Topic $2$: Steady$-$State Temperature Distribution in a Rod with Internal Heat Generation. The problem is worth $25$ points. Consider a rod of length $L$ generating heat internally at a uniform rate of $q[\frac{(W)}{m^3}].$ The thermal conductivity of the material is $k,$ and the surface at both ends of the rod $($at $x=0$ and $x=L)$ is maintained at a constant temperature $T_s.$ Assume steady$-$state conditions and that heat transfer occurs only along the length of the rod $(1$D problem$).$

Derive the equation for the temperature distribution $T(x)$ inside the rod, considering uniform internal heat generation.

Solve for the temperature distribution $T(x)$ subject to the boundary conditions $T(0)=T_s$ and $T(L)=T_s.$

Find the location of the maximum temperature inside the rod and express it in terms of $q,k,L,$ and $T_s.$