Asymptotic integration of a second order ordinary differential equation

Abstract

Equation (1) ( r ( t ) x ′ ) ′ + f ( t ) x = 0 (r(t)x’)’ + f(t)x = 0 is regarded as a perturbation of (2) ( r ( t ) y ′ ) ′ + g ( t ) y = 0 (r(t)y’)’ + g(t)y = 0 , where the latter is nonoscillatory at infinity. It is shown that if a certain improper integral involving f − g f - g converges sufficiently rapidly (but perhaps conditionally), then (1) has a solution which behaves for large t t like a principal solution of (2). The proof of this result is presented in such a way that it also yields as a by-product an improvement on a recent related result of Trench.