PDE7

First, notice that there are two derivates of w.r.t in the differential equation and two inital conditions. There are also two derivatives on and and two boundary conditions on each. Now, separate variables.
The new boundary conditions are:
Plug back into the original DE.
Separate variables. For simplicity, keep higher derivatives on top, keep the term with the psi function, and let the constant be .
The solution for the DE involving is:
Don't worry about the inital conditions until later. Work is done on this function for now.
Separate variables. Keep higher derivatives on top.
The choice of is for convenience.
Write the new ODEs and BCs.
The solution of is:
Letting would give no interesting solutions.
The case would not add any information to the sum in the final answer.
Letting would give no interesting solutions.
 The eigenvalues are
The eigenfunctions are
No coefficients are needed in the formula for the eigenfunctions. They are absorbed by the other constants and . The solution is
The first initial condition gives
This is the sinesine double Fourier series for f(x,y), so the coefficients are given by