Last time math was involved, we saw a very simple differential equation.

Note that the derivative of a function, if it exists, is a function itself. In the above, we have identified the velocity as a function of time. Nothing stops us from finding another derivative. The rate of change of velocity is the acceleration. We can now say that velocity is the derivative of the position as a function of time, and that acceleration is the second derivative of the position as a function of time.

The point is that we can have differential equations that involve higher derivatives. For example,

y''(x) = -k*k*y(x)

is a very simple second-order (meaning the second derivative y'' shows up) differential equation. To solve for a solution, we now need two initial conditions. These are sometimes called bounadry conditions, if we know the value of the function in two different places. In this case, suppose that I know that y(0) = 0 and y(1) = 0. This allows me to find a solution in the region between x = 0 and x = 1. I would need more information to solve the problem outside of that region, but it is often the case (in my current work, it is pretty much always the case) that we are only interested in some finite region.

In the second-order equation above, the solutions look like

y(x) = a*sin(k*x) + b*cos(k*x)

where a, b and k are constants to be determined by the boundary conditions. Since we have required that the function be zero at the endpoints, we must have b = 0 and also require sin(k) = 0. This last equation has solutions for every integer multiple of pi, each giving a different solution. So we have a situation here where there are an infinite number of solutions when we take k = n*pi, where n is any integer greater than zero. Values of k where there are solutions are called eigenvalues, and the collection of all the eigenvalues is called the spectrum of the problem. The resulting functions, called eigenfunctions, are in this case just scaled versions of the original sine function.

There is a lot of established theory about what sort of eigenvalues you can expect, if any, given a type of differential equation. My line of work involves a particular form called the Sturm-Liouville problem, which I will get into next time.