Asymptotic analysis of some classes of ordinary differential equations with large high-frequency terms

A complete asymptotic expansion is constructed for solutions of the Cauchy problem for nth order linear ordinary differential equations with rapidly oscillating coefficients, some of which may be proportional to ω n/2, where ω is oscillation frequency. A similar problem is solved for a class of systems of n linear first-order ordinary differential equations with coefficients of the same type. Attention is also given to some classes of first-order nonlinear equations with rapidly oscillating terms proportional to powers ω d . For such equations with d ∈ (1/2, 1], conditions are found that allow for the construction (and strict justification) of the leading asymptotic term and, in some cases, a complete asymptotic expansion of the solution of the Cauchy problem.