Asymptotic solution for a class of nonlinear singularly perturbed equation

A class of weakly nonlinear ordinary equation for second order is considered.By using the Lindatedt-Poincare method,introducing the transformation of parameter and eliminating the secular terms in the formal solution,the first order uniformly valid asymptotic expansion is obtained.And by using the multi-scales method,introducing many scale variables and in light of the principle for without appearing the secular terms,the asymptotic expansion of the solution is constructed too.Comparison between the two expansions leads to the same results.