Many differential equations in mathematical physics take the Sturm-Liouville form $$-\frac{d}{dx}\left[p(x)\frac{dy}{dx}\right]+q(x)y=\lambda w(x)y$$ with boundary conditions at the ends of an interval. Here $\lambda$ is an eigenvalue and $w(x)>0$ is a weight function.

Under suitable regularity and self-adjoint boundary conditions, the eigenvalues are real and eigenfunctions belonging to distinct eigenvalues are orthogonal with respect to the weighted inner product $$\langle f,g\rangle_w=\int_a^b f^(x)g(x)w(x),dx.$$ Thus, for $m\ne n$, $$\int_a^b y_m^(x)y_n(x)w(x),dx=0.$$

The vibrating string with fixed endpoints is a simple example: $$-\frac{d^2y}{dx^2}=\lambda y,\qquad y(0)=y(L)=0.$$ Its eigenfunctions are $$y_n(x)=\sin\frac{n\pi x}{L},$$ with $$\lambda_n=\left(\frac{n\pi}{L}\right)^2.$$ These are exactly the normal-mode shapes of the string.

A central consequence is eigenfunction expansion: sufficiently well-behaved functions can be expanded in the orthogonal eigenfunctions, $$f(x)=\sum_n c_n y_n(x),$$ with coefficients obtained by projection. Fourier series are one special case.

Sturm-Liouville theory explains why the same pattern—discrete eigenvalues, orthogonal modes and expansions—appears in waves, heat conduction, quantum bound states and separation of variables.