Consider the following Sturm-Liouville problem: $$X''+\lambda X=0,\quad X'(0) = 0,\, X(\pi) = 0,$$ where $X = X(x)$.
- Find all positive eigenvalues and corresponding eigenfunctions of the problem.
- Is $\lambda = 0$ an eigenvalue for this problem? If yes, find its eigenfunction. If no, explain why it is not.