In part A I am not sure how to verify that the analytic solution is of the form described? Additionally I have done Laplace's transforms before but i have no idea how to do part B as well. Thanks for all your help!
On the first page you checked that the Dirichlet boundary conditions and . This shows that 2 boundary conditions are satisfied. You need to compute the Laplacian of the function to show that the equation is true everywhere in the domain.