Open course ware

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? I want to talk about a slightly different way to solve a linear first-order equation. And if you look at the equation-- I'll do an example. That's the best. Do you notice what's different from our favorite equation? The change is 2t. The interest rate a is increasing with time, changing with time. So we still have a linear equation, still just y. But the coefficient is varying. We have a variable coefficient 2t. And if we think here of applications to economy, to banks, that would be rampant inflation, the interest rate 2t climbing and climbing forever. But we want to see that this is a class of equations that we can solve. OK. And the new method is called an integrating factor. It's a magic factor that makes the equation simple. So that's another nice way to solve all the problems that we've dealt with so far, plus this new one. So what is this factor? Well, for this 2t problem, the right factor is e to the minus t squared. And...