You're going to have fun with that one. The overall procedure (you can find this in the bookThe Mathematics of Diffusion,by J. Crank) is to take the Laplace Transform of the DE, which gives you a second-order ODE. You'll also take the LT of the boundary conditions, as you've done. Solve the resulting system. The result is a function of and the LT variable

Then you have to take the inverse LT. This is where the fun and games begin, because most likely, you can't use a table to just find the inverse LT. You'll have to go back to the definition of the inverse LT using the complex line integral, and then use residue calculus to compute the integrals.

This is a fairly involved problem, it looks like. Is this for a class?