Here is a statement (or theorem) from my book:
$\pmb{\text{A neat trick: Turning nonlinear separable equations into linear separable equations}}$
$\mathcal{\text{Knowing when to substitute:}}$
You can use the trick of setting $y=xv$ when you have differential equation that is of the form:
$\dfrac{dy}{dx}=f(x,y)$
when $f(x,y)=f(tx,ty)$ where $t$ is a constant.
I understand this. But the book gives no proof. I mean how can we ensure that the trick of setting $y=xv$ will always work whenever $f(x,y)=f(tx,ty)$
Just plug $y=xv$ in the equation.
$$\frac{d(xv)}{dx}=x\frac{dv}{dx}+v=f(x,xv)$$
and
$$\frac{dv}{f(1,v)-v}=\frac{dx}x.$$
Alternatively,
$$f(tx,ty)=f(x,y)\iff f(x,y)=g\left(\frac yx\right)=g(v)$$ and the equation is
$$xv'+v=g(v).$$