Recently at our school we was asked to try solve the following (our calculus course is rather basic so this kind of tasks is out of our scope, however the were considered as interesting and useful): Transform equation $x^2 \frac{dz}{dx}-xy \frac{dz}{dy}=2$, assuming u and v as new independent variables taking $u=xy, v=\frac{y}{x}$.
Unfortunately our course was compressed due to pandemic and our teacher is out of time, so we were not told the solution of this task. I was not able to cope it, so would be very interested in solution. Thanks!
Hint:
$$ \frac{\partial z}{\partial x}=\frac{\partial z}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial z}{\partial v}\frac{\partial v}{\partial x}$$ $$\frac{\partial z}{\partial y}=\frac{\partial z}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial z}{\partial v}\frac{\partial v}{\partial y}$$