Analytic approximation to the harmonic oscillator equation with a sub-period time dependent parameter

Analytic solutions are presented for the time dependent harmonic oscillator equation as well as for the corresponding amplitude (Ermakov) and phase equations. The solutions are adequate approximations for an initially constant amplitude and a time dependent parameter that varies continuously in a time much shorter than the characteristic period. The amplitude and phase representation of the solution is shown not to be unique. The relationship between the phase independent amplitudes is derived from the orthogonal functions invariant.