Harmonic functions and real valued function related to it

Question

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.

Answer 1

We are looking for radial harmonic functions.

We have $u_x=2xh'(x^2+y^2)$ hence $u_{xx}=2h'(x^2+y^2)+4x^2h''(x^2+y^2)$, and in a similar way we can compute $u_{yy}$. This gives, letting $t:=x^2+y^2$, a differential equation satisfied by $h$.