In this and the following chapter the solution of differential equations of the types typically encountered in physical science and engineering is extended to situations involving more than one independent variable. Only linear equations will be considered.

The most commonly occurring independent variables are those describing position and time, and we will couch our discussion and examples in notations which suggest these variables; of course the methods are not restricted to such cases. To this end we will, in discussing partial differential equations (p.d.e.), use the symbols u, v, w, for the dependent variables and reserve x, y, z, t for independent ones. For reasons explained in the preface, the partial equations will be written out in full rather than employ a suffix notation, but unless they have a particular significance at any point in the development, the arguments of the dependent variables, once established, will be generally omitted.

As in other chapters we will concentrate most of our attention on second-order equations since these are the ones which arise most often in physical situations. The solution of first-order p.d.e. will necessarily be involved in treating these, and some of the methods discussed can be extended without difficulty to third- and higher-order equations.

The method of ‘separation of variables’ has been placed in a separate chapter, but the division is rather arbitrary and has really only been made because of the general usefulness of that method.