 Solution

On an interval with $x\ne 0$, set $y=xv$. The product rule gives $y^{\prime }=v+x{v}^{\prime }$, and substitution givesThis is a separable differential equation. Since $1+v^2>0$ for real $v$, separation loses no real constant solution:Thus the general solution on any interval avoiding zero and the poles of the tangent function isAt $x=0$ 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 $x\to 0$.