On the asymptotic integration of systems of linear differential equations with oscillatory decreasing coefficients

V. Sh. Burd ; V. A. Karakulin

Matematičeskie zametki, Tome 64 (1998) no. 5, pp. 658-666

Résumé

A system of linear differential equations with oscillatory decreasing coefficients is considered. The coefficients have the form $t^{-\alpha}a(t)$, $\alpha>0$ where $a(t)$ is a 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: $$ \frac{d^2x}{dt^2}+\biggl(1+\frac{\sin\lambda t}{t^\alpha}\biggr)x=0, $$ where $\lambda$ and $\alpha$, $0\alpha\le1$, are real numbers.