g(r)=f(x,y) とおくと
fx=g'(r)(∂r/∂x)
fxx=g''(r)(∂r/∂x)^2+g'(r)(∂^2r/∂x^2)=g''(r)(x^2/r^2)+g'(r)(y^2/r^3)
fyy=g''(r)(y^2/r^2)+g'(r)(x^2/r^3)

fxx+fyy=g''(r)+(1/r)g'(r)=0 から
rg'(r)=C_1
g(r) = C_1*logr + C_2