Problem $2(25$ Points$)$

Consider one$-$dimensional, steady state heat transfer through the plane wall with heat generation in Figure $2.$ If heat transfer through the wall occurs in the $x-$direction by conduction only, apply the Heat Equation to answer the questions below. The dimensions and surface temperatures in the system are as indicated in Figure $2.$

Note: clearly state all simplifying assumptions, boundary conditions and if any of these are used in simplifying equations clearly indicate how and why. Show all your steps.

$($a$)$ Using the Heat Equation and appropriate assumptions and parameters, prove that the temperature distribution is given by$,$

$T(x)=-\frac{(q^{})x^2}{2k}+C_1*x+C_2$

Where, $C_1$ and $C_2$ are constants $(15$ points$).$

$($b$)$ Consider boundary conditions of the first kind describing the surface temperatures $T_{S,1}$ and $T_{S,2}at+L$ and $-L$ respectively. Prove that the constant $C_1$ in problem $2(a)$ is given by the expression below $(10$ points$).$

$C_1=\frac{T_{S,2}-T_{S,1}}{2L}$