Damped oscillations modeled by an n-th order time dependent quasi-linear differential system

A general formula based on the extended (by Popov [4]) Krylov-Bogoliubov-Mitropolskii method [1], [2] is presented for obtaining asymptotic solution of an n-th order time dependent quasi-linear differential equation with damping. The method of determination of the solution is simple and easier than the classical formulae developed by several authors as well as the technique initiated by the original contributors [1], [2]. The general solution can be used arbitrarily for different values of n = 2, 3. The method can be used not only for periodic forcing terms, but also for some non-periodic (bounded) forces. All the solutions can be determined from a single trial solution. On the contrary, at least two trial solutions are needed to investigate time-dependent differential equations; one is for the resonance case and the other for the non-resonance case. The later solution is sometimes used in the case of non-periodic external forces. However, the resonance cases (including damped forced vibrations [7]) are mainly considered in this paper, since these are important in vibration problems.