= Slope parameter for a homogeneous first-order equation
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 $y=xF(p)$ gives
$$
p=F(p)+xF'(p)\frac{dp}{dx},\qquad
\frac{dx}{x}=\frac{F'(p)}{p-F(p)}\,dp.
$$
Where 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 $2xp'=-p(1-p^2)$ and its nonconstant solutions obey $xp^2/(1-p^2)=C$. Substitution recovers the <parabolas> $y^2=4C(x+C)$, together with the constant solution $y=0$. Real graph branches must have $p^2=C/(x+C)\geq0$, and the original restrictions $x\ne0$, $p\ne\pm1$ still apply. The constant slopes $p=\pm1$ solve the reduced equation but are excluded from the original equation.
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