Exact solution of the quadratic mixed-parity Helmholtz–Duffing oscillator

In this paper, the exact solution of the quadratic mixed-parity Helmholtz–Duffing oscillator is derived by using Jacobi elliptic functions. It is also shown that the exact period of oscillation is given as a function of the complete elliptic integral of the first kind. At the end of the paper, we examine the stability of the system and determine the regions for which periodic and unbounded motions take place.