Slope substitution for equations involving x y'-y ($v=y/x⟹x{y}^{\prime }-y=x^2{v}^{\prime }$)

On an interval with $x\ne 0$, the substitution $y=xv$ and the product rule give the displayed identity. An equation $x{y}^{\prime }-y=g(y/x)$ becomes $v^{\prime }=g(v)/x^2$ and permits separation of variables wherever $g(v)\ne 0$. Roots $g(v_0)=0$ also give solutions $y=v_{0}x$ and must be retained if division by $g$ was used. This substitution works for this displayed class even though its explicit factor $1/x$ means that $y^{\prime }=y/x+g(y/x)/x$ need not be a homogeneous equation of the usual form $y^{\prime }=F(y/x)$.