What Is a PDE?

A partial differential equation relates a function of several variables to its partial derivatives. The great second-order linear family $A u_{xx} + B u_{xy} + C u_{yy} = \cdots$ is sorted by the sign of $B^2 - 4AC$: negative is elliptic, zero is parabolic, positive is hyperbolic.

The classification is not cosmetic — it dictates behavior. Elliptic equations (Laplace) describe equilibrium; parabolic (heat) describe diffusion that smooths and forgets; hyperbolic (wave) describe propagation that remembers. Boundary and initial conditions differ per type.

The wave equation $u_{tt} = c^2 u_{xx}$ in $(x, t)$ has $A = c^2$, $B = 0$, $C = -1$ (with $t$ as the second variable): $B^2 - 4AC = 4c^2 > 0$ — hyperbolic, as expected for propagation.