Use the method of eigenfunction expansion to solve with boundary conditions and initial condition

I think I might have stuffed my solution here, but I don't know.

Would someone please be able to help me out?

Base a trial solution on the solution to the homogeneous problem

Now represent as a series of the particular eigenfunctions

Let

i.e. Let

Sub into PDE

The only way for this to be true for all is if

This is a first order ODE.....so use an integrating factor

where

Multiply by

In order for choose

Skipping a few lines of calculation...