In the second equation, the solution is where is any real number. In the first equation, things are more serious. There are three possible cases: . Two of these cases would lead to only trivial solutions ( ), I leave that up to you to try to do. Only in the case do we get interesting solutions. If then . Thus, the product solutions are (we absorbed the constant ). We are told that this should be zero when , substiting we get . Thus, the solutions must have the form . Then we are told at we get zero, so . Therefore, . Therefore, the functions satisfy the PDE and the boundary value problem.
The idea is that solves this PDE and we try to pick so that at we get i.e. . Can you solve for the coefficients to make this statement true?