Unified Krylov–Bogoliubov–Mitropolskii method for solving nth order non-linear systems with slowly varying coefficients

The unified Krylov–Bogoliubov-Mitropolskii (KBM) method is extended for obtaining the transient response of an n-th order (n⩾2) non-linear system with slowly varying coefficients. The method is a generalization of KBM method and covers all the three cases when the eigenvalues of the unperturbed equation are real, complex conjugate, or purely imaginary. It is shown that by suitable substitution for the eigenvalues in the general result that the solution corresponding to each of the three cases can be obtained. The method is illustrated by examples.