Families of separable partial differential equations

Abstract

In this paper the authors investigate classes of partial differential equations (mostly in two independent variables, of order at most 4) which can be solved by the well‐known technique of separation of variables. In particular, several classical ordinary differential equations, some of which have been generalized (by means of certain parameters), are used to assist in the construction of such partial differential equations. The process gives rise to (potentially) many families of equations which appear to be difficult to solve. The article not only illustrates the technique used in the determination of solutions to such equations, but also contrasts the effect of the nature of solutions as certain parameters effect them. In addition, the paper indicates how other classes of solvable equations might be obtained; equations with more than two variables and of any order. Only analytical methods are discussed; however it would not be difficult to use computing machinery as the number of independent variables ...