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

Abstract: The Ermakov-Pinney Equation, $$\ddot{x}+\omega² x=\frac{h<sup>2}{x<sup>3},$$ has a varied provenance which we briefly delineate. We introduce time-dependent functions in place of the $\omega<sup>2$ and $h<sup>2$. 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.