The paper discusses an elementary spring model representing the motion of a magnet suspended from the ceiling at one end of a vertical spring which is held directly above a second magnet fixed on the floor. There are two cases depending upon the north-south pole orientation of the two magnets. The attraction or repelling force induced by the magnets follow an inverse quartic law and thus we are led to a nonlinear model suitable for discussion in a beginning differential equations course. Spring models are common fare in such courses, but usually only linear models with simple sinusoidal forcing are considered. The resultant model is autonomous and thus an energy approach permits a full phase portrait of the resultant motions in the phase plane. These phase portraits show interesting behaviour of the system, reinforcing one's natural physical intuition. The computer algebra system "Mathematica" is employed here, although almost any other system would suffice. Such a system permits almost effortless calculations and can generate the graphics needed to thoroughly investigate the model. (Contains 6 figures.)