matching the initial condition
Separation of variables hands you a whole family of building-block solutions — one separated mode for each allowed eigenvalue — each already satisfying the PDE and the boundary conditions. But there are infinitely many of them, and on their own they don't know your particular starting state. The final, decisive step is to pick the right COMBINATION of modes so that, at the initial instant, the sum reproduces the given initial condition exactly. That fitting step is matching the initial condition.
Here is the mechanism. Superpose the modes with unknown amplitudes: u(x,t) = sum over n of c_n X_n(x) T_n(t). Set t = 0; the time factors collapse to constants (often T_n(0) = 1), leaving u(x,0) = sum of c_n X_n(x), which must equal the prescribed initial function f(x). But that equation is exactly the demand that f be expanded in the eigenfunctions X_n — a Fourier (sine, cosine, or general) series! So the unknown amplitudes c_n are nothing but the Fourier coefficients of f, computed by the coefficient integrals using orthogonality. Match the coefficients on both sides and the solution is pinned down.
This is where every earlier piece pays off at once: the eigenfunctions provide the basis, orthogonality lets you extract each c_n by an integral, and Fourier convergence guarantees the sum really equals f. Once the c_n are known, you simply restore the time factors T_n(t) and you have the full solution for all later times — each mode evolving on its own clock. For the wave equation, which is second order in time, there are two initial conditions (displacement and velocity), so you match two sets of coefficients, typically one expanding the initial shape and one the initial speed.
Heat on 0 < x < L, zero at both ends, initial temperature f(x). The solution is u(x,t) = sum of b_n sin(n pi x / L) e^(-k (n pi/L)^2 t). At t = 0 this is sum of b_n sin(n pi x / L) = f(x), so b_n = (2/L) integral from 0 to L of f(x) sin(n pi x / L) dx — exactly the sine-series coefficients. The initial data chooses the b_n; the exponentials then carry each mode forward in time.
The mode amplitudes ARE the Fourier coefficients of the initial data — found by orthogonality, then carried forward by the time factors.
This step needs HOMOGENEOUS boundary conditions: the modes only form a valid basis when they all satisfy the same zero boundary data. If the boundary conditions are inhomogeneous, subtract off a steady state first to make them homogeneous, then match the initial condition of the leftover part.