I'm curious about the intervals of existence of the DE in question. It is solvable via separation of variables, upon which one receives $$ y = \tan\left(-\frac1x - x + C\right) $$ for arbitrary constant $C$. The problem is that the intervals of existence of this equation are quite ugly to express neatly. Namely, they are the simply-connected components of the set $\{x\in\mathbb{R}\ |\ -\frac1x-x+C\neq \pi(n+1/2)\ \forall n\in\mathbb{N}\}$. One can find these intervals explicitly quite simply using the quadratic formula, but I dislike this solution because
- It leads to really ugly intervals (though I'm not sure there's any nicer way of getting the intervals of existence).
- It doesn't give much intuition for how the interval of existence is related to the initial DE. Why can't $x\in\mathbb{R}$? In the original DE, why should we expect the solution not to exist at these rather odd points?