I'm working on the following problem:
Find all harmonic functions on $\mathbb R^2$ of the form $f(x^2+y^2)$, where $f$ is of class $C^2$.
What I've done so far: Define $u(x,y)=f(x^2+y^2)$. Then the second-order partial derivatives of $u$ are $$ u_{xx}=2f'(x^2+y^2)+4x^2f''(x^2+y^2),\\ u_{yy}=2f'(x^2+y^2)+4y^2f''(x^2+y^2). $$ If $u$ is harmonic then $u_{xx}+u_{yy}=0$, so we have $$ f'(x^2+y^2)+(x^2+y^2)f''(x^2+y^2)=0. $$ At this point I'm not quite sure what to do. Any help would be appreciated!
Substitute $r= x^2 +y^2$ then you get $$f'(r) +r f''(r) =0$$ and $$\frac{f''(r)}{f'(r)} dr = -\frac{1}{r} dr$$ hence $$\ln |f'(r)| =\ln|r| +C$$ and $$f'(r)= \frac{C}{r}$$ therefore $$f(r) =C\ln |r| +D$$
and hence $$u(x,y) =C\ln (x^2 +y^2 ) +D$$