Modern Introduction to Differential Equations (eBook)

Henry J. Ricardo (Autor)

eBook Download: EPUB

2009 | 2. Auflage
536 Seiten
Elsevier Science (Verlag)
978-0-08-088603-9 (ISBN)

Infos

A Modern Introduction to Differential Equations, Second Edition, provides an introduction to the basic concepts of differential equations. The book begins by introducing the basic concepts of differential equations, focusing on the analytical, graphical, and numerical aspects of first-order equations, including slope fields and phase lines. The discussions then cover methods of solving second-order homogeneous and nonhomogeneous linear equations with constant coefficients; systems of linear differential equations; the Laplace transform and its applications to the solution of differential equations and systems of differential equations; and systems of nonlinear equations.
Each chapter concludes with a summary of the important concepts in the chapter. Figures and tables are provided within sections to help students visualize or summarize concepts. The book also includes examples and exercises drawn from biology, chemistry, and economics, as well as from traditional pure mathematics, physics, and engineering. This book is designed for undergraduate students majoring in mathematics, the natural sciences, and engineering. However, students in economics, business, and the social sciences with the necessary background will also find the text useful.
- student friendly readability- assessible to the average student
- early introduction of qualitative and numerical methods
- large number of exercises taken from biology, chemistry, economics, physics and engineering
- Exercises are labeled depending on difficulty/sophistication
- Full ancillary package including; Instructors guide, student solutions manual and course management system
- end of chapter summaries
- group projects

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A Modern Introduction to Differential Equations, Second Edition, provides an introduction to the basic concepts of differential equations. The book begins by introducing the basic concepts of differential equations, focusing on the analytical, graphical, and numerical aspects of first-order equations, including slope fields and phase lines. The discussions then cover methods of solving second-order homogeneous and nonhomogeneous linear equations with constant coefficients; systems of linear differential equations; the Laplace transform and its applications to the solution of differential equations and systems of differential equations; and systems of nonlinear equations. Each chapter concludes with a summary of the important concepts in the chapter. Figures and tables are provided within sections to help students visualize or summarize concepts. The book also includes examples and exercises drawn from biology, chemistry, and economics, as well as from traditional pure mathematics, physics, and engineering. This book is designed for undergraduate students majoring in mathematics, the natural sciences, and engineering. However, students in economics, business, and the social sciences with the necessary background will also find the text useful. - Student friendly readability- assessible to the average student- Early introduction of qualitative and numerical methods- Large number of exercises taken from biology, chemistry, economics, physics and engineering- Exercises are labeled depending on difficulty/sophistication- End of chapter summaries- Group projects

Inhalt

Front Cover 1
A Modern Introduction to Differential Equations 4
Copyright Page 5
Dedication 6
Table of Contents 8
Preface 12
Acknowledgments 16
Chapter 1. Introduction to Differential Equations 18
Introduction 18
1.1 Basic Terminology 19
1.2 Solutions of Differential Equations 25
1.3 Initial-Value Problems and Boundary-Value Problems 30
Summary 42
Project 1-1 43
Chapter 2. First-Order Differential Equations 44
Introduction 44
2.1 Separable Equations 45
2.2 Linear Equations 55
2.3 Compartment Problems 65
2.4 Slope Fields 73
2.5 Phase Lines and Phase Portraits 85
2.6 Equilibrium Points: Sinks, Sources, and Nodes 91
*2.7 Bifurcations 98
*2.8 Existence and Uniqueness of Solutions 105
Summary 112
Project 2-1 113
Project 2-2 114
Chapter 3. The Numerical Approximation of Solutions 116
Introduction 116
3.1 Euler’S Method 116
3.2 The Improved Euler Method 135
3.3 More Sophisticated Numerical Methods: Runge-Kutta and Others 139
Summary 144
Project 3-1 146
Chapter 4. Second- and Higher-Order Equations 148
Introduction 148
4.1 Homogeneous Second-Order Linear Equations with Constant Coefficients 148
4.2 Nonhomogeneous Second-Order Linear Equations with Constant Coefficients 158
4.3 The Method of Undetermined Coefficients 161
4.4 Variation of Parameters 169
4.5 Higher-Order Linear Equations with Constant Coefficients 176
4.6 Higher-Order Equations and Their Equivalent Systems 181
4.7 The Qualitative Analysis of Autonomous Systems 190
4.8 Spring-Mass Problems 205
*4.9 Existence and Uniqueness 220
4.10 Numerical Solutions 224
Summary 233
Project 4-1 236
Chapter 5. Systems of Linear Differential Equations 238
Introduction 238
5.1 Systems and Matrices 238
5.2 Two-Dimensional Systems of First-Order Linear Equations 244
5.3 The Stability of Homogeneous Linear Systems: Unequal Real Eigenvalues 259
5.4 The Stability of Homogeneous Linear Systems: Equal Real Eigenvalues 271
5.5 The Stability of Homogeneous Linear Systems: Complex Eigenvalues 278
5.6 Nonhomogeneous Systems 287
5.7 Generalizations :The n × n Case (n = 3) 298
Summary 316
Project 5-1 318
Project 5-2 319
Chapter 6. The Laplace Transform 320
Introduction 320
6.1 The Laplace Transform of Some Important Functions 321
6.2 The Inverse Transform and The Convolution 328
6.3 Transforms of Discontinuous Functions 340
6.4 Transforms of Impulse Functions—The Dirac Delta Function 348
6.5 Transforms of Systems of Linear Differential Equations 353
6.6 A Qualitative Analysis Via The Laplace Transform 358
Summary 366
Project 6-1 368
Chapter 7. Systems of Nonlinear Differential Equations 370
Introduction 370
7.1 Equilibria of Nonlinear Systems 370
7.2 Linear Approximation at Equilibrium Points 375
7.3 The Poincaré-Lyapunov Theorem 384
7.4 Two Important Examples of Nonlinear Equations and Systems 392
*7.5 Van Der Pol’S Equation and Limit Cycles 402
Summary 411
Project 7-1 413
Appendix A. Some Calculus Concepts and Results 416
A.1 Local Linearity: The Tangent Line Approximation 416
A.2 The Chain Rule 417
A.3 The Taylor Polynomial/Taylor Series 417
A.4 The Fundamental Theorem Of Calculus (FTC) 420
A.5 Partial Fractions 421
A.6 Improper Integrals 422
A.7 Functions of Several Variables/Partial Derivatives 425
A.8 The Tangent Plane: The Taylor Expansion of F(X,Y) 426
Appendix B. Vectors and Matrices 428
B.1 Vectors and Vector Algebra Polar Coordinates
B.2 Matrices and Basic Matrix Algebra 431
B.3 Linear Transformations and Matrix Multiplication 432
B.4 Eigenvalues and Eigenvectors 438
Appendix C. Complex Numbers 442
C.1 Complex Numbers: The Algebraic View 442
C.2 Complex Numbers: The Geometric View 444
C.3 The Quadratic Formula 445
C.4 Euler’S Formula 445
Appendix D. Series Solutions of Differential Equations 446
D.1 Power Series Solutions of First-Order Equations 446
D.2 Series Solutions of Second-Order Linear Equations: Ordinary Points 448
D.3 Regular Singular Points: The Method of Frobenius 451
D.4 The Point at Infinity 456
D.5 Some Additional Special Differential Equations 457
Answers/Hints to Odd-Numbered Exercises 460
Index 522

Textauszug

CHAPTER 1

Introduction to Differential Equations

INTRODUCTION

What do the following situations have in common?

 An arms race between nations

 Tracking of the rate at which HIV-positive patients come to exhibit AIDS

 The dynamics of supply and demand in an economy

 The interaction between two or more species of animals on an island

The answer is that each of these areas of investigation can be modeled with differential equations. This means that the essential features of these problems can be represented using one or several differential equations, and the solutions of the mathematical problems provide insights into the future behavior of the systems being studied.

This book deals with change, with flux, with flow, and, in particular, with the rate at which change takes place. Every living thing changes. The tides fluctuate over the course of a day. Countries increase and diminish their stockpiles of weapons. The price of oil rises and falls. The proper framework of this course is dynamics—the study of systems that evolve over time.

The origin of dynamics (originally an area of physics) and of differential equations lies in the earliest work by the English scientist and mathematician Sir Isaac Newton (1642–1727) and the German philosopher and mathematician Gottfried Wilhelm Leibniz (1646–1716) in developing the new science of calculus in the seventeenth century. Newton in particular was concerned with determining the laws governing motion, whether of an apple falling from a tree or of the planets moving in their orbits. He was concerned with rates of change. However, you mustn’t think that the subject of differential equations is all about physics. The same types of equations and the same kind of analysis of dynamical systems can be used to model and understand situations in biology, economics, military strategy, and chemistry, for example. Applications of this sort will be found throughout this book.

In the next section, we will introduce the language of differential equations and discuss some applications.

1.1 Basic Terminology

1.1.1 Ordinary and Partial Differential Equations

Ordinary Differential Equations

Definition 1.1.1

An ordinary differential equation (ODE) is an equation that involves an unknown function of a single variable, its independent variable, and one or more of its derivatives.

Example 1.1.1

An Ordinary Differential Equation

Here’s a typical elementary ODE, with some of its components indicated:

This equation describes an unknown function of t that is equal to three times its own derivative. Expressed another way, the differential equation describes a function whose rate of change is proportional to its size (value) at any given time, with constant of proportionality one-third.

The Leibniz notation for a derivative, , is helpful because the independent variable (the fundamental quantity whose change is causing other changes) appears in the denominator, the dependent variable in the numerator. The three equations

leave no doubt about the relationship between independent and dependent variables. But in an equation such as , we must infer that the unknown function w is really w(t), a function of the independent variable t.

In many dynamical applications, the independent variable is time, represented by t, and we may denote the function’s derivative using Newton’s dot notation,1 as in the equation . You should be able to recognize a differential equation no matter what letters are used for the independent and dependent variables and no matter what derivative notation is employed. The context will determine what the various letters mean, and it’s the form of the equation that should be recognized. For example, you should be able to see that the two ordinary differential equations

are the same—that is, they are describing the same mathematical or physical behavior. In Equation (A) the unknown function u depends on t, whereas in Equation (B) the function y is a function of the independent variable x, but both equations describe the same relationship that involves the unknown function, its derivatives, and the independent variable. Each equation is describing a function whose second derivative equals three times its first derivative minus seven times itself.

The Order of an Ordinary Differential Equation

One way to classify differential equations is by their order.

Definition 1.1.2

An ordinary differential equation is of order n, or is an nth-order equation, if the highest derivative of the unknown function in the equation is the nth derivative.

The equations

are all first-order differential equations because the highest derivative in each equation is the first derivative. The equations

and

are second-order equations, and is of order 5.

A General Form for an Ordinary Differential Equation

If y is the unknown function with a single independent variable x, and y(k) denotes the kth derivative of y, we can express an nth-order differential equation in a concise mathematical form as the relation

or often as

The next example shows what these forms look like in practice.

Example 1.1.2

General Form for a Second-Order ODE

If y is an unknown function of x, then the second-order ordinary differential equation can be written as or as

Note that F denotes a mathematical expression involving the independent variable x, the unknown function y, and the first and second derivatives of y.

Alternatively, in this last example we could use ordinary algebra to solve the original differential equation for its highest derivative and write the equation as

Partial Differential Equations

If we are dealing with functions of several variables and the derivatives involved are partial derivatives, then we have a partial differential equation (PDE). (See Section A.7 if you are not familiar with partial derivatives.) For example, the partial differential equation , which is called the wave equation, is of fundamental importance in many areas of physics and engineering. In this equation we are assuming that u = u(x, t), a function of the two variables x and t. However, in this text, when we use the term differential equation, we’ll mean an ordinary differential equation. Often we’ll just write equation, if the context makes it clear that an ordinary differential equation is intended.

Linear and Nonlinear Ordinary Differential Equations

Another important way to categorize differential equations is in terms of whether they are linear or nonlinear.

Definition 1.1.3

If y is a function of x, then the general form of a linear ordinary differential equation of order n is

(1.1.1)

What is important here is that each coefficient function ai, as well as f, depends on the independent variable x alone and doesn’t have the dependent variable y or any of its derivatives in it. In particular, Equation (1.1.1) involves no products or quotients of y and/or its derivatives.

Example 1.1.3

A Second-Order Linear Equation

The equation , where ω is a constant, is linear. We can see the form of this equation as follows:

The coefficients of the various derivatives of the unknown function x are functions (sometimes constant) of the independent variable t alone.

The next example shows that not all first-order equations are linear.

Example 1.1.4

A First-Order Nonlinear Equation (an HIV Infection Model)

The equation models the growth and death of T cells, an important component of the immune system. 2 Here T(t) is the number of T cells present at time t. If we rewrite the equation by removing parentheses, we get , and we see that there is a term involving the square of the unknown function. Therefore, the equation is not linear.

2E. K. Yeargers, R. W. Shonkwiler, and J. V. Herod, An Introduction to the Mathematics of Biology: With Computer Algebra Models (Boston: Birkhäuser, 1996): 341.

In general, there are more systematic ways to analyze linear equations than to analyze nonlinear equations, and we’ll see some of these methods in Chapters 2, 5, and 6. However, nonlinear equations are important and appear throughout this book. In particular, Chapter 7 is devoted to their analysis.

1.1.2 Systems of Ordinary Differential Equations

In earlier mathematics courses, you have had to deal with systems of algebraic equations, such as

Similarly, in working with differential equations, you may find yourself confronting...

Produkt-Details

• Erscheint lt. Verlag: 24.2.2009
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• ISBN-10: 0-08-088603-5 / 0080886035
• ISBN-13: 978-0-08-088603-9 / 9780080886039