An initial-value problem specifies enough conditions at one point to select a trajectory. A boundary-value problem instead imposes conditions at different points of the independent variable.

For example,

$$y''+\lambda y=0,$$

with

$$y(0)=0,\qquad y(L)=0,$$

asks for a function satisfying both the differential equation and endpoint constraints.

A boundary-value problem may have no solution, one solution or several solutions. For some equations, nontrivial solutions exist only for special parameter values such as particular $\lambda$.

This structure appears in beam deflection, heat conduction, wave modes and eigenvalue problems. Boundary conditions are therefore part of the mathematical problem itself, not values applied after solving an unrestricted ODE.