Non-monotone positive solutions of second-order linear differential equations: existence, nonexistence and criteria

We study non-monotone positive solutions of the second-order linear differential equations: $(p(t)x')' + q(t) x = e(t)$, with positive $p(t)$ and $q(t)$. For the first time, some criteria as well as the existence and nonexistence of non-monotone positive solutions are proved in the framework of some properties of solutions $\theta (t)$ of the corresponding integrable linear equation: $(p(t)\theta')'=e(t)$. The main results are illustrated by many examples dealing with equations which allow exact non-monotone positive solutions not necessarily periodic. Finally, we pose some open questions.