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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.

An implicit homogeneous first-order differential equation $y/x=F(p)$, where $p=dy/dx$, can be reduced by treating the slope as an auxiliary variable. Differentiating $y=xF(p)$ givesWhere the divisions are valid, this is a separable differential equation. Constant slopes lost when dividing by $p-F(p)$ must be checked in the original equation.

For $F(p)=2p/(1-p^2)$, the reduced equation is $2x{p}^{\prime }=-p(1-p^2)$ and its nonconstant solutions obey $x{p}^2/(1-p^2)=C$. Substitution recovers the parabolas $y^2=4C(x+C)$, together with the constant solution $y=0$. Real graph branches must have $p^2=C/(x+C)\ge 0$, and the original restrictions $x\ne 0$, $p\ne \pm 1$ still apply. The constant slopes $p=\pm 1$ solve the reduced equation but are excluded from the original equation.