Solutions of the Laplace equation How do I find solutions $u=f(r)$ of the two-dimensional Laplace equation $u_{xx}+u_{yy}=0$ that depend only on the radial coordinate $r= \sqrt{x^2+y^2}$

You should also use the polar coordinate version of the Laplace operator 
$$
\Delta u 
= {\partial^2 u \over \partial r^2}
+{1 \over r} {\partial u \over \partial r}
+ {1 \over r^2} {\partial^2 u \over \partial \theta^2}
$$
and trying for $u(r, \phi) = f(r)$. This reduces to solving the ODE
$$
0 = g'(r) +{1 \over r} {g}
$$
for $g(r) = f'(r)$.
