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GILBERT STRANG: OK. This video is about the third of the great trio of partial differential equations. Laplace's equation was number one. That's called an elliptic equation. The heat equation was number two. That's called a parabolic equation. Now we reach the wave equation. That's number three, and it's called a hyperbolic equation. So somehow the three equations remind us of ellipses, parabolas, and hyperbolas. They have different types of solutions. Laplace's equation, you solve it inside a circle or inside some closed region. The heat equation and the wave equation, time enters, and you're going forward in time. The heat equation is first order in time, du dt. And the wave equation, the full-scale wave equation, is second order in time. That stands for the second derivative, d second u dt squared. And it matches the second derivative in space with a velocity coefficient c squared. I'm in one-dimensional space. If I were in three dimensi...

GILBERT STRANG: OK. So today is unforced-- that means zero on the right-hand side, looking for null solutions-- damped-- that means there is a coefficient B in the first derivative. And what's the solution? This is really a basic, basic equation. In many applications, A would be the mass. In a spring, for example, A would be a mass. B is the damping, the friction. And C is the spring constant, the force that pulls the mass back. Or in electronics, B would be the resistance. It's giving some friction, giving some heat. So that's our equation. Just we have to be able to solve it. And we want to look for exponentials. A pure exponential is just right for a constant coefficient equation like that. So I'll substitute y equals e to the st. And what happens? Well, so there's a C e to the st. That's a Cy. The derivative brings down an s. Two derivatives bring down s squared. So I simply have As squared, Bs, and C all together, multiplying e to the st. And ...

GILBERT STRANG: OK. A third video about stability for second order, constant coefficient equations. But we'll move on to matrices here. So this is a rather special video. So this is our familiar equation. And I took a to b1, I just divided out a. No problem. So that's one second order equation. But we know how to convert it to two first order equations. And here they are. So this is two equations. That's a 2 by 2 matrix there. And so let me read the top equation. It says that dy dt is 0y plus 1dy dt. So that equation is a triviality. dy dt equals dy dt. The second equation is the real one. The derivative of y prime is y double prime. So this is second derivative here, equals minus cy and minus b y prime. And that's my equation y double prime, when I bring the minus cy over as plus cy, and I bring the minus b y prime over as plus b y prime. I have my equation. So that equation is the same as that one. It's just written with a vector unknown. It's a ...

GILBERT STRANG: This is a topic I think is interesting. I like this one. It's about stability or instability of a steady state. So let me show you the differential equation. It could be linear, but might be non linear. dy dt is f of y. I'm going to-- I keep it that right hand side not depending on t, so just a function of y. And when do I have a steady state? There's a steady state when the derivative is 0. So if the derivative is 0 when f of y equals 0, let me call those special y's by a capital letter. So capital Y is a number, a starting value, where the right hand side of the equation is 0. And if the right hand side of the equation is 0, the left side of the equation is 0, and dy dt is 0, and we don't go anywhere. So the solution-- if f or y is equal to 0, then we have y stays at y. It's a constant for all time, and my question is, if we start near capital Y, do we approach capital Y as time goes on? It's, in that case, I would say, stable...

GILBERT STRANG: OK. We're still solving systems of differential equations with a matrix A in them. And now I want to create the exponential. It's just natural to produce e to the A, or e to the A t. The exponential of a matrix. So if we have one equation, small a, then we know the solution is an e to the A t, times the starting value. Now we have n equations with a matrix A and a vector y. And the solution should be, at time t, e to the A t, times the starting value. It should be a perfect match with this one, where this had a number in the exponent and this has a matrix in the exponent. OK. No problem. We just use the series for e to the A t. We plug in a matrix instead of a number. So the identity, plus A t, plus 1/2 A t squared, plus 1/6 of A t cubed, forever. It's the same. It's the exponential series. The most important series in mathematics, I think. And it gives us an answer. And that answer is a matrix. Everything here, every term, is a matrix. OK....

GILBERT STRANG: OK? Now, finally, a nonlinear equation. Growth, but it's-- the growth is cut off by competition-- slowed down my competition. Let me show you the equation. In a way, it's the simplest nonlinear equation I could think of. We have the usual dy/dt equal ay. That would give exponential growth. The growth would go on forever. But then, as y gets large-- if we think of y as the population, so the growth is exponential when the birth rates higher than the death rate, and we grow. We have-- and that's what's happening. But our world population is not growing pure exponentially, forever. And this is a simple term, a simple model not, very accurate, but it's the right place to start. For competition, when you have too many people, somehow the multiplying y times y gives you a number of interactions of all the y people with themselves. And those interactions-- that competition slows down the growth, and so the coefficient there is negative. We wan...

GILBERT STRANG: OK. We're coming to the point where we need matrices. That's the point when we have several equations, several differential equations instead of just one. And it's a matrix that does that coupling. So can I-- this won't be a full course in linear algebra. That would be available, you may know on, open courseware for 18.06. That's the linear algebra course. But [INAUDIBLE] facts, and why not just say them here in a few minutes? So I have a matrix. Well there's a matrix. That's a 3 by 3 matrix. And first I want to ask how does it multiply a vector. So there it is multiplying a vector, v1, v2, v3. And what's the result, key idea? It takes the answer on the right-hand side is this number v1, times that column, plus this number, that number times the second column, plus the third number, the third number times the third column, combination of the columns of a. That's what a times v is. That's what the notation of matrix m...

GILBERT STRANG: OK well, here we're at the beginning. And that I think it's worth thinking about what we know. Calculus. Differential equations is the big application of calculus, so it's kind of interesting to see what part of calculus, what information and what ideas from calculus, actually get used in differential equations. And I'm going to show you what I see, and it's not everything by any means, it's some basic ideas, but not all the details you learned. So I'm not saying forget all those, but just focus on what matters. OK. So the calculus you need is my topic. And the first thing is, you really do need to know basic derivatives. The derivative of x to the n, the derivative of sine and cosine. Above all, the derivative of e to the x, which is e to the x. The derivative of e to the x is e to the x. That's the wonderful equation that is solved by e to the x. Dy dt equals y. We'll have to do more with that. And then the inverse fun...

GILBERT STRANG: I would like you to see the big picture of linear algebra. We're not doing, in this set of videos, a full course on linear algebra. That's already on OpenCourseWare 1806. And now I'm concentrating on differential equations, but you got to see linear algebra this way. And this way means subspaces. And there are four of them in the big picture. And we-- previous video described the column space and the null space. Now, we've got two more, making four. And let me look at this matrix-- it's for subspaces-- and put them into the big picture. So the first space I'll look at is the row space. Now, the row space has these rows-- has the vector 1, 2, 3 and the vector 4, 5, 6, two vectors there, and all their combinations. That's the key idea in linear algebra, linear combinations. So 1, 2, 3 is a vector in three dimensional space. 4, 5, 6 is another one. Now, if I take all their combinations, do you visualize that if I have two vectors, ...

GILBERT STRANG: OK, this is the video about two neat functions-- the step function and its derivative the delta function. So if I can just introduce you to those functions and show you that they're very natural inputs to a differential equation. They happen all the time in real life. And so we need to understand how to compute these formulas and compute with them. OK, so the first one is the step function and it's-- I'll call it h after its inventor who was an engineer named Heaviside, started with an H. And the step function, let me write the formula. h of t is 0 for t negative and 1 for t greater or equal to 0. OK. So, that's the step function. It just has two values and it has a jump. You could say jump function also. Jump function, step function. All right. And notice I've also graphed the shifted step function. What happens to any function including this one if I change from t, which jumps at 0, to h of t minus t? If I put in t minus some fixed nu...

GILBERT STRANG: So this is the key video about solving a system of n linear constant coefficient equations. So how do I write those equations? Y is now a vector, a vector with n components. Instead of one scalar, just a single number y-- do you want me to put an arrow on y? No, I won't repeat it again. But that's to emphasize that y is a vector. Its first derivative, it's a first order system. System meaning that there can be and will be more than one unknown, y1, y2, to yn. So how do we solve such a system? Then the matrix is multiplying that y and they equate. The y's are coupled together by that matrix. They're coupled together, and how do we uncouple them? That is the magic of eigenvalues and eigenvectors. Eigenvectors are vectors that go in their own way. So when you have an eigenvector, it's like you have a one by one problem and the a becomes just a number, lambda. So for a general vector, everything is a mixed together. But for an eigenvect...

GILBERT STRANG: OK, thanks. Here's a second video that involves the matrix exponential. But it has a new idea in it, a basic new idea. And that idea is two matrices being called "similar." So that word "similar" has a specific meaning, that a matrix A, is similar to another matrix B, if B comes from A this way. Notice this way. It means there's some matrix M-- could be any invertible matrix. So that I take A, multiply on the right by M and on the left by M inverse. That'd probably give me a new matrix. Call it B. That matrix is called "similar" to B. I'll show you examples of matrices that are similar. But first is to get this definition in mind. So in general, a lot of matrices are similar to-- if I have a certain matrix A, I can take any M, and I'll get a similar matrix B. So there are lots of similar matrices. And the point is all those similar matrices have the same eigenvalues. So there's a little family of matric...