In either case we are left with an equation in only two independent variables, x and y. It should thus be no more complicated than the earlier equations in the variables t and x. The auxiliary conditions that are imposed are quite different from those for the wave equation. In electromagnetic theory, for example, they are determined if the charge is measured on the boundary. In thermodynamics, they are the temperature determined by the temperature at the edge of the rectangle.
Separation of variables works much as it did for the heat equation and the wave equation .
We use first information from the partial differential equation, then the information from homogeneous boundary conditions, and lastly information from inhomogeneous boundary conditions, which are treated much like the initial conditions for the earlier PDE's. A good rule of thumb in this subject is to deal with the most homogeneous boundaries first. Use the solution we already obtained as a springboard, as follows. Let us turn our attention now to a fully multidimensional problem. There will be few new concepts, though as we shall see the additional dimensions require some extra book-keeping with several indices to label the pieces of the solution correlating with the various dimensions.