I am doing the following problem:
Transform the equation $$y\left(\frac{\partial z}{\partial x}\right) - x \left(\frac{\partial z}{\partial y} \right) = (y-x)z$$
by introuducing new independent variables:
$$u = x^2 + y^2, v = \frac{1}{x} + \frac{1}{y}$$
and a new function:
$$w = \ln(z) - (x + y)$$
I did everything as it is in the example I have (system of first order diferentials) and I got to this equation as the new one:
$$\left(\frac{xz}{y^2}- \frac{yz}{x^2}\right) \frac{\partial w}{\partial v} = 0$$
The solutions say this is the result: $\frac{\partial w}{\partial v} = 0$
I checked multiple times what I've done and I cannot find a mistake. Am I allowed to divide by the term in the braces and treat it as a constant that's different from 0? If I can, why? It doesn't make sense to me. If we try to write it in terms of v and w, we cannot divide it at all.
How to do this properly and is there another way at approaching it?
Making
$$ z(x,y) = e^{x+y}e^{w(x,y)} $$
and proceeding with the change of variables, after some symbolic effort we get at
$$ \frac{v \sqrt{u v^2+1} \left(\sqrt{u v^2+1}+1\right) \sqrt{\frac{u v^2-2 \left(\sqrt{u v^2+1}+1\right)}{v^2}} \exp \left(w(u,v)+\frac{u v^2+\sqrt{u v^2+1}+1}{v \sqrt{u v^2+1}}\right)\frac{\partial w}{\partial v}(u,v)}{u v^2+2 \sqrt{u v^2+1}+2}=0\Rightarrow \frac{\partial w}{\partial v}(u,v)=0 $$