The steady-state temperature distribution $u(x)$ in a uniform rod of finite length satisfies the boundary value problem

$$\begin{gathered} -D \frac{d^{2}}{d x^{2}} u(x)=f(x), \quad 0<x<l \\ u(0)=0, \quad u(l)=0 \end{gathered}$$

where $D>0$ is the (constant) diffusion coefficient. Determine the Green's function $G(x, \xi)$ for this problem. Now replace the above homogeneous boundary conditions with the inhomogeneous boundary conditions $u(0)=\alpha, \quad u(l)=\beta$ and give a solution to the new boundary value problem. Hence, obtain the steady-state solution for the following problem with the specified boundary conditions:

$$\begin{aligned} &-D \frac{\partial^{2}}{\partial x^{2}} u(x, t)+\frac{\partial}{\partial t} u(x, t)=x, \quad 0<x<1 \\ &u(0, t)=1 / D, \quad u(1, t)=2 / D, \quad t>0 \end{aligned}$$

[You may assume that a steady-state solution exists.]