This video sets the theme of the course. To solve a coupled system of evolution equations, switch to coordinates where the equations decouple. We go over an example where this is fairly simple. In general, decoupling involves the eigenvalues and eigenvectors of a matrix, which we will learn about later.

Sometimes the evolution equations involve derivatives. To solve a system of first order linear ordinary differential equations (ODEs), we do a change of variables to decouple the modes. Most of this video is about solving the resulting scalar equations, all of the form $\frac{dy}{dt} = \lambda y$. You should have seen this in calculus, but in case you haven't, here it is.

Next we show how to solve the second order differential equation $\frac{d^2y}{dt^2} = \lambda y$, when $\lambda$ is negative or zero. This is called the elliptic case, and the solutions are most easily expressed in terms of the trig functions $\cos(t)$ and $\sin(t)$.

Don't worry too much if you haven't seen second order ODEs before. We're going to be doing a lot with them in Chapter 5, which gives you a decent amount of time to get the hang of them. You may want to review this video when we get there.

Finally we turn to the hyperbolic case, where $\lambda>0$. When $\lambda>0$, the solutions to $\frac{d^2y}{dt^2} = \lambda y$ are most easily expressed in terms of the hyperbolic trig functions $\cosh(t)$ and $\sinh(t)$. As with the previous video, most of this material will be used heavily in Chapter 5, and you may want to review this video then.