= Homogeneous tangent differential equation
{title2=$y'=y/x+a\tan(y/x)$}
For real $a\ne0$ and $x\ne0$, set $u=y/x$ to obtain $xu'=a\tan u$. On nonconstant branches, separation gives $\log|\sin u|=a\log|x|+C$, hence $\sin(y/x)=C_1|x|^a$. Zeros of the divided-out tangent give $y=k\pi x$, already represented by $C_1=0$. Points where $\cos(y/x)=0$ remain excluded because the original <differential equation> is undefined. An implicit trigonometric identity alone does not specify a connected branch or its domain.
Back to article page
