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Math Help - An inhomogeneous Sturm-Liouville system

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    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.
    Last edited by xinglongdada; July 31st 2011 at 05:16 PM.
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