Find all real-valued functions $h$, defined and of class $C^2$ on the positive real line, such that the function $u(x,y)=h(x^2 + y^2)$ is harmonic.
I was thinking that it will be harmonic only if $h$ will be constant.
My attempt:
Since $u$ is harmonic, $$ u_{xx}+u_{yy}=0 $$ $ u_{xx}=4x^2h_{xx}+2h_x $ and $u_{yy}=4y^2h_{yy}+2h_y$. So, $$ 4\Big(x^2h_{xx}+y^2 h_{yy}\Big)+2\Big( h_x+h_y \Big) =0$$
After that I stuck. Any hint will be appreciated.
On
For every $z>0$, $z=x^2+y^2$ describes a circle with center $(0,0)$. The mean value of $u(x,y)$ over this circle is $h(z)$, and this must be equal to $u(0,0)=h(0)$ due to the mean value property of harmonic functions.
$ u=h(x^2+y^2) $ \begin{eqnarray*} \frac{ \partial u} {\partial x} =2x h'(x^2+y^2) \\ \frac{ \partial^2 u} {\partial x^2} = 4x^2 h''(x^2+y^2)+ 2 h'(x^2+y^2). \\ \end{eqnarray*} Substitute this into $u_{xx}+u_{yy}=0$ and let $z=x^2+y^2$ \begin{eqnarray*} \color{blue}{zh''(z)+h'(z)=0}. \end{eqnarray*}