Find the eigenvalues $\displaystyle \lambda_n$ and eigenfunctions $\displaystyle X_n$ of the boundary value problem

$\displaystyle X'' + \lambda X = 0 \ \ \ X'(0) = 0 \ \ \ X'(L) = 0$

Using the explicit form of $\displaystyle X_n(x)$ show that $\displaystyle \int_0^L X_nX_m \ dx = 0 \ (m \neq n)$ and $\displaystyle \int_0^L X_n X_m \ dx = \frac{L}{2} \ (m = n)$