the Sturm-Liouville problem
/ STURM lee-oo-VEEL /
The simple vibrating-string problem y'' + lambda y = 0 is the friendly first example of a much wider family. Real rods are not uniform, real drums are not flat, real materials vary in density and stiffness from point to point. The Sturm-Liouville problem is the master template that handles all such variable, non-uniform situations while keeping every beautiful property — real frequencies, orthogonal modes, complete expansions — that made the simple case so useful.
It is the eigenvalue problem for a self-adjoint second-order equation written in the standard form (p(x) y')' + (q(x) + lambda w(x)) y = 0 on an interval a <= x <= b, together with suitable boundary conditions at the two ends. Here p(x) > 0 and w(x) > 0 are given weight-like coefficient functions, q(x) is another given function, and lambda is the eigenvalue parameter. The pieces have meaning: p controls how the equation diffuses or transmits, q is a potential-like term, and w is the weight that says how the orthogonality bookkeeping is done. Most named special equations — Legendre's, Bessel's, Hermite's, Chebyshev's — are Sturm-Liouville problems in disguise, each with its own p, q, w.
What makes the framework so important is that, under mild conditions, it guarantees a whole package of results at once: the eigenvalues are all real and form an increasing sequence lambda_1 < lambda_2 < lambda_3 < ... marching off to infinity; the eigenfunctions are orthogonal with respect to the weight w; and they are complete, so any reasonable function expands in them. This is the abstract heart that explains why Fourier series, Bessel-function series, and Legendre-polynomial series all work for the same structural reason.
The plain problem y'' + lambda y = 0 is Sturm-Liouville with p(x) = 1, q(x) = 0, w(x) = 1, since (1·y')' + (0 + lambda·1) y = y'' + lambda y. Bessel's equation x^2 y'' + x y' + (x^2 - n^2) y = 0 rewrites as (x y')' + (-n^2/x + lambda x) y = 0, so there p(x) = x, w(x) = x.
Reading off p, q, w from a familiar equation reveals it as a Sturm-Liouville problem and hands you its orthogonality weight for free.
The structural guarantees (real, increasing eigenvalues; orthogonal, complete eigenfunctions) hold cleanly for the regular case; in singular cases (p or w vanishing at an end, or an infinite interval) the conclusions still largely hold but need extra care, and the spectrum can even become continuous.