A class of time-varying differential equations for vibration research and application

As a pivotal branch within the realm of differential equations, the theory of oscillation holds a crucial position in the exploration of natural sciences and the construction of modern control theory frameworks. Despite the extensive research conducted globally, focusing on individual or combined analyses of key elements such as explicit damping terms, positive and negative coefficients, time-varying delays, and nonlinear neutral terms, systematic investigations into the oscillatory behavior of even-order differential equations that concurrently embody these four complex characteristics remain scarce. This paper, by establishing reasonable assumptions, innovatively presents two crucial criteria, aiming to preliminary delve into the oscillation patterns of even-order differential equations under specific complex settings. In the course of the study, a variety of mathematical techniques, such as Riccati transformation, calculus scaling methods, and partial integration, have been utilized by the researchers to perform the necessary derivations and confirmations.