A note on the method of multiple scales

A modification of the method of multiple scales makes use of the expansion of parameters of the system in order to remove undesirable characteristics of the solution obtained by the usual multiple-scale method. When applied to the damped harmonic oscillator, the modification leads to the exact solution. For the Duffing equation it leads to an approximation which can be reduced to the solution reported by Nayfeh [1], However, the solution derived here appears to be more accurate and the frequency takes on a form without nonuniformities.