Asymptotic expansions in a nonhomogeneous differential equation

Abstract

In recent papers [2]-[8] asymptotic behavior of the solutions of an ordinary differential equation has been discussed by the development of an asymptotic expansion for the solution that is valid to one term. In [5], [7], asymptotic expansions valid to two terms were obtained for certain solutions of a differential equation. These results which were valid to two terms were derived under slightly stronger conditions than those results which were valid for a single term. In this note, under an appropriate strengthening of the hypothesis which yields an asymptotic expansion valid to one term, we will show that an asymptotic expansion of solutions of a nonhomogeneous nonlinear differential equation may be developed which is valid to N < oo terms. The method used here differs from those in [1] to the extent that a singular integral equation is not used. We consider, as in [3], the nonhomogeneous differential equation