If $u(x, y)=f(\frac{x}{y})$ is a harmonic function,
a) solve the ODE satisfied by $f$. ($U_{xx} +U_{yy}=0$)
b) Show that $∂u/∂r≡0$, where $r=\sqrt{x^2 +y^2}$ as usual.
I have solved part a as in picture but can’t get zero for part b.
Could anyone help me out?
In polar coordinates is $f(x/y)=f(\tan\theta)$, independent from $r$. But we can follow the hard way through the partial derivatives. First, the relation between derivatives and recprocals of derivatives doesn't hold: $\dfrac{\partial r}{\partial x}$ has not the simple relation with his reciprocal as in one variable.
$x=r\cos\theta$ then $\dfrac{\partial x}{\partial r}=\cos\theta=\dfrac{x}{r}$
$y=r\sin\theta$ then $\dfrac{\partial y}{\partial r}=\sin\theta=\dfrac{y}{r}$
$$\frac{\partial u}{\partial r}=\frac{\partial u}{\partial x}\dfrac{\partial x}{\partial r}+\frac{\partial u}{\partial y}\dfrac{\partial y}{\partial r}=$$
$$=\dfrac{f'}{y}\dfrac{x}{r}+\dfrac{-xf'}{y^2}\dfrac{y}{r}=0$$