Separation of variables and Sturm-Liouville problems

Separation of variables is a guess, made precise: for a PDE like the heat or wave equation in one space dimension, assume the solution has the product form u(x, t) = X(x)T(t). Substituting that guess in and dividing through by XT splits the equation into two pieces — one depending only on x, one depending only on t — and since a function of x alone can only equal a function of t alone if both are the same constant, both pieces must equal a shared separation constant, usually written −λ. That turns one PDE into two ODEs.

The spatial ODE, together with the boundary conditions carried over from the original problem, is a Sturm-Liouville problem: it doesn't have a solution for every λ, only for a discrete set of eigenvalues, each with its own eigenfunction. This is exactly analogous to a matrix only having nontrivial null-space solutions at its eigenvalues — here the "matrix" is a differential operator, and the boundary conditions are what force the spectrum to be discrete rather than continuous.

The eigenfunctions from a Sturm-Liouville problem are always orthogonal to each other with respect to a weight function that's built into the problem's own structure (for the simplest textbook cases that weight is just 1, which is why it's easy to forget it exists at all). That orthogonality is what makes the whole method work: any solution satisfying the boundary conditions can be written as a series — a generalised Fourier series — of these eigenfunctions, and the orthogonality is exactly what lets you solve for each series coefficient independently by projecting the initial condition onto each eigenfunction in turn.