On the Solution of the Unforced Damped Duffing Oscillator with No Linear Stiffness Term

Using a series of functional transformations we reduce the unforced,damped Duffing oscillator to equivalent equations of the Abel andEmden–Fowler classes. Taking into account the known exact analyticsolutions of these equivalent equations we prove that there does notexist an exact analytic solution of the damped, unforced Duffingoscillator in terms of known (tabulated) analytic functions. It followsthat a new class of solutions must be defined for solving this problem`exactly'. Finally, a new approximate solution of the intermediateintegral of the damped Duffing oscillator with weak damping isconstructed.