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

Article type: Research Article

Authors: Gingold, H.^;

Affiliations: Department of Mathematics, West-Virginia University, Morgantown, WV 26506, U.S.A.

Note: [] This research was supported in part by Grant NAG-1-741 from the National Aeronautics and Space Administration.

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.

DOI: 10.3233/ASY-1988-1404

Journal: Asymptotic Analysis, vol. 1, no. 4, pp. 317-350, 1988