How much is going out? 00:29:35.000 --> 00:29:41.000 Two there and two here. Four is going out, 00:29:39.000 --> 00:29:45.000 three is coming in. This also has to be one. 00:29:43.000 --> 00:29:49.000 Those are just the flow rates of water or the liquid that is 00:29:48.000 --> 00:29:54.000 coming in. Now, the only thing I am going 00:29:52.000 --> 00:29:58.000 to specify is the concentration of what is coming in. 00:29:58.000 --> 00:30:04.000 Here the concentration is 5 e to the minus t. 00:30:02.000 --> 00:30:08.000 And that is what makes the problem inhomogeneous. 00:30:07.000 --> 00:30:13.000 Here the concentration is going to be zero. 00:30:10.000 --> 00:30:16.000 In other words, pure water is flowing in here 00:30:14.000 --> 00:30:20.000 to create the liquid balance. Here, on the other hand, 00:30:18.000 --> 00:30:24.000 salt solution is flowing in but with a steadily declining 00:30:23.000 --> 00:30:29.000 concentration. So, what is the system? 00:30:28.000 --> 00:30:34.000 Well, you have set it up exactly the way you did when you 00:30:34.000 --> 00:30:40.000 studied first order equations. It is inflow minus outflow. 00:30:41.000 --> 00:30:47.000 What is the outflow? The outflow is all in this 00:30:46.000 --> 00:30:52.000 pipe. The flow rates are liters per 00:30:50.000 --> 00:30:56.000 hour. Three liters per hour flowing 00:30:54.000 --> 00:31:00.000 out. How much salt does that 00:30:57.000 --> 00:31:03.000 represent? It is negative three times the 00:31:02.000 --> 00:31:08.000 concentration of salt. But the concentration, 00:31:06.000 --> 00:31:12.000 notice, equals x divided by one. 00:31:10.000 --> 00:31:16.000 In other words, x represents both the 00:31:13.000 --> 00:31:19.000 concentration and the amount. So I don't have to distinguish. 00:31:18.000 --> 00:31:24.000 If I had made it two liter tanks then I would have had to 00:31:23.000 --> 00:31:29.000 divide this by two. I am cheating, 00:31:26.000 --> 00:31:32.000 but it is enough already. x prime equals minus 3x. 00:31:32.000 --> 00:31:38.000 That is what is going out. What is coming in? 00:31:36.000 --> 00:31:42.000 Well, 2y is coming in. Concentration here. 00:31:39.000 --> 00:31:45.000 What is coming in? Is it y 2 liter? 00:31:43.000 --> 00:31:49.000 Plus what is coming in from the outside. 00:31:46.000 --> 00:31:52.000 We have to add that in, and that will be plus 5 e to 00:31:51.000 --> 00:31:57.000 the negative t. How about y? 00:31:54.000 --> 00:32:00.000 y prime is changing. What comes in from x? 00:32:00.000 --> 00:32:06.000 That is 3x. What goes out? 00:32:01.000 --> 00:32:07.000 Well, two is leaving here and two is leaving here. 00:32:05.000 --> 00:32:11.000 It doesn't matter that they are going out through separate 00:32:10.000 --> 00:32:16.000 pipes. They are both going out. 00:32:12.000 --> 00:32:18.000 It is minus 4, 2 and 2. 00:32:14.000 --> 00:32:20.000 How about the inhomogeneous term? 00:32:16.000 --> 00:32:22.000 There is one coming in, but there is no salt in it. 00:32:20.000 --> 00:32:26.000 Therefore, that is not changing. 00:32:23.000 --> 00:32:29.000 What is coming through that pipe is necessary for the liquid 00:32:27.000 --> 00:32:33.000 balance. But it has no effect 00:32:31.000 --> 00:32:37.000 whatsoever. I will put a zero here but, 00:32:35.000 --> 00:32:41.000 of course, you don't have to put that in. 00:32:38.000 --> 00:32:44.000 This is now an inhomogeneous system. 00:32:42.000 --> 00:32:48.000 In other words, the system is x prime equals 00:32:46.000 --> 00:32:52.000 this matrix, negative 3, the same sort of stuff we 00:32:50.000 --> 00:32:56.000 always had, plus the inhomogeneous term which is the 00:32:55.000 --> 00:33:01.000 column vector 5 e to the minus t and zero. 00:33:00.000 --> 00:33:06.000 It is the presence of this term 00:33:04.000 --> 00:33:10.000 that makes this system inhomogeneous. 00:33:06.000 --> 00:33:12.000 And what that corresponds to is this little closed system being 00:33:11.000 --> 00:33:17.000 attacked from the outside by these external pipes which are 00:33:15.000 --> 00:33:21.000 bringing salt in. Without those, 00:33:17.000 --> 00:33:23.000 of course the balance would be all wrong. 00:33:20.000 --> 00:33:26.000 I would have to change this to three and cut that out, 00:33:24.000 --> 00:33:30.000 I guess. But then, it would be just a 00:33:27.000 --> 00:33:33.000 simple homogenous system. It is these pipes that make it 00:33:32.000 --> 00:33:38.000 inhomogeneous. Now, I should start to solve 00:33:35.000 --> 00:33:41.000 that. I did this just to illustrate 00:33:38.000 --> 00:33:44.000 where a system might come from. Before I solve that, 00:33:42.000 --> 00:33:48.000 what I want to do is, of course, is solve it in 00:33:45.000 --> 00:33:51.000 general. In other words, 00:33:47.000 --> 00:33:53.000 how do you solve this in general? 00:33:49.000 --> 00:33:55.000 Because I promised you that you would be able to do in general, 00:33:54.000 --> 00:34:00.000 regardless of what sort of functions were in the r of t, 00:33:58.000 --> 00:34:04.000 that column vector. So let's do it. 00:34:10.000 --> 00:34:16.000 First of all, you have to learn the name of 00:34:15.000 --> 00:34:21.000 the method. This method is for solving x 00:34:20.000 --> 00:34:26.000 prime equals Ax. It is a method for finding a 00:34:26.000 --> 00:34:32.000 particular solution. 00:34:36.000 --> 00:34:42.000 Of course, to actually solve it then you have to add the 00:34:39.000 --> 00:34:45.000 complimentary function. We are looking for a particular 00:34:43.000 --> 00:34:49.000 solution for this system. Now, the whole cleverness of 00:34:47.000 --> 00:34:53.000 the method, which I think was discovered a couple hundred 00:34:51.000 --> 00:34:57.000 years ago by, I think, Lagrange, 00:34:53.000 --> 00:34:59.000 I am not sure. The method is called variation 00:34:57.000 --> 00:35:03.000 of parameters. I am giving you that so that 00:35:00.000 --> 00:35:06.000 when you forget you will be able to look it up and be indexes to 00:35:05.000 --> 00:35:11.000 some advanced engineering mathematics book or something, 00:35:08.000 --> 00:35:14.000 whatever is on your shelf. But, if course, 00:35:11.000 --> 00:35:17.000 you won't remember the name either so maybe this won't work. 00:35:15.000 --> 00:35:21.000 Variation of parameters, I will explain to you why it is 00:35:19.000 --> 00:35:25.000 called that. All the cleverness is in the 00:35:22.000 --> 00:35:28.000 very first line. If you could remember the very 00:35:25.000 --> 00:35:31.000 first line then I trust you to do the rest yourself. 00:35:30.000 --> 00:35:36.000 I don't know any motivation for this first step, 00:35:34.000 --> 00:35:40.000 but mathematics is supposed to be mysterious anyway. 00:35:40.000 --> 00:35:46.000 It keeps me eating. It says, look for a solution 00:35:45.000 --> 00:35:51.000 and there will be one of the following form. 00:35:49.000 --> 00:35:55.000 Now, it will look exactly like -- 00:36:00.000 --> 00:36:06.000 Look carefully because it is going to be gone in a moment. 00:36:04.000 --> 00:36:10.000 It will look exactly like this. But, of course, 00:36:08.000 --> 00:36:14.000 it cannot be this because this solves the homogeneous system. 00:36:13.000 --> 00:36:19.000 If I plug this in with these as constants it cannot possibly be 00:36:18.000 --> 00:36:24.000 a particular solution to this because it will stop there and 00:36:23.000 --> 00:36:29.000 satisfy that with r equals zero. The whole trick is you think of 00:36:29.000 --> 00:36:35.000 these are parameters which are now variable. 00:36:32.000 --> 00:36:38.000 Constants that are varying. That is why it is called 00:36:35.000 --> 00:36:41.000 variation of parameters. You think of these, 00:36:38.000 --> 00:36:44.000 in other words, as functions of t. 00:36:41.000 --> 00:36:47.000 We are going to look for a solution which has the form, 00:36:45.000 --> 00:36:51.000 since they are functions of t, I don't want to call them c1 00:36:49.000 --> 00:36:55.000 and c2 anymore. I will call them v because that 00:36:52.000 --> 00:36:58.000 is what most people call them, v or u, sometimes. 00:37:02.000 --> 00:37:08.000 The method says look for a solution of that form. 00:37:06.000 --> 00:37:12.000 The variation parameters, these are the parameters that 00:37:10.000 --> 00:37:16.000 are now varying instead of being constants. 00:37:14.000 --> 00:37:20.000 Now, if you take it in that form and start trying to 00:37:18.000 --> 00:37:24.000 substitute into the equation you are going to get a mess. 00:37:23.000 --> 00:37:29.000 I think I was wrong in saying I could trust you from this point 00:37:28.000 --> 00:37:34.000 on. I will take the first step from 00:37:32.000 --> 00:37:38.000 you, and then I could trust you to do the rest after that first 00:37:38.000 --> 00:37:44.000 step. The first step is to change the 00:37:42.000 --> 00:37:48.000 way this looks by using the fundamental matrix. 00:37:46.000 --> 00:37:52.000 Remember what the fundamental matrix was? 00:37:50.000 --> 00:37:56.000 Its entries were the two columns of solutions. 00:37:54.000 --> 00:38:00.000 These are solutions to the homogeneous system. 00:38:00.000 --> 00:38:06.000 And I am going to write it using the fundamental matrix as, 00:38:05.000 --> 00:38:11.000 now thinks about it. The fundamental matrix has 00:38:09.000 --> 00:38:15.000 columns x1 and x2. Your instinct might be using 00:38:13.000 --> 00:38:19.000 matrix multiplication to put the v1 and the v2 here, 00:38:18.000 --> 00:38:24.000 but that won't work. You have to put them here. 00:38:30.000 --> 00:38:36.000 This says the same thing as that. 00:38:33.000 --> 00:38:39.000 Let's just take a second out to calculate. 00:38:37.000 --> 00:38:43.000 The x is going to look like (x1, y1). 00:38:40.000 --> 00:38:46.000 That is my first solution. My second solution, 00:38:44.000 --> 00:38:50.000 here is the fundamental matrix, is (x2, y2). 00:38:49.000 --> 00:38:55.000 And I am multiplying this on the right by (v1, 00:38:53.000 --> 00:38:59.000 v2). Does it come out right? 00:38:56.000 --> 00:39:02.000 Look. What is it? 00:38:59.000 --> 00:39:05.000 The top is x1 v1 plus x2 v2. 00:39:02.000 --> 00:39:08.000 The top, x1 v1 plus x2 v2. 00:39:06.000 --> 00:39:12.000 It is in the wrong order, but multiplication is 00:39:10.000 --> 00:39:16.000 commutative, fortunately. And the same way the bottom 00:39:14.000 --> 00:39:20.000 thing will be v1 y1 plus v2 y2. 00:39:18.000 --> 00:39:24.000 If I had written it on the other side instead, 00:39:22.000 --> 00:39:28.000 which is tempting because the v's occur on the left here, 00:39:27.000 --> 00:39:33.000 that won't work. What will I get? 00:39:31.000 --> 00:39:37.000 I will get v1 x1 plus v2 y1, 00:39:35.000 --> 00:39:41.000 which is not at all what I want. 00:39:37.000 --> 00:39:43.000 You must put it on the right. But this is a very important 00:39:42.000 --> 00:39:48.000 thing. This is going to plague us on 00:39:45.000 --> 00:39:51.000 Monday, too. It must be written on the right 00:39:49.000 --> 00:39:55.000 and not on the left as a column vector. 00:39:52.000 --> 00:39:58.000 The rest of the program is very simple. 00:39:55.000 --> 00:40:01.000 I will write it out as a program. 00:40:00.000 --> 00:40:06.000 Substitute into the system, into that, in other words, 00:40:04.000 --> 00:40:10.000 and see what v has to be. That is what we are looking 00:40:08.000 --> 00:40:14.000 for. We know what the x1 and x2 are. 00:40:11.000 --> 00:40:17.000 It is a question of what those coefficients are. 00:40:14.000 --> 00:40:20.000 And see what v is. Let's do it. 00:40:32.000 --> 00:40:38.000 Let's substitute. Let's see. 00:40:35.000 --> 00:40:41.000 The system is x prime equals Ax plus r. 00:40:41.000 --> 00:40:47.000 I want to put in (x)p, this proposed particular 00:40:46.000 --> 00:40:52.000 solution. And it is a fundamental matrix, 00:40:50.000 --> 00:40:56.000 and the v is unknown. How do I differentiate the 00:40:56.000 --> 00:41:02.000 product of two matrices? You differentiate the product 00:41:02.000 --> 00:41:08.000 of two matrices using the product rule that you learned 00:41:07.000 --> 00:41:13.000 the first day of 18.01. Trust me. 00:41:10.000 --> 00:41:16.000 Let's do it. I am going to substitute in. 00:41:14.000 --> 00:41:20.000 In other words, here is my (x)p, 00:41:17.000 --> 00:41:23.000 (x)p, and I am going to write in what that is. 00:41:21.000 --> 00:41:27.000 The left-hand side is the derivative of, 00:41:25.000 --> 00:41:31.000 X prime times v, plus X times the derivative of 00:41:29.000 --> 00:41:35.000 v. Notice that one of these is a 00:41:34.000 --> 00:41:40.000 column vector and the other is a square matrix. 00:41:37.000 --> 00:41:43.000 That is perfectly Okay. Any two matrices which are the 00:41:39.000 --> 00:41:45.000 rate shape so you can multiply them together, 00:41:42.000 --> 00:41:48.000 if you want to differentiate their product, 00:41:44.000 --> 00:41:50.000 in other words, if the entries are functions of 00:41:46.000 --> 00:41:52.000 t it is the product rule. The derivative of this times 00:41:49.000 --> 00:41:55.000 time plus that times the derivative of this. 00:41:52.000 --> 00:41:58.000 You have to keep them in the right order. 00:41:54.000 --> 00:42:00.000 You are not allowed to shuffle them around carelessly. 00:41:57.000 --> 00:42:03.000 So that is that. What is it equal to? 00:42:00.000 --> 00:42:06.000 Well, the right-hand side is A. And now I substitute just (x)p 00:42:08.000 --> 00:42:14.000 in, so that is X times v plus r. Is this progress? 00:42:14.000 --> 00:42:20.000 What is v? It looks like a mess but it is 00:42:20.000 --> 00:42:26.000 not. Why not? 00:42:22.000 --> 00:42:28.000 It is because this is not any old matrix X. 00:42:29.000 --> 00:42:35.000 This is a matrix whose columns are solutions to the system. 00:42:34.000 --> 00:42:40.000 And what does that do? That means X prime satisfies 00:42:39.000 --> 00:42:45.000 that matrix differential equation. 00:42:42.000 --> 00:42:48.000 X prime is the same as Ax. 00:42:45.000 --> 00:42:51.000 And, by a little miracle, the v is tagging along in both 00:42:50.000 --> 00:42:56.000 cases. This cancels that and now there 00:42:54.000 --> 00:43:00.000 is very little left. The conclusion, 00:42:58.000 --> 00:43:04.000 therefore, is that Xv is equal to r. 00:43:02.000 --> 00:43:08.000 What is v? It is v that we are looking 00:43:05.000 --> 00:43:11.000 for, right? You have to solve a matrix 00:43:09.000 --> 00:43:15.000 equation, now. This is a square matrix so you 00:43:13.000 --> 00:43:19.000 have to do it by inverting the matrix. 00:43:16.000 --> 00:43:22.000 You don't just sloppily divide. You multiply on which side by 00:43:21.000 --> 00:43:27.000 what matrix? Choice of left or right. 00:43:25.000 --> 00:43:31.000 You multiply by the inverse matrix on the left or on the 00:43:29.000 --> 00:43:35.000 right? It has to be on the left. 00:43:35.000 --> 00:43:41.000 Multiply both sides of the equation by X inverse on the 00:43:42.000 --> 00:43:48.000 left, and then you will get v is equal to X inverse r. 00:43:50.000 --> 00:43:56.000 How do I know the X inverse 00:43:55.000 --> 00:44:01.000 exists? Does X inverse exist? 00:44:00.000 --> 00:44:06.000 For a matrix inverse to exist, the matrix's determinant must 00:44:05.000 --> 00:44:11.000 be not zero. Why is the determinant of this 00:44:10.000 --> 00:44:16.000 not zero? Because its columns are 00:44:13.000 --> 00:44:19.000 independent solutions. 00:44:22.000 --> 00:44:28.000 Of course this is not right. I forgot the prime here. 00:44:31.000 --> 00:44:37.000 I am not failing this course after all. 00:44:34.000 --> 00:44:40.000 v prime equals that. 00:44:43.000 --> 00:44:49.000 This is done by differentiating each entry in the column vector. 00:44:47.000 --> 00:44:53.000 And, therefore, we should integrate it. 00:44:50.000 --> 00:44:56.000 It will be the integral, just the ordinary 00:44:53.000 --> 00:44:59.000 anti-derivative of x inverse times r. 00:44:57.000 --> 00:45:03.000 This is a column vector. The entries are functions of t. 00:45:02.000 --> 00:45:08.000 You simply integrate each of those functions in turn. 00:45:06.000 --> 00:45:12.000 So integrate each entry. 00:45:12.000 --> 00:45:18.000 There is my v. Sorry, you cannot tell the v's 00:45:19.000 --> 00:45:25.000 from the r's here. And so, finally, 00:45:25.000 --> 00:45:31.000 the particular solution is (x)p is equal to -- 00:45:34.000 --> 00:45:40.000 It is really not bad at all. It is equal to X times v. 00:45:37.000 --> 00:45:43.000 It's equal to X times the integral of X inverse r dt. 00:45:40.000 --> 00:45:46.000 Now, actually, 00:45:43.000 --> 00:45:49.000 there is not much work to doing that. 00:45:45.000 --> 00:45:51.000 Once you have solved the homogeneous system and gotten 00:45:49.000 --> 00:45:55.000 the fundamental matrix, taking the inverse of a 00:45:52.000 --> 00:45:58.000 two-by-two matrix is almost trivial. 00:45:54.000 --> 00:46:00.000 You flip those two and you change the signs of these two 00:45:57.000 --> 00:46:03.000 and you divide by the determinant. 00:46:01.000 --> 00:46:07.000 You multiply it by r. And the hard part is if you can 00:46:04.000 --> 00:46:10.000 do the integration. If not, you just leave the 00:46:07.000 --> 00:46:13.000 integral sign the way you have learned to do in this silly 00:46:11.000 --> 00:46:17.000 course and you still have the answer. 00:46:14.000 --> 00:46:20.000 What about the arbitrary constant of integration? 00:46:17.000 --> 00:46:23.000 The answer is you don't need to put it in. 00:46:20.000 --> 00:46:26.000 Just find one particular solution. 00:46:22.000 --> 00:46:28.000 It is good enough. You don't have to put in the 00:46:25.000 --> 00:46:31.000 arbitrary constants of integration. 00:46:29.000 --> 00:46:35.000 Because they are already in the complimentary function here. 00:46:33.000 --> 00:46:39.000 Therefore, you don't have to add them. 00:46:35.000 --> 00:46:41.000 I am sorry I didn't get a chance to actually solve that. 00:46:39.000 --> 00:46:45.000 I will have to let it go. The recitations will do it on 00:46:42.000 --> 00:46:48.000 Tuesday, will solve that particular problem, 00:46:45.000 --> 00:46:51.000 which means you will, in effect.