Solve Wave Equation Initial-Boundary-Value-Problem

Question

I am trying to solve the following problem and this is my working so far. I'm struggling to get to the general solution for $X(x)$ as I'm not sure of the $\lambda$ value. Please could someone point me in the right direction?

Answer 1

For non-positive values of $\lambda$ there is no solution. Using the hint of Paul, to obtain:

$$X(x) = C \cos (n\pi x)$$

For $$T(t)= Ae^{-4\lambda t}$$

Now combining....

$$u(x,t)= \sum_{n=1}^\infty C_n \cos (n \pi x) e^{ (-4n^2\pi^2 t)}$$