the bridge to separation of variables
It can feel mysterious why so much effort goes into the abstract eigenvalue theory of a single variable. The payoff is this: that whole theory is the machinery that solves the great partial differential equations of physics — the heat equation, the wave equation, Laplace's equation. The bridge to separation of variables is the realisation that solving a partial differential equation on a nice region is, at its core, a Sturm-Liouville problem in disguise.
The method works like this. To solve, say, the heat equation u_t = u_xx on 0 <= x <= L with the ends held at zero, you guess that a solution factors as a product u(x, t) = X(x) T(t), one factor depending only on space, the other only on time. Substituting and dividing by X T separates the variables: X''/X = T'/T. The left side depends only on x, the right only on t, yet they are equal, so both must equal a single constant, conventionally written minus lambda. This splits the one partial differential equation into two ordinary ones: X'' + lambda X = 0 with the boundary conditions (a Sturm-Liouville eigenvalue problem!) and T' + lambda T = 0 (a simple decay). The boundary conditions select the eigenvalues lambda_n and eigenfunctions X_n; each time factor evolves on its own; and you superpose the products and match the initial data using an eigenfunction expansion.
So everything in this field converges here. The eigenvalue problem supplies the allowed spatial modes; orthogonality lets you fit the initial condition; completeness guarantees the fit is exact; and the generalized Fourier series assembles the final answer. This is why Sturm-Liouville theory is not an isolated curiosity but the backbone of mathematical physics — the same X'' + lambda X = 0 reappears, with different p, q, w and geometry, behind heat flow, vibrating membranes, electrostatics, and the quantum wavefunctions of an atom.
For u_t = u_xx on [0, pi] with u(0, t) = u(pi, t) = 0 and u(x, 0) = f(x), separation gives X'' + lambda X = 0 (eigenvalues n^2, eigenfunctions sin(n x)) and T' + n^2 T = 0 (so T = e^(-n^2 t)). The full solution is u(x, t) = sum of b_n e^(-n^2 t) sin(n x), with b_n the Fourier sine coefficients of f.
The spatial half is a Sturm-Liouville eigenvalue problem; the time half is a simple decay; an eigenfunction expansion glues them to the initial data.
Separation of variables is powerful but not universal: it relies on a separable geometry (intervals, rectangles, disks, spheres) and homogeneous, separated boundary conditions. On an awkwardly shaped region, or with mixed boundary data, the product guess fails and you must turn to other methods.