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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 ...

PROFESSOR: OK. Here's an example that's more or less for fun. Because you'll see me try to do it. You can do it better. I call the problem the tumbling blocks. Only in this example, in my demonstration, it's going to be a tumbling book. I'm going to take a book, the sacred book, and throw it in the air. And I'll throw it three different ways. And the question is, is the spinning book stable or not? And let me tell you the three ways and then give you the three equations that came from Euler. So those are the three equations. You see that they're not linear. And those are for the angular momentum. So there's a little physics behind the equations. But for us, those are the three equations. So the first throw will spin around the very short axis, just the thickness of the book, maybe an inch. So when I toss that, as I'll do now, you will see if I can toss it not too nervously I hope. It came-- it was stable. The book came back to me withou...

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. 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...

PROFESSOR: OK. Finally, I'm going to solve this first order linear differential equation with a formula that works for any source term. So we've solved it for specific, nice, special source terms I'll remember later. But now we want a formula for the solution to that equation, period. And we want to understand the formula. So now, write the formula down, and then let's see why it's right. And then of course, we could put it into the equation and confirm that it's right. OK the formula is going to be-- this is the big formula you could say, for first order linear equations. So y of t. First we have the result of the amount-- I'm thinking of this as balance. The money in the bank is why it's increasing at this rate, because of interest being added. And it's increasing at this rate because of new deposits being added. And oh, maybe I should say about those deposits, I'm not thinking of like deposit once a year, or once a month, or once...

PROFESSOR: OK. We'll go through these together. Repeat after me. Civil. AUDIENCE: Civil. PROFESSOR: Civility. AUDIENCE: Civility. PROFESSOR: Civilized. AUDIENCE: Civilized. PROFESSOR: Civilization. . AUDIENCE: Civilization. PROFESSOR: Equal. AUDIENCE: Equal. PROFESSOR: Equality. AUDIENCE: Equality. PROFESSOR: Equalize. AUDIENCE: Equalize. PROFESSOR: Equalization. AUDIENCE: Equalization. PROFESSOR: Fertile. AUDIENCE: Fertile. PROFESSOR: Fertility. AUDIENCE: Fertility. PROFESSOR: Fertilize. AUDIENCE: Fertilize. PROFESSOR: Fertilization. AUDIENCE: Fertilization. PROFESSOR: Final. AUDIENCE: Final. PROFESSOR: Finality. AUDIENCE: Finality. PROFESSOR: Finalize. AUDIENCE: Finalize. PROFESSOR: Finalization. AUDIENCE: Finalization. PROFESSOR: General. AUDIENCE: General. PROFESSOR: Generality. AUDIENCE: Generality. PROFESSOR: Generalized. AUDIENCE: Generalized. PROFESSOR: Generalization. AUDIENCE: Generalization. PROFESSOR: Hospital. AUDIENCE: Hospital. PROFESSOR: Hospitality. A...