Effect of higher approximation of Krylov-Bogoliubov-Mitropolskii's solution and matched asymptotic solution of a differential system with slowly varying coefficients and damping near to a turning point

Abstract

Second approximate solution of a second order differential equation with slowly varying coefficients and damping is obtained by Krylov-Bogoliubov-Mitropolskii method. The method is illustrated by an example. The second or higher order approximate solution is able to give better results than first approximate solution when the reduced frequency is many times larger than the small parameter. On the contrary, higher order solution diverges faster than the lower order solution when the reduced frequency becomes small (i .e., near to a turning point). In these situations matched asymptotic solution is important. An example is made to illustrate the matter.