When you separate the equations you should get
So solving the t equation gives
This is a 2nd order ODE with constant coefficients this gives
So the solutions are
In the question below I dont understand where the e^(-kt) comes from? In general when I have been using laplace and fourier transforms the answers are in the form 4/(n^3)*(Pi^3) before the series and not an exponential. Any help on how e^(-kt) is derived would be much appriciated.