The concern of this work is the steady state periodic response having the same period as the excitation of strongly non-linear oscillators u d u mu e 1 u 2 u e 1 u u 2 e 2 u 3 P cos Ot, where m = 1, 0 or ร1, e 1 and e 1 are positive parameters which may be arbitrarily large. Single-mode and two-mode harmonic balance (HB) approximations, and second order perturbationmultiple time scales (MMS) with reconstitution version I and version II approximations to the steady state amplitude frequency response curves are compared, for the case m = 1 with each other, and with those obtained by numerically integrating the equation of motion. The transformation of time T = Ot and detuning in the square of forcing frequency are used in the MMS with reconstitution version I and version II. The objective here is to assess the accuracy of these approximate solutions in predicting the system response over some range of system parameters by examining their ability or failure in establishing the correct qualitative behavior of the actual (numerical) solution. The cases m = 0 and m = ร1, are studied for selected range of system parameter, using the single and two modes harmonic balance method and compared to those obtained numerically. It was shown that MMS version II, in addition to being appreciably simpler than MMS version I, leads to more accurate qualitative and quantitative results even when the non-linearity is not necessarily small.