Let’s try a problem that is a bit more involved. We solve the partial differential equation

Where , such that .

Solution

We will follow the same procedure used in the first example. First we identify a differential operator L and eigenfunctions  such that

The are numbers called eigenvalues whose form we will determine. Once we have the eigenfunctions, we write f in a series expansion

then plug it into the differential equation, and solve mode by mode for the coefficient functions .  Let’s see how to apply the method to the equation we are trying to solve in this case. We have . Looking at the right hand side, we see that the differential operator (acting with respect to x) is

Taking this together with the boundary conditions specified in the problem, the eigenvalue problem we need to solve is

with      . Now, since we are only dealing with a function of a single variable x, we can write the equation in terms of ordinary derivatives. Moving all terms to the right hand side, we obtain the familiar simple harmonic oscillator equation

The solution of this equation is given in terms of sines and cosines. Taking the most general linear combination, we try

Notice that for the first derivative of this function, we have

The second derivative is

So this function satisfies . To find out what is, we apply the boundary conditions. The first of these is that . Now