Hidden symmetries and linearization of the modified Painlevé–Ince equation

The linearization and hidden symmetries of the modified Painlevé–Ince equation, y″+σyy′+βy3=0, where σ and β are constants, are presented. The linearization of this equation by a nonlocal transformation yields a damped (stable) or growing (unstable) harmonic oscillator equation for β≳0. Hidden symmetries are analyzed by transforming the modified Painlevé–Ince equation to a third-order ordinary differential equation (ODE) which, in general, is invariant under a three-parameter group by a Riccati transformation. A type I hidden symmetry is found of a second-order ODE found from the third-order ODE where a symmetry is lost in the reduction of order by the non-normal subgroup invariants. A type II hidden symmetry occurs in the third-order ODE because the symmetries of a second-order ODE, reduced from the third-order ODE by another set of normal subgroup invariants, are increased.