Separation of Variables

Separation of variables is the central technique for linear PDEs on bounded domains. The introduction ran it once for the heat equation; here we lay it out as a general method, because the same four moves solve the heat, wave, and Laplace equations alike.

Assume a product. Look for solutions of the form u(x, t) = X(x)\,T(t) — one factor per variable.
Substitute and separate. Plug in and rearrange until each side depends on only one variable.
Introduce a separation constant. A function of x can equal a function of t only if both equal the same constant -\lambda.
Solve the resulting ODEs and superpose the pieces to match the data.

The eigenvalue problem at its heart

Separation turns the spatial part into a boundary-value problem for X:

X'' = -\lambda X, \qquad X(0) = X(L) = 0.

This is an eigenvalue problem — the continuous cousin of eigenvectors and eigenvalues. Only special values \lambda_n = (n\pi/L)^2 admit a nonzero solution satisfying the boundary conditions, and the solutions are the eigenfunctions

X_n(x) = \sin\frac{n\pi x}{L}, \qquad n = 1, 2, 3, \dots

The boundary conditions act as a sieve, selecting a discrete spectrum of allowed modes — exactly the sines that Fourier series are built from. That is no coincidence: it is why Fourier series are the natural language for these problems.

Seek u = X(x)T(t); substitute to get \frac{X''}{X} = \frac{1}{\alpha}\frac{T'}{T} = -\lambda (heat) or similar.
The homogeneous boundary conditions make X'' = -\lambda X an eigenvalue problem with eigenvalues \lambda_n = (n\pi/L)^2 and eigenfunctions \sin(n\pi x/L).
Superpose the product solutions and use the initial data to fix the coefficients (a Fourier series).

The allowed shapes

On [0, \pi] the eigenfunctions are \sin(nx). Step n: each is zero at both ends (satisfying the Dirichlet conditions) and packs in one more half-wave. These are the only spatial shapes separation allows — the building blocks every solution is assembled from.