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The temperature distribution in the plane wall of area $A=1m^2,$ thickness, $L=0\mathrm{.}35m,$ thermal conductivity $k=15\frac{(W)}{(m)*(K)}$ is governed by the $1$ D heat conduction equation:

$\frac{d^{2}T}{d{x}^2}+b=0$

where, $b={10}^5\frac{C}{m^2}.$

The left boundary is maintained at isothermal condition, $T_1=100C,$ while the right boundary is exposed to hot gas at $T_{}=1000C$ with a convective heat transfer coefficient of $h=500\frac{W}{m^2}.K$ as shown in the diagram.

Answer the following questions:

$($a$)$ Describe the heat transfer equation given in equation $1$ completely.

$($b$)$ Determine the general solution of equation $(1)$ for the temperature distribution $T(x).$

$($c$)$ Provide the two boundary conditions for the problem with respect to the given $x$ coordinate axis shown.

$($d$)$ Use the boundary conditions to determine the particular solution for the problem.

$($e$)$ Find the location where the maximum temperature occurs and determine its value.

$($f$)$ Determine the conduction heat rate at the mid$-$plane.

$($g$)$ What is the surface temperature of the right boundary?