ABSTRACT
Having studied several solution procedures for the heat, wave, and Laplace equations, we need to explain why we have chosen these particular models in preference to others. In Chapter 4 we mentioned that these were typical examples of what we called parabolic, hyperbolic, and elliptic equations, respectively. Below we present a systematic discussion of the general second-order linear PDE in two independent variables and show how such an equation can be reduced to its simplest form. It will be seen that if the equation has constant coefficients, then its dominant part-that is, the sum of the terms containing the highest-order derivatives with respect to each of the variables-consists of the same terms as one of the above three equations. This gives us a good indication of what solution technique we should use, and what kind of behavior to expect from the solution.
