Solutions Manual to accompany Ordinary Differential Equations E-Book

Michael D. Greenberg

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2014

Beschreibung

Features a balance between theory, proofs, and examples and provides applications across diverse fields of study Ordinary Differential Equations presents a thorough discussion of first-order differential equations and progresses to equations of higher order.

Leseprobe

Contents

CHAPTER 1 First-Order Differential Equations

Section 1.1

Section 1.2

Section 1.3

Section 1.4

Section 1.5

Section 1.6

Section 1.7

Section 1.8

Section 1.9

CHAPTER 2 Higher-Order Linear Equations

Section 2.2

Section 2.3

Section 2.4

Section 2.5

Section 2.6

Section 2.7

Section 2.8

Section 2.9

Section 2.10

CHAPTER 3 Applications of Higher-Order Equations

Section 3.2

Section 3.3

Section 3.4

Section 3.5

Section 3.6

CHAPTER 4 Systems of Linear Differential Equations

Section 4.1

Section 4.2

Section 4.3

Section 4.4

Section 4.5

Section 4.6

Section 4.7

Section 4.8

Section 4.9

Section 4.10

CHAPTER 5 Laplace Transform

Section 5.2

Section 5.3

Section 5.4

Section 5.5

Section 5.6

CHAPTER 6 Series Solutions

Section 6.2

Section 6.3

Section 6.4

Section 6.5

CHAPTER 7 Systems of Nonlinear Differential Equations

Section 7.2

Section 7.3

Section 7.4

Section 7.5

Section 7.6

CAS TUTORIAL

1. MAPLE

2. MATLAB

3. MATHEMATICA

4. MAPLE for this Text, by Chapter

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

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Library of Congress Cataloging-in-Publication Data:

Greenberg, Michael D.

Ordinary differential equations / Michael D. Greenberg.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-118-39899-9

1. Differential equations—Textbooks. 2. Differential equations, Partial—Textbooks. I. Title.

QA372.G725 2012

515'.352—dc23       2011042287

CHAPTER 1

First-Order Differential Equations

Section 1.1

This first section is simply to introduce you to differential equations: what they look like, some ideas as to how they arise in applications, and some important definitions. We see that the complete problem might be not just the differential equation, but also one or more "initial conditions." If such conditions are prescribed, the problem is called an initial value problem, or IVP. For instance, (6) [that is, equation (6) in the text] is an IVP because in addition to the DE (differential equation) there are two initial conditions, given by (6b), so that the solution of the IVP must satisfy not only the DE (6a), but also those two initial conditions.

Chapter 1 is about first-order equations; that is, equations in which the highest derivative is of first order. In that case, hence all through Chapter 1, there will be only one initial condition. In later chapters we will find that the "appropriate" number of initial conditions for a DE is the same as the order of the equation. For instance, (6a) is of second order and, sure enough, there are two initial conditions in (6b).

The distinction between linear and nonlinear differential equations will be of great importance, so it is necessary to be able to tell if a given equation is linear or nonlinear. Later, we will find that the key is whether or not a certain linearity property is satisfied, but for now it will suffice not to know about that property, but simply to say that an nth-order equation is linear if it is in, or can be put into, the form (14). What is the form of (14)? First, put all occurrences of the unknown, that is, the dependent variable such as y in (14), on the LHS (left-hand side of the equation); anything else goes on the right. If the LHS is a linear combination of y, y',…, y(n), then the DE is linear. Actually, the "constants" that multiply y, y',…, y(n) in (14) are permitted to be functions of x; the point is that they don't depend on y or its derivatives.

EXAMPLES

Example 1. (Definitions) State the order of the

whether it is linear or nonlinear, homogeneous or nonhomogeneous, and determine whether or not the given functions are solutions, that is, whether or not they "satisfy" the DE: