1. The problem statement, all variables and given/known data Given X''(x) + lambda*X(x) = 0 X(0) = X'(0), X(pi) = X'(pi) Find all eigenvalues and eigenfunctions. 2. Relevant equations Case lambda = 0 Case lambda > 0 Case lambda < 0 3. The attempt at a solution First case, X(x) = Ax + B but the function doesn't satisfy the boundary condition at pi. Second case, lambda = k^2 X(x) = Acos(kx) + Bsin(kx) X'(x) = Bkcos(kx) - Aksin(kx) X(pi) - X'(pi) = Bsin(kpi) + Bk^2 sin(kpi) => 1+ k^2 = 0 so lambda = -1 a) I was told the eigenfunction for that is X(x) = Ce^x. How? b) Also, is that just the lambda_0 or does it satisfy all possibly eigenfunctions (i.e. eigenvalue lambda_n)?