In the partial differential equation $$(x+y)z_x-(x-y)z_y=0$$ change the variables $(x,y)$ to $(u,v)$, $u=ln(\sqrt{x^2+y^2}), v=tan^{-1}(\frac{y}{x})$
Can someone please help with this, don't where to start. Explanations for each step would be much appreciated. Thank you.
More verbosely, what they mean is the following.
Consider a (smooth enough) solution $z=z(x,y)$ of the partial differential equation (PDE). $z$ satisfies, for all $x,y$ in the relevant domain, $$(x+y)z_x(x,y)-(x-y)z_y(x,y)=0\tag{**}.$$ Now, let $w=w(u,v)$ be such that $$z(x,y)=w(\ln((x^2+y^2)^{1/2}),\,\tan^{-1}(y/x))\tag{*}.$$ What PDE does $w$ satisfy?
Hint: compute the partial derivatives of $z$ in (*) and plug them into (**).