# Thread: An inhomogeneous Sturm-Liouville system

1. ## An inhomogeneous Sturm-Liouville system

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

Consider the following inhomogeneous S-L system:
$y''+[\lambda+q(x)]y=0;y(0)\cos\alpha-y'(0)\sin\alpha=0,y(1)\cos\beta-y'(1)\sin\beta=0.$

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

2. when $\lambda=\lambda_m$ for some eigenvalue, then a solution exists if and only if
$\int_0^1 f(x)\varphi_m(x)\,dx=0,$
where $\varphi_m(x)$ is the eigenfunction correponding to $\lambda_m.$