= Solution
On an interval with $x\ne0$, set $y=xv$. The <product rule> gives $y'=v+xv'$, and substitution gives
$$
x^2(v+xv')=x^2+x^2v+x^2v^2,
\qquad\boxed{xv'=1+v^2.}
$$
This is a <separable differential equation>. Since $1+v^2>0$ for real $v$, separation loses no real constant solution:
$$
\frac{dv}{1+v^2}=\frac{dx}{x},
\qquad \arctan v=\log|x|+C.
$$
Thus the <general solution> on any interval avoiding zero and the poles of the <tangent function> is
$$
\boxed{y(x)=x\tan(\log|x|+C).}
$$
At $x=0$ the original <differential equation> forces $y(0)=0$, but no real solution from this family extends over a whole interval through zero: $\log|x|+C$ encounters infinitely many poles as $x\to0$.
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