the general second-order linear PDE
Most of the partial differential equations that physics hands you — heat spreading, waves travelling, a soap film settling into its shape — are second-order: the highest derivatives that appear are second derivatives, like u_xx or u_xy or u_yy. If you also insist the equation be linear (the unknown u and its derivatives appear only to the first power, never multiplied together, never inside a function like sin), you arrive at a single grand template that captures almost the whole classical subject. This entry is that template.
In two independent variables x and y, the general second-order linear PDE is A u_xx + 2B u_xy + C u_yy + D u_x + E u_y + F u + G = 0, where the coefficients A, B, C, D, E, F and the term G may depend on x and y but not on u. The first three terms — A u_xx + 2B u_xy + C u_yy — are the principal part, the highest-order block that carries the second derivatives; the factor of 2 on the cross term is a convention that makes the algebra below tidy. The remaining terms D u_x + E u_y + F u are the lower-order part, and G is a source (when G is zero the equation is homogeneous). The whole behaviour of the equation — what data you may impose, how fast effects travel, how smooth the answer is — turns out to be decided by just the three principal coefficients A, B, C.
Why fuss over this exact form? Because it is the object that the entire classification theory acts on. The combination B^2 - AC built from the principal part (its discriminant) sorts every such equation into one of three families — elliptic, parabolic, hyperbolic — and that single label predicts almost everything that follows. Be honest about the scope, though: this is the LINEAR, second-order, two-variable picture. Quasilinear or fully nonlinear equations, first-order equations, and higher-order equations all need their own discussion, and in more than two variables the classification is read off the matrix of second-order coefficients rather than a single number.
Laplace's equation u_xx + u_yy = 0 has A=1, B=0, C=1; the heat equation u_t = u_xx, read with (x,t) as the two variables, has A=1 (on u_xx), B=0, C=0 (no u_tt term); the wave equation u_tt - u_xx = 0 has A=-1, B=0, C=1. Three famous equations, all fitting the one template, differing only in their principal coefficients.
The three model equations are not exotic special cases — they are the same general form with different A, B, C.
It is the principal part (the second-derivative terms) alone that fixes the type and the qualitative behaviour; the lower-order terms D u_x + E u_y + F u and the source G can change details but never the elliptic / parabolic / hyperbolic character.