a homogeneous boundary value problem
Imagine a string fixed at both ends, with no one plucking it and no force driving it — held flat and still. Nothing in the rules pushes it, and the ends are pinned at zero. One answer is obvious: it just stays flat, y = 0 everywhere. The interesting question is whether any other shape can satisfy the same all-zero conditions. That question — when does the do-nothing problem secretly have a non-trivial answer — is the heart of the homogeneous boundary value problem.
Formally, a homogeneous boundary value problem has both a homogeneous equation, with zero on the right-hand side, and homogeneous boundary conditions, with zero on the right of each end condition — for instance y'' + p y' + q y = 0 with y(a) = 0 and y(L) = 0. Because everything is zero, the function that is identically zero, y(x) = 0, always satisfies it; this is called the trivial solution. The decisive question is whether a non-trivial solution also exists. For most equations it does not, and y = 0 is the only solution. But for special equations — or, crucially, equations carrying an adjustable parameter — a genuine nonzero solution can appear.
This setup matters because it is the seedbed of eigenvalue theory. When the homogeneous equation contains a parameter lambda, as in y'' + lambda y = 0 with y(0) = y(L) = 0, the trivial solution is all you get for almost every value of lambda — except at a special discrete set of lambda values, where a non-trivial solution suddenly exists. Those special values are the eigenvalues and the non-trivial solutions are the eigenfunctions. So studying when a homogeneous BVP has more than the zero solution is exactly the doorway into Sturm-Liouville theory.
For y'' = 0 with y(0) = 0, y(1) = 0, the general solution is y = c1 + c2 x. Forcing y(0) = 0 gives c1 = 0, and y(1) = 0 gives c2 = 0, so only y = 0 survives — the trivial solution alone. But for y'' + pi^2 y = 0 with y(0) = 0, y(1) = 0, the function y = c sin(pi x) works for any c, a non-trivial solution.
Most homogeneous BVPs admit only the zero solution; the special equations that admit a nonzero one are exactly the eigenvalue cases.
The trivial solution y = 0 is always present, so it never counts as 'finding a solution'. The whole point is to detect a non-trivial solution; if only y = 0 works, the homogeneous BVP is said to have only the trivial solution.