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Energy and Contour Plots for Qualitative Analysis of Nonlinear Differential Equations
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| Organization: | Technikon Pretoria and the University of Southern Mississippi |
| Organization: | Tshwane University of Technology, Arcadia Campus, South Africa |
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| Mathematics and Computer Education |
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Currently there is a "reform" taking place in the teaching of beginning differential equations. Rather than focusing upon various types of equations and the techniques involved in obtaining exact solutions, more emphasis is being placed graphical analysis and modeling, and consequently, on systems of first order equations. Modern computer hardware and sophisticated computer algebra software have changed the problems which can be discussed in these beginning courses. Almost every beginning differential equations text discusses the classical problem of the motion of a mass attached to a vibrating spring. Understanding this simple model is justified as it can be viewed as the first step in the investigation of more complex vibrating systems such as the Duffing oscillator. and, of course, the principles involved are common to many applied problems. However, these texts generally only consider the linear case. In this article, in keeping with the reform, we consider the nonlinear case which provides an intuitive and natural motivating example to introduce the beginning student to phase portraits and the qualitative study of solutions to the model. Topics commonly taught in a second semester ordinary differential equations course. Closed form solutions to nonlinear equations can be difficult to compute and numerical solutions can be inaccurate, even using the classical Runge-Kutta methods. Accordingly, we discuss an energy approach which permits the global structure of the phase plane to be seen. This approach is well known to physicists but generally appears parenthetically in most beginning texts. The energy approach yields trajectories in the phase plane as constant energy level curves. Using the contour plot capabilities of readily available computer algebra systems such as Maple or Mathematica, we can display trajectories, detect oscillatory solutions, and introduce the notation of separatrix. This discussion is elementary and we feel is appropriate, for by doing so we can discuss even wider classes of nonlinear equations than is usually encountered in a beginning course.
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