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Ebook A first course in differential (10th edition): Part 2

(BQ) Part 2 book "A first course in differential" has contents: Modeling with higher order differential equations, series solutions of linear equations, the laplace transform, systems of linear first order differential equations, numerical solutions of ordinary differential equations. | 5 Modeling with Higher-Order Differential Equations 5.1 Linear Models: Initial-Value Problems 5.1.1 Spring/Mass Systems: Free Undamped Motion 5.1.2 Spring/Mass Systems: Free Damped Motion 5.1.3 Spring/Mass Systems: Driven Motion 5.1.4 Series Circuit Analogue 5.2 Linear Models: Boundary-Value Problems 5.3 Nonlinear Models Chapter 5 in Review We have seen that a single differential equation can serve as a mathematical model for diverse physical systems. For this reason we examine just one application, the motion of a mass attached to a spring, in great detail in Section 5.1. Except for terminology and physical interpretations of the four terms in the linear differential equation a dy d 2y ϩ b ϩ cy ϭ g(t), dt 2 dt the mathematics of, say, an electrical series circuit is identical to that of a vibrating spring/mass system. Forms of this linear second-order equation appear in the analysis of problems in many different areas of science and engineering. In Section 5.1 we deal exclusively with initial-value problems, whereas in Section 5.2 we examine applications described by boundary-value problems. In Section 5.2 we also see how some boundary-value problems lead to the important concepts of eigenvalues and eigenfunctions. Section 5.3 begins with a discussion on the differences between linear and nonlinear springs; we then show how the simple pendulum and a suspended wire lead to nonlinear models. 192 Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 5.1 LINEAR MODELS: INITIAL-VALUE PROBLEMS ● 193 LINEAR MODELS: INITIAL-VALUE PROBLEMS 5.1 REVIEW .