This is really more of a hint than a fully fledged answer, but the way to go is:

1. rewrite the equation as an eigenvalue problem $H\psi = \lambda \psi$

2. prove that $H$ is self adjoint (use integration by parts and the boundary conditions).

3. use the standard argument that says that selfadjoint operators in Hilbert space have real eigenvalues (see e.g http://planetmath.org/encyclopedia/EigenvaluesOfAHermitianMatrixAreReal.html)

This is really more of a hint than a fully fledged answer, but the way to go is:

1. rewrite the equation as an eigenvalue problem $H\psi = \lambda \psi$

2. prove that $H$ is self adjoint (use integration by parts and the boundary conditions).

3. use the standard argument that says that selfadjoint operators in Hilbert space have real eigenvalues (see e.g Link)