The Ermakov-Pinney Equation: its varied origins and the effects of the introduction of symmetry-breaking functions

2015/10/30 by Morris, Richard Michael, Leach, Peter Gavin Lawrence · 1 citation
#22E60 #34A05 #70B05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1510.08992

Abstract

The Ermakov-Pinney Equation, x+ω² x=(h²)/(x³), has a varied provenance which we briefly delineate. We introduce time-dependent functions in place of the ω² and h². The former has no effect upon the algebra of the Lie point symmetries of the equation. The latter destroys the sl(2,\Re) symmetry and a single symmetry persists only when there is a specific relationship between the two time-dependent functions introduced. We calculate the form of the corresponding autonomous equation for these cases.

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