= Solution
Again put $u=y/x$, requiring $x\ne0$. Substituting $y'=u+xu'$ gives $xu'=1+u^2$. For real $u$ the factor $1+u^2$ never vanishes, so separation loses no constant real solution:
$$
\arctan u=\log|x|+C\quad\text{locally},\qquad
\boxed{y=x\tan\left(\log|x|+C\right).}
$$
Each real solution is defined on a connected interval avoiding zero and every pole of the <tangent function>. Changing the constant by an integer multiple of $\pi$ gives the same branch. If complex-valued solutions are also allowed, the separated-out constant values $u=\pm i$ supply the additional solutions $y=\pm ix$; these do not occur in the real-valued family.
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