Past exam of the differential equations and applications course of the University of Oxford / 2008 / A / 2 / d / Solution

Expand a forcing in the complete orthonormal basis ${\varphi }_n$. Since $L{\varphi }_n=-{\lambda }_n{\varphi }_n$, its inverse multiplies the $n$th coefficient by $-1/{\lambda }_n$. ConsequentlyThe negative sign follows from the eigenvalue convention $Ly=-\lambda y$. This kernel satisfies the mixed endpoint conditions in its first argument and $L_{x}g=\delta (x-t)$.