This is really more of a hint than a fully fledged answer, but the way to go is:
rewrite the equation as an eigenvalue problem $H\psi = \lambda \psi$
prove that $H$ is self adjoint (use integration by parts and the boundary conditions).
use the standard argument that says that selfadjoint operators in Hilbert space have real eigenvalues (see e.g http://planetmath.org/encyclopedia/EigenvaluesOfAHermitianMatrixAreReal.html)