An integral of motion for the damped cubic-quintic Duffing oscillator with variable coefficients

In this short communication, integrals of motion for the undamped and damped cubic-quintic Duffing oscillators with time-dependent coefficients are obtained for the first time in the literature under appropriate analytical conditions. The integrals of motion are obtained using Noether’s theorem, and the conditions for their existence are directly related to the well-known Milne–Pinney equation, which is associated to the Ermakov–Lewis systems. We perform here some numerical simulations to illustrate the validity of our analytical results.