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Coupled spring equations for modelling the motion of two springs with weights attached, hung in series from the ceiling are described. For the linear model using Hooke's Law, the motion of each weight is described by a fourthorder linear differential equation. A nonlinear model is also described and damping and external forcing are considered. The model has many features that permit the meaningful introduction of many concepts including: accuracy of numerical algorithms, dependence on parameters and initial conditions, phase and synchronization, periodicity, beats, linear and nonlinear resonance, limit cycles, harmonic and subharmonic solutions. These solutions produce a wide variety of interesting motions and the model is suitable for study as a computer laboratory project in a beginning course on differential equations or as an individual or a small-group undergraduate research project. IntroductionThe classical syllabus for beginning differential equations is rapidly changing from emphasizing solution techniques for a variety of types of differential equations to emphasizing systems and more qualitative aspects of the theory of ordinary differential equations. In particular, there is an emphasis on nonlinear equations due largely to the wide availability of high powered numerical algorithms and almost effortless graphics capabilities that come with computer algebra systems such as Mathematica and Maple.In this article, we investigate an old problem that appears now to be relegated to the exercises in texts, if it appears at all (see for example [1, pp. 220-221]). This is the problem of two springs and two weights attached in series, hanging from the ceiling. Under the assumption that the restoring forces behave according to Hooke's Law, this two degrees of freedom problem is modelled by a pair of coupled, second-order, linear differential equations. By differentiating and substituting one equation into the other, the motion of each weight can be shown to be determined by a linear, fourth-order differential equation. We like this example for this very reason, most models in elementary texts are only of second order. Moreover, the questions about phase now have a very nice physical interpretation;

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