Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 54, pp. 1-6.

Title: Liapunov exponents for higher-order linear
       differential equations whose characteristic equations have
       variable real roots
 
Author: Michael I. Gil' (Ben Gurion Univ., Israel)

Abstract: 
 We consider  the  linear  differential equation
 $$
 \sum_{k=0}^n  a_k(t)x^{(n-k)}(t)=0\quad t\geq 0, \; n\geq 2,
 $$
 where $a_0(t)\equiv 1$, $a_k(t)$ are continuous bounded functions.
 Assuming that all the roots of the polynomial
 $z^n+a_1(t)z^{n-1}+ \dots +a_n(t)$ are real and satisfy the inequality
 $r_k(t)<\gamma$ for $t\geq 0$ and $k=1, \dots, n$,
 we prove that the solutions of the above equation satisfy
 $|x(t)|\leq  \mathop{\rm const} e^{\gamma t}$ for  $t\geq 0$.
 
Submitted December 27, 2007. Published April 15, 2008.
Math Subject Classifications: 34A30, 34D20.
Key Words: Linear differential equations; Liapunov exponents;
           exponential stability.