= Solution
For the <Dirichlet Green's function>, solve $-DG_{xx}=\delta(x-\xi)$ with zero endpoint values. Away from $\xi$, $G$ is linear. Continuity at $\xi$ and the <derivative> jump $G_x(\xi+,\xi)-G_x(\xi-,\xi)=-1/D$ give
$$
\boxed{G(x,\xi)=\begin{cases}\dfrac{x(l-\xi)}{Dl},&x\leq\xi,\\[4pt]\dfrac{\xi(l-x)}{Dl},&x\geq\xi.\end{cases}}
$$
For nonzero <Dirichlet boundary conditions>, subtract their linear interpolant. The resulting <boundary value problem> has solution
$$
\boxed{u(x)=\alpha+\frac{\beta-\alpha}{l}x+\int_0^lG(x,\xi)f(\xi)\,d\xi.}
$$
In the final <heat equation>, a <steady state> has zero time <derivative>. Here $l=1$, $f(x)=x$, $\alpha=1/D$, $\beta=2/D$. Integrating $u''=-x/D$ twice and imposing the endpoint values gives
$$
\boxed{u_{\mathrm{steady}}(x)=\frac1D\left(1+\frac76x-\frac16x^3\right).}
$$
Its second <derivative> and both <boundary conditions> verify the answer directly.
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