Elliptic, Parabolic, Hyperbolic

Nearly every second-order linear PDE in two variables can be written

A\,u_{xx} + B\,u_{xy} + C\,u_{yy} + (\text{lower-order terms}) = 0.

Astonishingly, a single number built from the top coefficients — the discriminant B^2 - 4AC — sorts all of them into three families with completely different behaviour. The names are borrowed from the conic sections Ax^2 + Bxy + Cy^2 = 1, which the same discriminant classifies.

The three types

Elliptic (B^2 - 4AC < 0) — e.g. Laplace's equation u_{xx} + u_{yy} = 0. Models steady states; smooth everywhere; determined by boundary values all around a region.
Parabolic (B^2 - 4AC = 0) — e.g. the heat equation u_t - \alpha u_{xx} = 0. Models diffusion; smooths data out and marches forward in time.
Hyperbolic (B^2 - 4AC > 0) — e.g. the wave equation u_{tt} - c^2 u_{xx} = 0. Models propagation; carries signals at finite speed along characteristics, preserving sharp features.

The type is not bureaucratic labelling — it dictates what conditions you must supply, whether solutions stay smooth, and which solution method applies.

For A u_{xx} + B u_{xy} + C u_{yy} + \cdots = 0, compute B^2 - 4AC:

< 0 → elliptic (steady state, e.g. Laplace).
= 0 → parabolic (diffusion, e.g. heat).
> 0 → hyperbolic (waves, e.g. the wave equation).

Worked classification

For the heat equation u_t = \alpha u_{xx}, treat t as the second variable: the only second-order term is u_{xx}, so A = -\alpha (or \alpha), B = 0, C = 0. Then B^2 - 4AC = 0 — parabolic. For the wave equation u_{tt} - c^2 u_{xx} = 0: A = -c^2, B = 0, C = 1, giving B^2 - 4AC = 4c^2 > 0 — hyperbolic.