Spring pendulum: Parametric excitation vs an external force

The method of multiple scales is applied to obtain an approximate solution to the nonlinear dynamic equations describing a spring pendulum with the vertical oscillations of the suspension point up to and including the fourth order corrections. The solutions of these equations, where an external force enters the equations multiplicatively, are compared with the solution considered earlier, for the behavior of a spring pendulum subject to an external force, which enters the appropriate equations additively. It turns out that in lower orders in small parameter, the two solutions coincide for the case where the external force and viscous damping force are equally small, but they differ when the damping is much smaller than the external force.