Eigenvalues of the 1D laplacian with mixed boundary conditions

Question

I am trying to find the eigenvalues and eigenvectors of the Laplacian with mixed boundary conditions on $[0,L]$:

More precisely:

$$X''(x) = \lambda X(x)$$ with $X'(0)=0$ and $X(L)=a$.

I know how to do it with pure Dirichlet or pure Neumann, but not for this mixture.

Could you help me or point me to the right reference ?

Thanks folk

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Just found part of the answer here: http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors_of_the_second_derivative#Mixed_Dirichlet-Neumann_boundary_conditions

but I am not sure how to relate it to parameter $a$ in the question.

any help welcome