 Solution

Choose a complete orthonormal system of eigenfunctions ${\varphi }_n$ for the self-adjoint problem. Expand $f={\sum }_n{f}_n{\varphi }_n$ and $y={\sum }_n{c}_n{\varphi }_n$, withSubstitution into $Ly=f$ gives ${\lambda }_n{c}_n=f_n$. Since no eigenvalue is zero, the Sturm-Liouville eigenfunction expansion isFor eigenfunctions not normalized to one, divide each term additionally by $⟨{\varphi }_n,{\varphi }_n⟩$. The expansion is understood in the weighted function space and, with sufficient data regularity, in the operator domain. A zero eigenvalue would instead require its corresponding source coefficient to vanish and would leave an arbitrary homogeneous component.