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.