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

In the distribution definition, the Green's function is characterized, for fixed $0<\xi <1$, byIn the classical definition, solve $L_{x}g=0$ separately for $0<x<\xi$ and $\xi <x<1$, with those endpoint conditions, and imposeA jump in $g$ would introduce a distributional derivative of the Dirac delta, absent from the right side. Continuity eliminates it; integrating the equation across $\xi$ leaves one quarter of the derivative jump equal to one. These Green function jump conditions make the definitions equivalent. The forced solution is $y(x)={\int }_0^{1}g(x,\xi )f(\xi )d\xi$.