V3I1P33

Qualitative Behaviour of Solutions of Second and Third Order Neutral Differential Equations: A Survey Study

Ajay kumar^1*, Dr. Ajay kumar²

Abstract

In this study, the qualitative characteristics exhibited by solutions to second and third-order neutral differential equations related to their oscillatory behaviour, stability, boundedness, and asymptotic behaviour have been examined. These equations are characterised by their combination of unpredictable time delays coupled with unpredictable time variability in both their state variable and derivatives. Further, this is leading to an intricate relationship in their dynamics. Therefore, the behaviour expressed by these equations depends heavily on the order of the Differential Equation, the assumptions about the coefficients, and the temporal delay structure associated with the equation. The paper reviews existing criteria used to establish whether second-order equations exhibit oscillation. Specifically, provides details on the influence of the delay parameter and neutral coefficients in terms of these behaviours as determined by monotonicity and sign conditions. Conversely, the response of the equations to neutral terms and delays is considerably greater than for second-order equations and exhibits mixed behaviours (e.g., oscillatory, non-oscillatory) to small parameter fluctuations. The study highlights the need for unified analytical frameworks capable of addressing nonlinearities and variable delays across different equation orders

Keywords:

Neutral differential equations, Oscillation theory, Stability analysis, Higher-order delay equations