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GILBERT STRONG: This is the second video on second order differential equations, constant coefficients, but now we have a right hand side. And the first one was free harmonic motion with a zero, but now I'm making this motion, I'm pushing this motion, but at a frequency omega. This is my forcing term. So I think I'm having a forcing frequency, omega, and remember that for this one, for the no solution, there was a natural frequency omega n. It's very important are those close, are those well separated? That governs whether the bridge that you're walking over oscillates too much and eventually falls. Or in the extreme case, are they equal? If omega n is equal to omega that's called resonance. Let me put that word in. Resonance. When omega equals omega n. And we're not going to deal with today, but you should know that always the formula has an omega minus omega n dividing by that. So if that is 0, if omega equals omega n our formula has to chang...

GILBERT STRONG: Well, you see spring has finally come to Boston. My sweater is gone and it's April the 16th, I think. It's getting late spring. So today, this video is the nice case, constant coefficient, linear equations, and the right hand side is an exponential. Those are the best. And we've seen that before. In fact, let me extend, we saw it for first order equations, here it is for second order equations, and it could be an nth order equation. We could have the nth derivative and all lower derivatives. The first derivative, the function itself, 0-th derivative with coefficients, constant coefficients, equalling e to the st. That's what makes it easy. And what do we do when the right hand side is e to the st? We look for a solution, a multiple of e to the st. Capital Y times e to the st, that's going to work. We just plug into the equation to find that transfer function capital Y. Can I just do that? I'll do it for the nth degree. Why not? So w...