Past exam of the differential equations 2 course of the University of Oxford / 2023 / 1 / c / i / Solution

At $a=\pi$, the nonzero homogeneous solution $e^{-x}\sin x$ satisfies both zero boundary conditions. Thus the boundary value problem has a nontrivial kernel and cannot have an inverse for arbitrary forcing.

More directly, the adjoint operator has the zero-endpoint homogeneous solution $w(x)=e^x\sin x$. If an ordinary Green's function existed, the integration by parts identity would implyBut $w(\xi )\ne 0$ for $0<\xi <\pi$. This contradiction rules out the Green's function for the unmodified operator. It is the Fredholm alternative obstruction.