Analytic solution of the system of two coupled differential equations with the fifth-order non-linearity

In this paper an approximate method for solving a system of two coupled second-order non-linear differential equation with strong and weak non-linearity is developed. The strong non-linearities are of the third and of the fifth order. The suggested new method is partially based on the elliptic-Krylov–Bogolubov method and on the power series method developed for solving the weak non-linear differential equations. The one-frequency solution is considered for the case of certain relations between the equation parameters p and m, and initial conditions. As a special case the oscillatory motion of the physical system described with a fourth-order strong non-linear differential equation is considered. A wide analysis of the corresponding equation is done. The results of the suggested analytic procedure are compared with numerical ones. They show a good agreement.