Asymptotic integration of a second order ordinary differential equation

Abstract:

Equation (1) $(r(t)x’)’ + f(t)x = 0$ is regarded as a perturbation of (2) $(r(t)y’)’ + g(t)y = 0$, where the latter is nonoscillatory at infinity. It is shown that if a certain improper integral involving $f - g$ converges sufficiently rapidly (but perhaps conditionally), then (1) has a solution which behaves for large $t$ 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.