On the asymptotic behavior of a second-order general differential equation

Studying ordinary or partial differential equations or integrals using traditional asymptotic analysis, unfortunately, fails to extract the exponentially small terms and fails to derive some of their asymptotic features. In this paper, we discuss how to characterize an asymptotic behavior of a singular linear differential equation by the methods in exponential asymptotics. This paper is particularly concerned with the formulation of the series representation of a general second-order differential equation. It provides a detailed explanation of the asymptotic behavior of the differential equation and its relation between the prefactor functions and the singulant of the expansion of the equation. Through having this relationship, one can directly uncover and investigate invisible exponentially small terms and Stokes phenomenon without doing more work for the particular type of equations. Here, we demonstrate how these terms and form of the expansion can be computed straight-away, and, in a manner, this can be extended to the derivation of the potential Stokes and anti-Stokes lines.

Automated summary

This paper discusses how to characterize the asymptotic behavior of a singular linear differential equation using exponential asymptotics. It focuses on the series representation of a general second-order differential equation and the relationship between the prefactor functions and the singulant of the expansion. By establishing this relationship, one can directly uncover and investigate invisible exponentially small terms and Stokes phenomenon without additional work for specific types of equations.