Transform the differential equation:
\begin{equation} x^2\frac{\partial{z}}{\partial{x}} + y^2\frac{\partial{z}}{\partial{y}}=0 \end{equation}
Taking as new variables $u=x$, $v=1/y-1/x$, $w=1/z-1/x$. Obtain a new differential equation in terms of $u$, $v$ and $w$.
This isn't an exercise from formal course of differential equations, this is an exercise to practice derivating in several variables, etc. I'm kind of confused in this one, as when I apply the Chain Rule I haven't been able to separate completely the terms involving $x$, $y$ and $z$ and the terms $u$, $v$,$w$.
Could you help me out?
Hint
$$z=z(x,y)=z(u,v)=\frac 1{w+1/x}=\frac1{w+1/u} \\ x^2=u^2\ \ and \ \ \ y^2=\{\frac 1{{v+1/u}}\}^2\\ \frac{\partial z}{\partial x}= \frac{\partial z}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial z}{\partial v} \frac{\partial v}{\partial x}$$ and $$\frac{\partial z}{\partial u}=-\left(\frac1{w+1/u}\right)^{-2}\left(\frac{\partial w}{\partial u}-1/u^2\right)$$