Write the solution as a sum of two functions, one independent of time.
The new boundary conditions are:
Substituting the sum of functions into the DE gives a simple ODE and a PDE:
And the PDE
Seperate variables to get two ODEs. Set them equal to a constant that will simplify calculations later:
This is a Fourier sine series for the function on the right. The coefficients are then given by:
The solution is