Solve in Mathematica

CHE $230$: Problem Set $11$

The steady$-$state one$-$dimensional heat conduction equation in a rod can be written as:

$-\frac{d}{dx}((x)\frac{dT}{dx})=q$

Here $T$ is the temperature, $x$ is the position along the rod, $$ is the thermal conductivity, and $q$

is the rate of heat generation in the rod.

Consider two rods, each of length $1,$ but with different "conductivity profiles":

${}_1(x)=\frac{1}{3}+x(1-x)$ versus ${}_2(x)=\frac{2}{3}-x(1-x).$

These have the same average $(1/2),$ but ${}_1$ is larger in the middle, and ${}_2$ is larger on the ends.

a$)$ Plot ${}_1$ and ${}_2$ in the same plot $($make ${}_1$ red and ${}_2$ blue$).$

b$)$ Solve the heat equation analytically for each rod using DSolve. Assume $q=1,T=T_0$ at

both ends of the rod, and compute $T$ relative to $T_0.$

c$)$ Plot the two solutions together $($make the first one red, the second blue$).$ Which one is

hotter in the middle? Why?