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We study linear differential equations whose coefficients consist of products of powers of natural logarithm and general continuous functions. We derive conditions that guarantee the non-oscillation of all non-trivial solutions of the treated type of equations. The conditions are formulated as a non-oscillation criterion, which is the counterpart of a previously obtained oscillation theorem. Therefore, from the presented main result, it follows that the analysed equations are conditionally oscillatory. The used method is based on averaging techniques for the combination of the generalized adapted Prüfer angle and the modified Riccati transformation. This article is finished by new corollaries and examples.

We deal with Euler‐type half‐linear second‐order differential equations, and our intention is to derive conditions in order their non‐trivial solutions are non‐oscillatory. This paper connects to the article P. Hasil, J. Šišoláková, M. Veselý: Averaging technique and oscillation criterion for linear and half‐linear equations, Appl. Math. Lett. 92 (2019), 62–69, where the corresponding oscillatory counterpart is studied and an oscillation criterion is established. The used effective technique for this investigation is the combination of the generalized adapted Prüfer angle and the modified Riccati transformation. This paper is completed by two corollaries and an example concerning the linear case when we obtain new results as well. [ABSTRACT FROM AUTHOR]