Motivating separation of variables

A physics student's first encounter with the method of separation of variables for solving linear partial differential equations can be confusing if the starting assumption—that the solution can be written as a product of functions, each of which depends on only one of the independent variables—is put forward without motivation. We describe a way of introducing the method that provides such motivation by presenting it first as an application of the general idea of expanding an unknown function in a basis, then showing the conventional approach as a way to streamline the calculation. The presentation is designed for students who have not yet studied function spaces, orthogonality, and the like.