Yes, using my method you would have to take a time step that would correspond to number of terms of the series expansion included in the solution. You could, however, use a far larger step than most of the numerical methods. If you could recognize the series expansion of the solution, you can find the corresponding closed form solution.
What makes you think that the new method is not general?
Well I quickly read through it and their proposed ansatz solution seemed to work just fine. However the algebra look fairly hardcore, I was summoning up the energy to verify it, when I scanned to the bottom of the paper and they plotted the analytic solution versus a numerical integration and another
perturbation approach. Just on one of the final oscillations before it decays away, the analytic solution deviates away and has a nonzero asymptotic solution.
I will read it a bit more carefully.I like the way if gives an explicit form for the decay of the frequency of the oscillations through the "omega" functions which go as the argument of the Jacobi elliptic functions (which is very much of interest to me). Need to rework the whole thing for the care of a constant force acting on it.