Heat Equation
GILBERT STRANG: This is heat equation video. So this is the second of the three basic partial differential equations. We had Laplace's equation, that was-- time was not there. Now time comes into the heat equation. We have a time derivative, and two-- matching with two space derivatives. So I have my function. My solution depends on t and on x, and I hope I can separate those two parts. This is exactly like the way we solved the ordinary systems of differential equations. We pulled out an e to the lambda t, where lambda was the eigenvalue, and then we had the eigenvector. Here, it's an eigenfunction because it depends on x. We didn't have x before, but now we have partial differential equations. X is also a coordinate here. So I look for solutions like that. And just as always, I substitute that into the differential equation to discover what-- what determines S, S of x. The time derivative brings down a lambda. The space derivative brings down-- has two space...