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

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 $t\to \mathrm{\infty }$ 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: $$\frac{d^{2}x}{d{t}^2}+(1+\frac{\sin \lambda t}{t^{\alpha }})x=0,$$ where $\lambda$ and $\alpha$,~ $0<\alpha \le 1$, are real numbers.