= Product substitution for a first-order equation
{title2=$xy'+y=F(xy),\quad v=xy$}
The <product rule> gives $v'=xy'+y$ for $v=xy$, reducing the displayed equation to the autonomous <separable differential equation> $v'=F(v)$. On an interval where $F(v)\ne0$, integrate $\int dv/F(v)=x+C$ and recover $y=v/x$ for $x\ne0$. 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.
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