On the Asymptotic Integration of a System of Linear Differential Equations with Oscillatory Decreasing Coefficients

TL;DR

This paper investigates the long-term behavior of solutions to linear differential systems with oscillatory, decreasing coefficients, introducing a transformation that simplifies the analysis by removing oscillatory terms.

Contribution

It develops an invertible change of variables to transform the system into a form without oscillatory principal parts, facilitating asymptotic analysis.

Findings

Constructed an invertible transformation for large t

Analyzed asymptotic behavior of solutions

Applied method to a specific oscillatory differential equation

Abstract

A system of linear differential equations with oscillatory decreasing coefficients is considered. The coefficients has the form $t^{-\alpha}a(t)$,~$\alpha>0$, where $a(t)$ is trigonometric polynomial with an arbitrary set of frequencies. The asymptotic behavior of the solutions of this system as $t\to\infty$ is studied. We construct an invertible (for sufficiently large $t$) change of variables that takes the original system to a system not containing oscillatory coefficients in its principal part. The study of the asymptotic behavior of the solutions of the transformed system is a simpler problem. As an example, the following equation is considered: $$ \frac{d^2x}{dt^2}+\left(1+\frac{\sin\lambda t} {t^\alpha}\right)x=0, $$ where $\lambda$ and $\alpha$,~ $0<\alpha\le 1$, are real numbers.