An analytical method for Mathieu oscillator based on method of variation of parameter

A simple, but very accurate analytical method for forced Mathieu oscillator is proposed, the idea of which is based on the method of variation of parameter. Assuming that the time-varying parameter in Mathieu oscillator is constant, one could easily obtain its accurately analytical solution. Then the approximately analytical solution for Mathieu oscillator could be established after substituting periodical time-varying parameter for the constant one in the obtained accurate analytical solution. In order to certify the correctness and precision of the proposed analytical method, the first-order and ninth-order approximation solutions by harmonic balance method (HBM) are also presented. The comparisons between the results by the proposed method with those by the numerical simulation and HBM verify that the results by the proposed analytical method agree very well with those by the numerical simulation. Moreover, the precision of the proposed new analytical method is not only higher than the approximation solution by first-order HBM, but also better than the approximation solution by the ninth-order HBM in large ranges of system parameters.