In solving this (which comes up in Maxwell-Boltzmann’s distribution), we find that on a sphere centred at origin, i.e., for $x^2+y^2+z^2=C$ where $C$ is some constant, the solution must take the form $$f(x)=A\exp\Bigr({F’(C)\over F(C)} x^2\Bigl)$$ And then we notice that this is also valid for all independent values of $x, y, z$.
Question: Is this the only solution? Or just some special case of a particular solution?