Product substitution for a first-order equation ($x{y}^{\prime }+y=F(xy),v=xy$)

The product rule gives $v^{\prime }=x{y}^{\prime }+y$ for $v=xy$, reducing the displayed equation to the autonomous separable differential equation $v^{\prime }=F(v)$. On an interval where $F(v)\ne 0$, integrate $\int dv/F(v)=x+C$ and recover $y=v/x$ for $x\ne 0$. Zeros of $F$ give additional constant solutions for $v$ and must be considered before division. A solution for $y$ extends through zero only if the recovered quotient has the necessary finite derivatives and satisfies the undivided original equation there.