Sturm-Liouville boundary conditions
An eigenvalue problem is half differential equation and half boundary conditions; change the conditions and you change the whole spectrum. For a Sturm-Liouville problem the boundary conditions are not arbitrary — they must be the kind that make the operator self-adjoint, because only then are eigenvalues real and eigenfunctions orthogonal. Picking the right class of boundary conditions is therefore as much a part of the problem as the equation.
Three families do the job. Regular (or separated) conditions impose a homogeneous relation at each endpoint independently, like alpha y(a) + alpha' y'(a) = 0 and beta y(b) + beta' y'(b) = 0; Dirichlet (y = 0), Neumann (y' = 0), and Robin (a mix) are the common cases. Periodic conditions tie the ends together, y(a) = y(b) and y'(a) = y'(b), as on a ring. Singular conditions arise when p(x) vanishes at an endpoint (as in Legendre at x = +/-1 or Bessel at x = 0) or the interval is infinite; there one replaces an explicit condition by a regularity demand — that y stay bounded, or square-integrable against the weight. In every case the requirement is the same underlying one: the boundary term p (u' v - u v') evaluated at both ends must cancel for all admissible u, v.
Get the conditions wrong and the guarantees evaporate: eigenvalues can go complex, eigenfunctions can fail to be orthogonal, and the expansion you build will not converge to the function you started with. This is why, for the sphere, the unspoken boundary condition is just boundedness at the poles, and for a heat-conducting bar an insulated end is a Neumann condition — recognizing which physical situation maps to which Sturm-Liouville boundary condition is where the modeling actually happens.
On a uniform bar [0, L] with both ends insulated, the heat-equation eigenvalue problem y'' + lambda y = 0 carries Neumann conditions y'(0) = y'(L) = 0. The eigenfunctions are cos(n pi x / L), n = 0, 1, 2, ..., including the constant mode lambda = 0 — which encodes the conserved total heat.
Same equation as the fixed string, but Neumann instead of Dirichlet boundaries flips the basis from sines to cosines and admits a zero eigenvalue.
Periodic boundary conditions can give eigenvalues with two independent eigenfunctions (degeneracy), unlike regular separated conditions where each eigenvalue is simple — a small but consequential difference when you count modes.