= Slope substitution for equations involving x y'-y
{title2=$v=y/x\ \Longrightarrow\ xy'-y=x^2v'$}
On an interval with $x\ne0$, the substitution $y=xv$ and the <product rule> give the displayed identity. An equation $xy'-y=g(y/x)$ becomes $v'=g(v)/x^2$ and permits <separation of variables> wherever $g(v)\ne0$. Roots $g(v_0)=0$ also give solutions $y=v_0x$ 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 $y'=y/x+g(y/x)/x$ need not be a homogeneous equation of the usual form $y'=F(y/x)$.
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