An invariant asymptotic formula for solutions of second-order linear ODE's

Abstract

An invariant matrix asymptotic formula for the approximation of solutions of second-order linear ordinary differential equations y″=φ(x)y is proposed and elaborated upon. This is utilized to develop scalar versions of asymptotic formulas for two linearly independent solutions and for their derivatives. This formula is shown to be valid in a half neighbourhood of a point x₀. The validity holds whether x₀ is an ordinary (regular) point for the ODE, whether x₀ is a singular regular point for the ODE, if some exceptional case is avoided, whether x₀ is a singular irregular point for the ODE, and whether or not x₀ is finite. The matrix version of our formulas is shown to be valid also at a turning point. The Liouville-Green approximation is extracted as a particular case of our formula. Examples are given. The formula has additional “globality” properties. Examples are given where the ODE is considered on an infinite interval (0, ∞) and its coefficient φ(x) is singular at x = 0 as well as at x = ∞. A uniformly valid approximation on the entire infinite interval is then provided.