Past exam of the differential equations 2 course of the University of Oxford / 2023 / 1 / a / ii / Solution

Use the unweighted inner product $⟨w,y⟩={\int }_0^a\overline{w}ydx$. Two applications of integration by parts giveThe given boundary conditions set $y=0$ at both ends, but leave $y^{\prime }(0)$ and $y^{\prime }(a)$ independently variable. The remaining boundary term vanishes for every admissible $y$ precisely when $w(0)=w(a)=0$. Consequently the formal adjoint and the boundary conditions specifying the adjoint operator are