= Homogeneous first-order differential equation
{title2=$y'=F(y/x)$}
A <homogeneous first-order differential equation> has the form $y'=F(y/x)$ on a domain where $x\ne0$. The substitution $v=y/x$, so $y'=v+xv'$, reduces it to
$$
xv'=F(v)-v.
$$
Where $F(v)-v\ne0$, this is a <separable differential equation>, with $\int dv/[F(v)-v]=\ln|x|+C$. Values $v_0$ satisfying $F(v_0)=v_0$ give the additional straight-line solutions $y=v_0x$, which must be checked before division. This scaling-based use of “homogeneous” is different from a <homogeneous linear differential equation>.
For example, $y'=y/x+x/y$ gives $F(v)=v+1/v$, so $vv'=1/x$ and $y^2=x^2(2\ln|x|+C)$. Real explicit branches are $y=\pm x\sqrt{2\ln|x|+C}$ on <intervals> where the expression under the square root is positive. Zeros of a denominator and endpoints of a branch must still be checked in the original <ordinary differential equation>.
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