KBM unified method for solving an nth order non-linear differential equation under some special conditions including the case of internal resonance

Abstract

The Krylov–Bogoliubov–Mitropolskii (KBM) unified method is used for obtaining the approximate solution of an nth order ( n ⩾ 4 ) ordinary differential equation with small non-linearities when a pair of eigen-values of the unperturbed equation is multiple (approximately or perfectly) of the other pair or pairs. The general solution can be used arbitrarily for over-damped, damped and undamped cases. In a damped or undamped case, one of the natural frequencies of the unperturbed equation may be a multiple of the other. Thus, the solution also covers the case of internal resonance which is an interesting and important part of non-linear oscillation. The determination of the solution is very simple and easier than the existing procedures developed by several authors (both in methods of averaging and multiple time scales) especially to tackle the case of internal resonance. The method is illustrated by an example of a fourth-order differential equation. The solution shows a good agreement with numerical solution in all of the three cases, e.g. over-damped, damped and undamped.