Past exam of the approximation theory and finite element methods course of the University of Oxford / 2007 / B / 7 / i / Solution

Taking $w=z$ in the dual problem, and using $e^x\ge 1$, givesHence $\Vert z{\Vert }_{H^1}\le \Vert \psi {\Vert }_2$. Testing with smooth compactly supported functions identifies $-z^{\prime \prime }+e^{x}z=\psi$ in distributions. Therefore $z^{\prime \prime }\in L^2$ and $z\in H^2(0,1)$, withintegration by parts with arbitrary endpoint test values also gives the natural Neumann boundary conditions $z^{\prime }(0)=z^{\prime }(1)=0$.