the Sturm-Liouville problem
/ shtoorm lyoo-VEEL /
When you separate variables in a PDE, the space part always boils down to the same kind of question: find the special shapes (functions) that a differential operator merely scales, together with the scaling factors. The Sturm-Liouville problem is the master template for that question. It is not one equation but a whole family, and almost every separated PDE on a finite region lands inside it.
Concretely, a Sturm-Liouville problem asks for numbers lambda and nonzero functions u(x) on an interval [a, b] solving -(p u')' + q u = lambda w u, together with boundary conditions at a and b. Here p(x) > 0, q(x), and the weight w(x) > 0 are given coefficients fixed by the PDE and geometry, and the prime means d/dx. Read it as: the operator L[u] = -(p u')' + q u, when applied to the special function u, gives back lambda times w times u. The lambda are the eigenvalues and the u are the eigenfunctions. This is exactly the matrix equation A v = lambda v of linear algebra, lifted to functions and a differential operator.
The point of the whole theory is that this innocent-looking ODE-with-boundary-conditions behaves like a symmetric matrix: it has a clean, real, discrete spectrum of eigenvalues and an orthogonal basis of eigenfunctions (see the Sturm-Liouville theorem). That is why separation of variables works at all, and why the resulting series (a generalized Fourier series) is guaranteed to converge and represent any reasonable function. The ordinary Fourier sine series, with -u'' = lambda u and u(0) = u(L) = 0, is just the simplest special case (p = w = 1, q = 0).
A vibrating string of length L, fixed at both ends, has space part -X'' = lambda X, X(0) = X(L) = 0. This is Sturm-Liouville with p = w = 1, q = 0. Its eigenvalues are lambda_n = (n*pi/L)^2 and eigenfunctions X_n = sin(n*pi*x/L), n = 1, 2, 3, ... — exactly the harmonics of the string.
The simplest Sturm-Liouville problem is the one behind the ordinary Fourier sine series.
Not every second-order ODE -(p u')' + q u = lambda w u is well-behaved: the good theorems require p > 0 and w > 0 on the interval and suitable boundary conditions. When p or w vanishes at an endpoint, or the interval is infinite, you are in the harder singular case.