Homogeneous tangent differential equation ($y^{\prime }=y/x+a\tan (y/x)$)

For real $a\ne 0$ and $x\ne 0$, set $u=y/x$ to obtain $x{u}^{\prime }=a\tan u$. 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.