# Boundary Value Problem with eigenvalue dependent boundary conditions

I have the following BVP:

$$(\alpha+x)y''+y'+\lambda^2 y=0$$, with the following boundary condition:

$$y(x=L)=0$$ and $$y'(x=0)+\lambda^2 y(0)=0$$,

where $$\alpha$$ is some constant, $$x$$ is a parameter and $$\lambda$$ is an eigenvalue. I tried to solve it but the orthogonality condition is not satisfied for this case. I assume it has to do with the (2) B.C. Any suggestions?, What is the physical meaning of this?

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