Past exam of the differential equations and applications course of the University of Oxford / 2010 / A / 1 / i / b

For each $0<\xi <2\pi$, the Green's function is a distribution in $x$ satisfyingEquivalently, $g$ is a classical homogeneous solution on each side of $\xi$, satisfies the same endpoint conditions, and obeys the Green function jump conditionsTo see equivalence, let $[g]$ and $[g_x]$ denote its jumps. The singular part of $L_{x}g$ is $[g]{\delta }^{\prime }(x-\xi )+([g_x]+2[g])\delta (x-\xi )$. Matching a unit Dirac delta distribution with no derivative first forces continuity, then the unit derivative jump. Conversely these conditions produce exactly that distribution. The solution for a regular forcing is $y(x)={\int }_0^{2\pi }g(x,\xi )f(\xi )d\xi$. The homogeneous endpoint problem has only the zero solution, which ensures uniqueness of this kernel.