 Solution

For the Dirichlet Green's function, solve $-D{G}_{xx}=\delta (x-\xi )$ with zero endpoint values. Away from $\xi$, $G$ is linear. Continuity at $\xi$ and the derivative jump $G_x(\xi +,\xi )-G_x(\xi -,\xi )=-1/D$ giveFor nonzero Dirichlet boundary conditions, subtract their linear interpolant. The resulting boundary value problem has solutionIn the final heat equation, a steady state has zero time derivative. Here $l=1$, $f(x)=x$, $\alpha =1/D$, $\beta =2/D$. Integrating $u^{\prime \prime }=-x/D$ twice and imposing the endpoint values givesIts second derivative and both boundary conditions verify the answer directly.