Late-order terms of second order ODEs in terms of pre-factors

Abstract

Factorial over a power approach is one of the fundamental techniques for deriving the late-order terms in the asymptotic approximation of integrals and differential equations. To our knowledge, although many differential equations depending on small or large parameters are addressed thoroughly and intensively by this approach in the literature to date, no explicit formula of the general representation of singularly-perturbed second order inhomogeneous ODEs in the form of this paper has yet been discussed generally in terms of their pre-factors. In this paper, we obtain a leading order asymptotic formula of the general asymptotic expansions suitable for the particular type of ODE by its pre-factors.