Asymptotic expansions in certain second order nonhomogeneous differential equations

Abstract

In the nonoscillation theory of ordinary differential equations the asymptotic behaviour of the solutions has often been described by exhibiting asymptotic expansions for these solutions. A fundamental illustration of this technique may be found in Hille [6] wherein the linearly independent solutions of a second order homogeneous differential equation were described by single termed asymptotic expansions. For the dominant solution, this result was successively extended by Waltman in [8] for a second order equation and in [9] for an n th order equation. A further generalization of these results appears in [3] where a complete n th order nonhomogeneous nonlinear differential equation was considered; again, asymptotic representations were given to describe the behaviour of the solutions of the differential equation. Moore and Nehari [7], Wong [10], [11], and Hale and Onuchic [2], also use asymptotic representations in discussing the behaviour of the solutions of certain differential equations. All of the above results are essentially perturbation problems with the unperturbed linear differential equation having the form y ( n ) = h ( t ) for some n and h ( t ).