SECOND APPROXIMATION OF THIRD-ORDER NONLINEAR SYSTEMS WITH SLOWLY VARYING COEFFICIENTS

摘要

To obtain the second approximate solution of a third order weakly nonlinear ordinary differential equation with slowly varying coefficients modelling a damped oscillatory process is considered based on the extension of Bogoliubov's asymptotic method. Asymptotic method is a strong candidate for approximate solutions of nonlinear differential equations. The second or higher order approximate solution is suitable to get better result than the first approximate solution when the reduced frequency is many times larger than the small parameter. On the other hand, second or higher order approximate solution diverges faster than the lower order approximate solution when the reduced frequency becomes small. The asymptotic solutions for different initial conditions show a good coincidence with those obtained by the numerical method. Comparison is made between the solutions obtained by the asymptotic method and those obtained by the numerical method in figures. The method is illustrated by an example.