The substituting $y=v x$ reduces the homogeneous differential equation $\frac{d y}{d x}=\frac{y}{x}+\tan \frac{y}{x}$ to the form

$(\tan v) d v=x d x$ $(\tan v) d v=\frac{d x}{x}$ $(\cot v) d v=x d x$ $\cot v d v=\frac{d x}{x}$

$\cot v d v=\frac{d x}{x}$

Substituting $y=v x$ and $\frac{d y}{d x}=v+x \frac{d v}{d x}$, we get $v+\frac{d v}{d x}=v+\tan v \Rightarrow(\cot v) d v=\frac{1}{x} d x$