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Ebook A first course in differential equations (9th edition): Part 2

(BQ) Part 2 book "A first course in differential equations" has contents: Modeling with higher order differential equations, modeling with higher order differential 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 equation ayЉ ϩ byЈ ϩ cy ϭ g(t), the mathematics of, say, an electrical series circuit is identical to that of vibrating spring/mass system. Forms of this linear second-order DE appear in the analysis of problems in many diverse 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. 181 182 CHAPTER 5 ● 5.1 MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS LINEAR MODELS: INITIAL-VALUE PROBLEMS REVIEW MATERIAL ● Sections 4.1, 4.3, and 4.4 ● Problems 29–36 in Exercises 4.3 ● Problems 27–36 in Exercises 4.4 INTRODUCTION In this section we are going to consider several linear dynamical systems in which each mathematical model is a second-order differential equation with constant coefficients along with initial conditions specified at a time that we shall take to be t ϭ 0: a dy d 2y ϩ b ϩ cy ϭ g(t), y(0) ϭ y0 , 2 dt dt yЈ(0) ϭ y1. Recall that the function g is the input, driving .