Approximate Solution of Slowly Varying Amplitude and Phase of Nonlinear Differential Systems Duffing Equation with External Forces

Objectives: To present a new approximate solution to the field equations obtained nonlinear differential Systems by using duffing equation with external forces which yields constant varying parameter. Methods: To determine an approximate solution of  nonlinear ordinary differential system with damping and slowly varying coefficients  a damped oscillatory process is considered based on the  Krylov-Bogoliubov (KB) method.  This paper develops a reliable algorithm based on the general Struble’s technique and extended KBM method for solving nonlinear differential systems. Moreover, we find a solution based on the KBM and general Struble’s technique of nonlinear autonomous systems with vary slowly with time, which is more powerful than the existing perturbation method. The method is illustrated by an example. Findings: Our aimed  to this paper of approximate Solution of Nonlinear Differential Systems duffing equation with external forces. However, in some cases it’s feasible to alternate nonlinear differential equations with an associated linear equation closely enough to give helpful results. Novelty: By applying this method in an example, we find a solution by considering initial conditions. The figures explained the nonlinear phenomena for the critical situation. Finally, results are discussed, primarily to enrich the physical prospects, and shown graphically by utilizing MATHEMATICA and MATLAB software.