Dear all, I've got a problem that I could not handle. It goes as follows.

Consider the following inhomogeneous S-L system:

$\displaystyle y''+[\lambda+q(x)]y=0;y(0)\cos\alpha-y'(0)\sin\alpha=0,y(1)\cos\beta-y'(1)\sin\beta=0.$

Prove

1. when $\displaystyle \lambda$ is not an eigenvalue of the corresponding homogeneous system, then there exactly exists one solution;

2. when $\displaystyle \lambda=\lambda_m$ for some eigenvalue, then a solution exists if and only if

$\displaystyle \int_0^1 f(x)\varphi_m(x)\,dx=0,$

where $\displaystyle \varphi_m(x)$ is the eigenfunction correponding to $\displaystyle \lambda_m.$