Determine the equilibrium temperature distribution inside a circular annulus ($r_1\le r\le r_2$) if the outer radius is at temperature $T_2$ and the inner radius is at temperature $T_1$.
The corresponding heat equation in polar coordinates can be expressed as follows:
$$\nabla^2u=\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right)+\frac{1}{r^2}\frac{\partial ^2 u}{\partial \theta^2}=0$$
In our case there is no dependence on $\theta$, so we can set the second derivative of $u$ with respect to $\theta$ in our equation to $0$. Thus we get an ordinary differential equation:
$$\frac{d}{dr}\left(r\frac{du}{dr}\right)=0,$$
which is an Euler-Cauchy equation. Solving with the boundary conditions $u(r_1)=T_1$ and $u(r_2)=T_2$ we get:
$$u(r)=\frac{(T_1 - T_2) \log(r) + T_2 \log(r_1) - T_1 \log(r_2)}{\log(r_1) - \log(r_2)}$$
Please let me know if this approach is correct.