Variable Change In A Differential Equation If I have the following differential equation:
$\dfrac{dy}{dx} = \dfrac{y}{x} - (\dfrac{y}{x})^2$ 
And if I make the variable change: $\dfrac{y}{x} \rightarrow z$
I know have $\dfrac{dy}{dx} = z-z^2$
What is $\dfrac{dx}{dy}$ after the variablechange?

Suppose you have a differential equation that looks like this: $$y'=F\left ( \frac{y}{x}\right )$$
then you can make a substitution $v(x)=\frac{y}{x} \iff y=vx \implies y'=v+xv'$ to transform your ODE into an ODE in $v$ $$\implies v+xv'=F(v) \iff \frac{dv}{F(v)-v}=\frac{dx}{x}$$
This equation is separated and you can solve it by the usual methods.
In your case you have: $$y'=\frac{y}{x}-\left (\frac{y}{x}\right)^2$$
A substitution $u=\frac{y}{x} \iff ux=y \iff y'=u+xu'$ will lead to the differential equation:
$$xu'=-u^2$$
Can you solve it from here?
