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The improved formulations of the Lindstedt Poincare perturbation method, theharmonic balance method and its modifications using the rational functions, theenergy balance method and the two terms Fourier series expansion have beenemployed to find approximate solution to the equation of motion for the cubicquartic potential AHO and its special cases (the cubic, the quartic, and thepurely cubic quartic potential AHOs), to assess the relative merits anddemerits of these techniques. Two new rational functions as approximatesolution for the harmonic balance method have been proposed and the rationalfunctions have been used as trial functions in the energy balance method forthe first time. The time periods T+ (for q(t) > 0), T- (for q(t) < 0) and thetotal period T so determined have been compared with the corresponding exactvalues for an extensive range of the AHO parameter values. The results havebeen thoroughly analysed in terms of their percentage difference from therelevant exact time period. It is concluded that the methods based on theLindstedt Poincare perturbation approach do not lead to reliable results for T+and T-, which makes their applicability for the asymmetric potential AHOsdoubtful. The harmonic balance technique together with its different variantsis the most successful, while the Fourier series approach also yieldscommendably accurate results. The outcome of the third-order energy balancemethod is reasonably good though the usage of the rational functions for thishas not been encouraging.

A comparative study of some analytic approximate methods for the anharmonic oscillator described by cubic quartic potential