Past exam of the differential equations and applications course of the University of Oxford / 2011 / A / 1 / b / iii / Solution

Under $y=e^{-2x}v$, the left Robin boundary condition becomes $v^{\prime }(0)-v(0)=0$ and the right Dirichlet boundary condition becomes $v(1)=0$. ChooseTheir Wronskian $v_L{v}_R^{\prime }-v_L^{\prime }{v}_R$ is the constant $\pi$. For $y_L=e^{-2x}{v}_L$ and $y_R=e^{-2x}{v}_R$, it is $W_y(x)=\pi e^{-4x}$. The coefficient enforcing the derivative jump four is $4/W_y(\xi )$. Writing $x_{<}=\min (x,\xi )$ and $x_{>}=\max (x,\xi )$, the Green's function isThe endpoint conditions, continuity and derivative jump follow directly from this Wronskian construction. Its exponential factor also shows why the kernel need not be symmetric in the unweighted inner product.