The connection between the characteristic roots and the corresponding solutions of a single linear differential equation with comparable coefficients

Given a linear differential equation of the form x((n)) + a(1)(t)x((n-1)) ...+ a(1)(t)x = 0 with variable coefficients defined on the positive semi-a xis for t >> 1. We denote its fundamental set of solutions (FSS) by {exp [i ntegral gamma(i) (t) dt]} (i = 1, 2,..., n). In this paper we look for the asymptotic connection (as t --> infinity) between the logarithmic derivativ es gamma(i)(t) of an FSS and of the roots of the characteristic equation y( n) + a(1)(t)y(n-1) +...+ a(n)(t) = 0. We mainly consider the case when the coefficients of the equation and the characteristic roots are comparable an d have the power order of growth for t --> infinity. We discuss the conditi ons when the functions gamma(i)(t) are equivalent to the corresponding root s lambda(i)(t) of the characteristic equation as t --> infinity.