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

الأصل · EN

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,) 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.