A unified Krylov–Bogoliubov–Mitropolskii method for solving nth order nonlinear systems

A unified theory is presented for obtaining the transient response of nth order nonlinear systems with small nonlinearities by Krylov–Bogoliubov–Mitropolskii method. The method is a generalization of Bogoliubovʹs asymptotic method and covers all three cases when the roots of the corresponding linear equation are real, complex conjugate, or purely imaginary. It is shown that by suitable substitution for the roots in the general result that the solution corresponding to each of the three cases can be obtained. The method is illustrated by examples.