boundary-value problem

Imagine a guitar string pinned at both ends. You are not told how it starts moving; you are told where it must sit at the two ends, and you want the shape it takes in between. That is the flavor of a boundary-value problem: a differential equation holds inside a region, and conditions are imposed on the boundary — at both ends of an interval, or all around the edge of a 2-D or 3-D region — rather than all at one starting instant.

Concretely, for a second-order ODE like y'' + p(x) y' + q(x) y = f(x) on [a, b], a boundary-value problem fixes data at both endpoints, say y(a) = A and y(b) = B (Dirichlet), or conditions on the slope (Neumann), or a mix (Robin). Contrast this with the initial-value problem, where both y(a) and y'(a) are given at the single point a. The split matters: an initial-value problem with nice coefficients always has exactly one solution, but a boundary-value problem can have one solution, none, or infinitely many, because you are threading a curve that must hit targets at two separated places at once.

Boundary-value problems are the natural home of steady-state physics: the equilibrium temperature in a rod whose ends are held fixed, the shape of a loaded beam clamped at its supports, the electrostatic potential in a region with charged walls. The Green's function is the tool built precisely to solve linear boundary-value problems for any source, and Sturm-Liouville theory is the spectral theory of the self-adjoint boundary-value problem.