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 n (t) x = 0 with variable coefficients defined on the positive semi ‐axis for t ≫ 1. We denote its fundamental set of solutions (FSS) by {exp [∫ ri (t) dt] } (i = 1, 2,…,n). In this paper we look for the asymptotic connection (as t → ∞) between the logarithmic derivatives ri (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 and have the power order of growth for t → ∞. We discuss the conditions when the functions λ i i(t) are equivalent to the corresponding roots λ i i(t) of the characteristic equation as t → ∞.