Summary of Chapter 6 : Interplay Between First Order Systems and Second Order Equations

In this chapter we extend the techniques we have developed for first order differential equations to more than one first order equation; namely, to systems of first order equations. We show how the existence-uniqueness theorems for such systems are natural extensions of our previous existence-uniqueness theorems, and that our previous numerical methods are easily extended to obtain numerical solutions of systems of first order equations.

Systems of first order equations are important because of the many situations that can be modeled by them. They are also important because of their relationship to second order differential equations. This relationship is exploited to obtain existence-uniqueness theorems and numerical solutions of second order differential equations.

Second order equations are important in their own right. First, they are used to model spring-mass systems, simple pendulums, and simple electrical circuits, which leads to the important notion of overdamped, critically damped, and underdamped motion. Second, we can obtain an explicit solution of one special class—linear with constant coefficients.

The ability to move backward and forward between systems and second order equations, taking advantage of the power of each, is the focus of this chapter. For example, we show that for second order differential equations, with the introduction of a phase plane, many of our graphical techniques carry over from the previous chapters. We describe how a curve in the phase plane can be constructed from a numerical, graphical, or analytical point of view.

The rest of this book is based on a thorough understanding of this chapter.

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