A rof of length `l` with thermally insulated lateral surface consists of material whose heat conductivity coefficient varies with temperature as `x = alpha//T`, where `alpha` is a constant. The ends of the rod are kept at temperatures `T_1` and `T_2`. Find the function `T (x)`, where `x` is the distance from the end whose temperature is `T_1`, and the heat flow density.

Explanation

By definition the heat flux (per unit area) is
`Q = - K (dT)/(dx) = - alpha (d)/(dx) 1n T = constant = + alpha (1n T_1 //T_2)/(l)`
Integrating `1n T = (x)/(l) 1n (T_2)/(T_1) + 1n T_1`
where `T_1` = temperature at the end `x = 0`
So `T = T_1 ((T_2)/(T_1))^(x//l)` and `Q = (alpha 1n T_1//T_2)/(l)`.