classification of second-order linear PDEs
There are infinitely many second-order linear PDEs, but they fall into just three families, and which family a given equation belongs to tells you almost everything about how its solutions behave — whether information spreads at a finite speed or instantly, whether disturbances stay sharp or smear out, whether the right data sit on the boundary of a region or at an initial instant. The three types are called elliptic, parabolic, and hyperbolic, and the names are borrowed from conic sections for a precise reason.
Write a general second-order linear equation in two variables as A u_xx + 2B u_xy + C u_yy + (lower-order terms) = 0, where the subscripts denote partial derivatives and A, B, C are the coefficients of the highest derivatives. The type is decided by the discriminant B^2 - A C, exactly the quantity that classifies the conic A x^2 + 2B x y + C y^2 = const. If B^2 - A C is less than 0 the equation is elliptic (think Laplace's equation, A = C = 1, B = 0, discriminant -1); if it equals 0 it is parabolic (the heat equation, with one second derivative missing); if it is greater than 0 it is hyperbolic (the wave equation, A = 1, C = -c^2, discriminant c^2 > 0). The classification can change from point to point if the coefficients vary, which is why transonic flow, for instance, is so delicate.
This is not bookkeeping — the type controls the physics. Hyperbolic equations have real characteristic curves along which signals travel at finite speed and along which sharp fronts and discontinuities propagate; they model waves. Parabolic equations describe diffusion: they smooth data instantly and have an infinite signal speed in idealized form, with an arrow of time built in. Elliptic equations describe equilibrium and steady states with no time direction at all, and their solutions are as smooth as possible. Each type also demands its own appropriate boundary or initial data for the problem to be well posed; mismatching the data to the type produces problems with no solution or wildly unstable ones.
Laplace u_xx + u_yy = 0 has B^2 - A C = 0 - (1)(1) = -1 < 0, elliptic. Heat u_t - u_xx = 0 has its (t, x) discriminant equal to 0, parabolic. Wave u_tt - c^2 u_xx = 0 has discriminant 0 - (1)(-c^2) = c^2 > 0, hyperbolic.
The three model equations are exactly one of each type — which is why they are taught together.
The type is fixed by the highest-order (principal) coefficients alone; the lower-order terms and the source do not change it, though for variable coefficients the type can differ from point to point.