Exact solutions of a generalized autonomous Duffing-type equation

Exact analytical solutions for a family of autonomous ordinary differential equations arising in connection with many important applied problems such as a cubic-quintic Duffing oscillator, Helmholtz-Duffing oscillator, and nonlinear Schrödinger equation is considered. We obtained several new classes of solutions for 8th and 10th degrees Duffing oscillatory problems. These new non-singular solutions are given in terms of Jacobi elliptic, Weierstrass, algebraic, exponential, and trigonometric functions and general functional forms of the coefficients of Duffing-type equations. We have provided a set of inequalities that helps in finding transformation to obtain solutions to other Duffing equations from a known set of solutions to a related equation in a systematic manner.