Elementary differential equations 11th edition boyce pdf

Differential Equations

and Boundary Value

Problems

B O Y C E | D I P R I M A | M E A D E

11th Edition

Elementary

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Elementary Differential Equations and Boundary Value Problems

Eleventh Edition

WILLIAM E. BOYCE

Edward P. Hamilton Professor Emeritus Department of Mathematical Sciences

Rensselaer Polytechnic Institute

RICHARD C. DIPRIMA

formerly Eliza Ricketts Foundation Professor Department of Mathematical Sciences

Rensselaer Polytechnic Institute

DOUGLAS B. MEADE

Department of Mathematics University of South Carolina - Columbia

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To Elsa, Betsy, and in loving memory of Maureen

To Siobhan, James, Richard Jr., Carolyn, Ann, Stuart,

Michael, Marybeth, and Bradley

And to the next generation:

Charles, Aidan, Stephanie, Veronica, and Deirdre

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The Authors

WILLIAM E. BOYCE received his B.A. degree in

Mathematics from Rhodes College, and his M.S. and Ph.D.

degrees in Mathematics from Carnegie-Mellon University.

He is a member of the American Mathematical Society,

the Mathematical Association of America, and the Society

for Industrial and Applied Mathematics. He is currently the

Edward P. Hamilton Distinguished Professor Emeritus of

Science Education (Department of Mathematical Sciences)

at Rensselaer. He is the author of numerous technical papers

in boundary value problems and random differential equations

and their applications. He is the author of several textbooks

including two differential equations texts, and is the coauthor

(with M.H. Holmes, J.G. Ecker, and W.L. Siegmann) of a text

on using Maple to explore Calculus. He is also coauthor (with

R.L. Borrelli and C.S. Coleman) of Differential Equations

Laboratory Workbook (Wiley 1992), which received the

EDUCOM Best Mathematics Curricular Innovation Award

in 1993. Professor Boyce was a member of the NSF-sponsored

CODEE (Consortium for Ordinary Differential Equations

Experiments) that led to the widely-acclaimed ODE Architect.

He has also been active in curriculum innovation and reform.

Among other things, he was the initiator of the “Computers

in Calculus” project at Rensselaer, partially supported by the

NSF. In 1991 he received the William H. Wiley Distinguished

Faculty Award given by Rensselaer.

RICHARD C. DIPRIMA (deceased) received his B.S., M.S.,

and Ph.D. degrees in Mathematics from Carnegie-Mellon

University. He joined the faculty of Rensselaer Polytechnic

Institute after holding research positions at MIT, Harvard,

and Hughes Aircraft. He held the Eliza Ricketts Foundation

Professorship of Mathematics at Rensselaer, was a fellow of

the American Society of Mechanical Engineers, the American

Academy of Mechanics, and the American Physical Society.

He was also a member of the American Mathematical Society,

the Mathematical Association of America, and the Society

for Industrial and Applied Mathematics. He served as the

Chairman of the Department of Mathematical Sciences at

Rensselaer, as President of the Society for Industrial and

Applied Mathematics, and as Chairman of the Executive

Committee of the Applied Mechanics Division of ASME.

In 1980, he was the recipient of the William H. Wiley

Distinguished Faculty Award given byRensselaer. He received

Fulbright fellowships in 1964--65 and 1983 and a Guggenheim

fellowship in 1982--83. He was the author of numerous

technical papers in hydrodynamic stability and lubrication

theory and two texts on differential equations and boundary

value problems. Professor DiPrima died on September 10,

1984.

DOUGLAS B. MEADE received B.S. degrees in

Mathematics and Computer Science from Bowling Green

State University, an M.S. in Applied Mathematics from

Carnegie Mellon University, and a Ph.D. in mathematics

from Carnegie Mellon University. After a two-year stint

at Purdue University, he joined the mathematics faculty at

the University of South Carolina, where he is currently an

Associate Professor of mathematics and the Associate Dean for

Instruction, Curriculum, and Assessment in the College of Arts

and Sciences. He is a member of the American Mathematical

Society, Mathematics Association of America, and Society

for Industrial and Applied Mathematics; in 2016 he was

named an ICTCM Fellow at the International Conference on

Technology in Collegiate Mathematics (ICTCM). His primary

research interests are in the numerical solution of partial

differential equations arising from wave propagation problems

in unbounded domains and from population models for

infectious diseases. He is also well-known for his educational

uses of computer algebra systems, particularly Maple.

These include Getting Started with Maple (with M. May,

C-K. Cheung, and G. E. Keough, Wiley, 2009, ISBN 978-0-

470-45554-8), Engineer’s Toolkit: Maple for Engineers (with

E. Bourkoff, Addison-Wesley, 1998, ISBN 0-8053-6445-5),

and numerous Maple supplements for numerous calculus,

linear algebra, and differential equations textbooks - including

previous editions of this book. He was a member of the

MathDL New Collections Working Group for Single Variable

Calculus, and chaired the Working Groups for Differential

Equations and Linear Algebra. The NSF is partially supporting

his work, together with Prof. Philip Yasskin (Texas A&M), on

the Maplets for Calculus project.

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Preface

As we have prepared an updated edition our first priorities

are to preserve, and to enhance, the qualities that have made

previous editions so successful. In particular, we adopt the

viewpoint of an applied mathematician with diverse interests

in differential equations, ranging from quite theoretical to

intensely practical--and usually a combination of both. Three

pillars of our presentation of the material are methods

of solution, analysis of solutions, and approximations of

solutions. Regardless of the specific viewpoint adopted, we

have sought to ensure the exposition is simultaneously correct

and complete, but not needlessly abstract.

The intended audience is undergraduate STEM students

whose degree program includes an introductory course in

differential equations during the first two years. The essential

prerequisite is a working knowledge of calculus, typically a

two- or three-semester course sequence or an equivalent.While

a basic familiarity with matrices is helpful, Sections 7.2 and 7.3

provide an overview of the essential linear algebra ideas needed

for the parts of the book that deal with systems of differential

equations (the remainder of Chapter 7, Section 8.5, and

Chapter 9).

A strength of this book is its appropriateness in a

wide variety of instructional settings. In particular, it allows

instructors flexibility in the selection of and the ordering of

topics and in the use of technology. The essential core material

is Chapter 1, Sections 2.1 through 2.5, and Sections 3.1 through

3.5. After completing these sections, the selection of additional

topics, and the order and depth of coverage are generally at

the discretion of the instructor. Chapters 4 through 11 are

essentially independent of each other, except that Chapter 7

should precede Chapter 9, and Chapter 10 should precede

Chapter 11.

A particularly appealing aspect of differential equations

is that even the simplest differential equations have a direct

correspondence to realistic physical phenomena: exponential

growth and decay, spring-mass systems, electrical circuits,

competitive species, and wave propagation. More complex

natural processes can often be understood by combining and

building upon simpler and more basic models. A thorough

knowledge of these basic models, the differential equations

that describe them, and their solutions--either explicit solutions

or qualitative properties of the solution--is the first and

indispensable step toward analyzing the solutions of more

complex and realistic problems. The modeling process is

detailed in Chapter 1 and Section 2.3. Careful constructions

of models appear also in Sections 2.5, 3.7, 9.4, 10.5, and

10.7 (and the appendices to Chap er 10). Various problem sets

throughout the book include problems that involve modeling

to formulate an appropriate differential equation, and then

to solve it or to determine some qualitative properties of its

solution. The primary purposes of these applied problems are

to provide students with hands-on experience in the derivation

of differential equations, and to convince them that differential

equations arise naturally in a wide variety of real-world

applications.

Another important concept emphasized repeatedly

throughout the book is the transportability of mathematical

knowledge. While a specific solution method applies to only a

particular class of differential equations, it can be used in any

application in which that particular type of differential equation

arises. Once this point is made in a convincing manner, we

believe that it is unnecessary to provide specific applications of

every method of solution or type of equation that we consider.

This decision helps to keep this book to a reasonable size, and

allows us to keep the primary emphasis on the development

of more solution methods for additional types of differential

equations.

From a student’s point of view, the problems that are

assigned as homework and that appear on examinations define

the course. We believe that the most outstanding feature of

this book is the number, and above all the variety and range,

of the problems that it contains. Many problems are entirely

straightforward, but many others are more challenging, and

some are fairly open-ended and can even serve as the basis

for independent student projects. The observant reader will

notice that there are fewer problems in this edition than in

previous editions; many of these problems remain available

to instructors via the WileyPlus course. The remaining 1600

problems are still far more problems than any instructor can

use in any given course, and this provides instructors with a

multitude of choices in tailoring their course to meet their own

goals and the needs of their students. The answers to almost all

of these problems can be found in the pages at the back of the

book; full solutions are in either the Student’s Solution Manual

or the Instructor’s Solution Manual.

While we make numerous references to the use of

technology, we do so without limiting instructor freedom to

use as much, or as little, technology as they desire. Appropriate

technologies include advanced graphing calculators (TI

Nspire), a spreadsheet (Excel), web-based resources (applets),

computer algebra systems, (Maple, Mathematica, Sage),

scientific computation systems (MATLAB), or traditional

programming (FORTRAN, Javascript, Python). Problems

marked with a G are ones we believe are best approached with

a graphical tool; those marked with a N are best solved with the

use of a numerical tool. Instructors should consider setting their

own policies, consistent with their interests and intents about

student use of technology when completing assigned problems.

Many problems in this book are best solved through

a combination of analytic, graphic, and numeric methods.

Pencil-and-paper methods are used to develop a model that

is best solved (or analyzed) using a symbolic or graphic

tool. The quantitative results and graphs, frequently produced

using computer-based resources, serve to illustrate and to

clarify conclusions that might not be readily apparent from

a complicated explicit solution formula. Conversely, the

vii

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viii PREFACE

implementation of an efficient numerical method to obtain

an approximate solution typically requires a good deal of

preliminary analysis--to determine qualitative features of the

solution as a guide to computation, to investigate limiting

or special cases, or to discover ranges of the variables or

parameters that require an appropriate combination of both

analytic and numeric computation. Good judgment may well

be required to determine the best choice of solution methods

in each particular case. Within this context we point out that

problems that request a “sketch” are generally intended to

be completed without the use of any technology (except your

writing device).

We believe that it is important for students to understand

that (except perhaps in courses on differential equations) the

goal of solving a differential equation is seldom simply to

obtain the solution. Rather, we seek the solution in order to

obtain insight into the behavior of the process that the equation

purports to model. In other words, the solution is not an end

in itself. Thus, we have included in the text a great many

problems, as well as some examples, that call for conclusions

to be drawn about the solution. Sometimes this takes the form

of finding the value of the independent variable at which the

solution has a certain property, or determining the long-term

behavior of the solution. Other problems ask for the effect of

variations in a parameter, or for the determination of all values

of a parameter at which the solution experiences a substantial

change. Such problems are typical of those that arise in the

applications of differential equations, and, depending on the

goals of the course, an instructor has the option of assigning as

few or as many of these problems as desired.

Readers familiar with the preceding edition will observe

that the general structure of the book is unchanged. The

minor revisions that we have made in this edition are in

many cases the result of suggestions from users of earlier

editions. The goals are to improve the clarity and readability of

our presentation of basic material about differential equations

and their applications. More specifically, the most important

revisions include the following:

1. Chapter 1 has been rewritten. Instead of a separate section

on the History of Differential Equations, this material

appears in three installments in the remaining three

section.

2. Additional words of explanation and/or more explicit

details in the steps in a derivation have been added

throughout each chapter. These are too numerous and

widespread to mention individually, but collectively they

should help to make the book more readable for many

students.

3. There are about forty new or revised problems scattered

throughout the book. The total number of problems has

been reduced by about 400 problems, which are still

available throughWileyPlus, leaving about 1600 problems

in print.

4. There are new examples in Sections 2.1, 3.8, and 7.5.

5. The majority (is this correct?) of the figures have been

redrawn, mainly by the use full color to allow for easier

identification of critical properties of the solution. In

addition, numerous captions have been expanded to clarify

the purpose of the figure without requiring a search of the

surrounding text.

6. There are several new references, and some others have

been updated.

The authors have found differential equations to be a

never-ending source of interesting, and sometimes surprising,

results and phenomena. We hope that users of this book, both

students and instructors, will share our enthusiasm for the

subject.

William E. Boyce and Douglas B. Meade

Watervliet, New York and Columbia, SC

29 August 2016

Supplemental Resources for

Instructors and Students

An Instructor’s Solutions Manual, ISBN 978-1-119-16976-5,

includes solutions for all problems not contained in the Student

Solutions Manual.

A Student Solutions Manual, ISBN 978-1-119-16975-8,

includes solutions for selected problems in the text.

A Book Companion Site, www.wiley.com/college/boyce,

provides a wealth of resources for students and instructors,

including

• PowerPoint slides of important definitions, examples, and

theorems from the book, as well as graphics for presentation

in lectures or for study and note taking.

• Chapter Review Sheets, which enable students to test their

knowledge of key concepts. For further review, diagnostic

feedback is provided that refers to pertinent sections in the

text.

• Mathematica, Maple, and MATLAB data files for selected

problems in the text providing opportunities for further

exploration of important concepts.

• Projects that deal with extended problems normally not

included among traditional topics in differential equations,

many involving applications from a variety of disciplines.

These vary in length and complexity, and they can be

assigned as individual homework or as group assignments.

A series of supplemental guidebooks, also published by John

Wiley & Sons, can be used with Boyce/DiPrima/Meade in

order to incorporate computing technologies into the course.

These books emphasize numerical methods and graphical

analysis, showing how these methods enable us to interpret

solutions of ordinary differential equations (ODEs) in the real

world. Separate guidebooks cover each of the three major

mathematical software formats, but the ODE subject matter is

the same in each.

• Hunt, Lipsman, Osborn, and Rosenberg, Differential

Equations with MATLAB , 3rd ed., 2012, ISBN 978-1-118-

37680-5

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http://www.wiley.com/college/boyce
PREFACE ix

• Hunt, Lardy, Lipsman, Osborn, and Rosenberg, Differential

Equations with Maple, 3rd ed., 2008, ISBN 978-0-471-

77317-7

• Hunt, Outing, Lipsman, Osborn, and Rosenberg,

Differential Equations with Mathematica, 3rd ed., 2009,

ISBN 978-0-471-77316-0

WileyPLUS WileyPLUS is an innovative, research-based online environment for effective teaching and learning.

WileyPLUS builds students’ confidence because it takes

the guesswork out of studying by providing students with a

clear roadmap: what to do, how to do it, if they did it right.

Students will take more initiative so you’ll have greater impact

on their achievement in the classroom and beyond.

WileyPLUS, is loaded with all of the supplements above,

and it also features

• The E-book, which is an exact version of the print text

but also features hyperlinks to questions, definitions, and

supplements for quicker and easier support.

• Guided Online (GO) Exercises, which prompt students to

build solutions step-by-step. Rather than simply grading

an exercise answer as wrong, GO problems show students

precisely where they are making a mistake.

• Homework management tools, which enable instructors

easily to assign and grade questions, as well as to gauge

student comprehension.

• QuickStart pre-designed reading and homework assign

ments. Use them as is, or customize them to fit the needs of

your classroom.

Acknowledgments It is a pleasure to express my appreciation to the many people who

have generously assisted in various ways in the preparation of this

book.

To the individuals listed below, who reviewed the manuscript

and/or provided valuable suggestions for its improvement:

Irina Gheorghiciuc, Carnegie Mellon University

Bernard Brooks, Rochester Institute of Technology

James Moseley, West Virginia University

D. Glenn Lasseigne, Old Dominion University

Stephen Summers, University of Florida

Fabio Milner, Arizona State University

Mohamed Boudjelkha, Rensselaer Polytechnic Institute

Yuval Flicker, The Ohio State University

Y. Charles Li, University of Missouri, Columbia

Will Murray, California State University, Long Beach

Yue Zhao, University of Central Florida

Vladimir Shtelen, Rutgers University

Zhilan Feng, Purdue University

Mathew Johnson, University of Kansas

Bulent Tosun, University of Alabama

Juha Pohjanpelto, Oregon State University

Patricia Diute, Rochester Institute of Technology

Ning Ju, Oklahoma State University

Ian Christie, West Virginia University

Jonathan Rosenberg, University of Maryland

Irina Kogan, North Carolina State University

To our colleagues and students at Rensselaer and The University

of South Carolina, whose suggestions and reactions through the years

have donemuch to sharpen our knowledge of differential equations, as

well as our ideas on how to present the subject.

To those readers of the preceding edition who called errors or

omissions to our attention.

To Tom Polaski (Winthrop University), who is primarily

responsible for the revision of the Instructor’s Solutions Manual and the Student Solutions Manual.

To Mark McKibben (West Chester University), who checked the

answers in the back of the text and the Instructor’s Solutions Manual for accuracy, and carefully checked the entire manuscript.

To the editorial and production staff of John Wiley & Sons, who

have always been ready to offer assistance and have displayed the

highest standards of professionalism.

The last, but most important, people we want to thank are our

wives: Elsa, for discussing questions both mathematical and stylistic

and above all for her unfailing support and encouragement, and Betsy,

for her encouragement, patience and understanding.

WILLIAM E. BOYCE AND DOUGLAS B. MEADE

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Brief Contents

PREFACE vii

1 Introduction 1

2 First-Order Differential Equations 24

3 Second-Order Linear Differential Equations 103

4 Higher-Order Linear Differential Equations 169

5 Series Solutions of Second-Order Linear Equations 189

6 The Laplace Transform 241

7 Systems of First-Order Linear Equations 281

8 Numerical Methods 354

9 Nonlinear Differential Equations and Stability 388

10 Partial Differential Equations and Fourier Series 463

11 Boundary Value Problems and Sturm-Liouville Theory 529

ANSWERS TO PROBLEMS 573

INDEX 60

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Contents

PREFACE vii

1 Introduction 1

1.1 Some Basic Mathematical Models; Direction

Fields 1

1.2 Solutions of Some Differential Equations 9

1.3 Classification of Differential Equations 16

2 First-Order Differential Equations 24

2.1 Linear Differential Equations; Method of

Integrating Factors 24

2.2 Separable Differential Equations 33

2.3 Modeling with First-Order Differential

Equations 39

2.4 Differences Between Linear and Nonlinear

Differential Equations 51

2.5 Autonomous Differential Equations and

Population Dynamics 58

2.6 Exact Differential Equations and Integrating

Factors 70

2.7 Numerical Approximations: Euler’s Method 76

2.8 The Existence and Uniqueness Theorem 83

2.9 First-Order Difference Equations 91

3 Second-Order Linear Differential Equations 103

3.1 Homogeneous Differential Equations with

Constant Coefficients 103

3.2 Solutions of Linear Homogeneous Equations; the

Wronskian 110

3.3 Complex Roots of the Characteristic Equation

120

3.4 Repeated Roots; Reduction of Order 127

3.5 Nonhomogeneous Equations; Method of

Undetermined Coefficients 133

3.6 Variation of Parameters 142

3.7 Mechanical and Electrical Vibrations 147

3.8 Forced Periodic Vibrations 159

4 Higher-Order Linear Differential Equations 169

4.1 General Theory of nth Order Linear Differential

Equations 169

4.2 Homogeneous Differential Equations with

Constant Coefficients 174

4.3 The Method of Undetermined Coefficients 181

4.4 The Method of Variation of Parameters 185

5 Series Solutions of Second-Order Linear Equations 189

5.1 Review of Power Series 189

5.2 Series Solutions Near an Ordinary Point,

Part I 195

5.3 Series Solutions Near an Ordinary Point,

Part II 205

5.4 Euler Equations; Regular Singular Points 211

5.5 Series Solutions Near a Regular Singular Point,

Part I 219

5.6 Series Solutions Near a Regular Singular Point,

Part II 224

5.7 Bessel’s Equation 230

6 The Laplace Transform 241

6.1 Definition of the Laplace Transform 241

6.2 Solution of Initial Value Problems 248

6.3 Step Functions 257

6.4 Differential Equations with Discontinuous Forcing

Functions 264

6.5 Impulse Functions 270

6.6 The Convolution Integral 275

7 Systems of First-Order Linear Equations 281

7.1 Introduction 281

7.2 Matrices 286

7.3 Systems of Linear Algebraic Equations; Linear

Independence, Eigenvalues, Eigenvectors 295

7.4 Basic Theory of Systems of First-Order Linear

Equations 304

7.5 Homogeneous Linear Systems with Constant

Coefficients 309

7.6 Complex-Valued Eigenvalues 319

7.7 Fundamental Matrices 329

7.8 Repeated Eigenvalues 337

7.9 Nonhomogeneous Linear Systems 345

8 Numerical Methods 354

8.1 The Euler or Tangent Line Method 354

8.2 Improvements on the Euler Method 363

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xii CONTENTS

8.3 The Runge-Kutta Method 367

8.4 Multistep Methods 371

8.5 Systems of First-Order Equations 376

8.6 More on Errors; Stability 378

9 Nonlinear Differential Equations and Stability 388

9.1 The Phase Plane: Linear Systems 388

9.2 Autonomous Systems and Stability 398

9.3 Locally Linear Systems 407

9.4 Competing Species 417

9.5 Predator-Prey Equations 428

9.6 Liapunov’s Second Method 435

9.7 Periodic Solutions and Limit Cycles 444

9.8 Chaos and Strange Attractors: The Lorenz

Equations 454

10 Partial Differential Equations and Fourier Series 463

10.1 Two-Point Boundary Value Problems 463

10.2 Fourier Series 469

10.3 The Fourier Convergence Theorem 477

10.4 Even and Odd Functions 482

10.5 Separation of Variables; Heat Conduction

in a Rod 488

10.6 Other Heat Conduction Problems 496

10.7 The Wave Equation: Vibrations of an Elastic

String 504

10.8 Laplace's Equation 514

11 Boundary Value Problems and Sturm-Liouville Theory 529

11.1 The Occurrence of Two-Point Boundary Value

Problems 529

11.2 Sturm-Liouville Boundary Value Problems 535

11.3 NonhomogeneousBoundaryValueProblems 545

11.4 Singular Sturm-Liouville Problems 556

11.5 Further Remarks on the Method of Separation

of Variables: A Bessel Series Expansion 562

11.6 Series of Orthogonal Functions: Mean

Convergence 566

ANSWERS TO PROBLEMS 573

INDEX 60

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CHAPTER1

Introduction

In this first chapter we provide a foundation for your study of differential equations in several

different ways. First, we use two problems to illustrate some of the basic ideas that we

will return to, and elaborate upon, frequently throughout the remainder of the book. Later,

to provide organizational structure for the book, we indicate several ways of classifying

differential equations.

The study of differential equations has attracted the attention of many of the world’s

greatest mathematicians during the past three centuries. On the other hand, it is important

to recognize that differential equations remains a dynamic field of inquiry today, with many

interesting open questions. We outline some of the major trends in the historical development

of the subject and mention a few of the outstanding mathematicians who have contributed to

it. Additional biographical information about some of these contributors will be highlighted

at appropriate times in later chapters.

1.1 Some Basic Mathematical Models;

Direction Fields

Before embarking on a serious study of differential equations (for example, by reading this

book ormajor portions of it), you should have some idea of the possible benefits to be gained by

doing so. For some students the intrinsic interest of the subject itself is enough motivation, but

for most it is the likelihood of important applications to other fields that makes the undertaking

worthwhile.

Many of the principles, or laws, underlying the behavior of the natural world

are statements or relations involving rates at which things happen. When expressed in

mathematical terms, the relations are equations and the rates are derivatives. Equations

containing derivatives are differential equations. Therefore, to understand and to investigate

problems involving the motion of fluids, the flow of current in electric circuits, the dissipation

of heat in solid objects, the propagation and detection of seismic waves, or the increase

or decrease of populations, among many others, it is necessary to know something about

differential equations.

A differential equation that describes some physical process is often called a

mathematical model of the process, and many such models are discussed throughout this

book. In this section we begin with two models leading to equations that are easy to solve. It

is noteworthy that even the simplest differential equations provide useful models of important

physical processes.

EXAMPLE 1 | A Falling Object

Suppose that an object is falling in the atmosphere near sea level. Formulate a differential equation

that describes the motion.

▼ 1

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2 CHAPTER 1 Introduction

▼ Solution:

We begin by introducing letters to represent various quantities that may be of interest in this problem.

The motion takes place during a certain time interval, so let us use t to denote time. Also, let

us use v to represent the velocity of the falling object. The velocity will presumably change with

time, so we think of v as a function of t ; in other words, t is the independent variable and v is the

dependent variable. The choice of units of measurement is somewhat arbitrary, and there is nothing

in the statement of the problem to suggest appropriate units, so we are free to make any choice that

seems reasonable. To be specific, let us measure time t in seconds and velocity v in meters/second.

Further, we will assume that v is positive in the downward direction---that is, when the object is

falling.

The physical law that governs the motion of objects is Newton’s second law, which states that

the mass of the object times its acceleration is equal to the net force on the object. In mathematical

terms this law is expressed by the equation

F = ma, (1)

where m is the mass of the object, a is its acceleration, and F is the net force exerted on the object. To

keep our units consistent, we will measure m in kilograms, a in meters/second2, and F in newtons.

Of course, a is related to v by a = dv/dt , so we can rewrite equation (1) in the form

F = m dv

dt . (2)

Next, consider the forces that act on the object as it falls. Gravity exerts a force equal to

the weight of the object, or mg, where g is the acceleration due to gravity. In the units we

have chosen, g has been determined experimentally to be approximately equal to 9.8 m/s2 near

the earth’s surface.

There is also a force due to air resistance, or drag, that is more difficult to model. This is not

the place for an extended discussion of the drag force; suffice it to say that it is often assumed that

the drag is proportional to the velocity, and we will make that assumption here. Thus the drag force

has the magnitude γ v , where γ is a constant called the drag coefficient. The numerical value of the drag coefficient varies widely from one object to another; smooth streamlined objects have much

smaller drag coefficients than rough blunt ones. The physical units for γ are mass/time, or kg/s for this problem; if these units seem peculiar, remember that γ v must have the units of force, namely, kg·m/s2.

In writing an expression for the net force F , we need to remember that gravity always acts in

the downward (positive) direction, whereas, for a falling object, drag acts in the upward (negative)

direction, as shown in Figure 1.1.1. Thus

F = mg − γ v (3)

and equation (2) then becomes

m dv

dt = mg − γ v. (4)

Differential equation (4) is a mathematical model for the velocity v of an object falling in the

atmosphere near sea level. Note that themodel contains the three constantsm, g, and γ . The constants m and γ depend very much on the particular object that is falling, and they are usually different for different objects. It is common to refer to them as parameters, since they may take on a range of

values during the course of an experiment. On the other hand, g is a physical constant, whose value

is the same for all objects.

γ υ

m

mg

FIGURE 1.1.1 Free-body diagram of the forces on a falling object.

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1.1 Some Basic Mathematical Models; Direction Fields 3

To solve equation (4), we need to find a function v = v( t) that satisfies the equation. It

is not hard to do this, and we will show you how in the next section. For the present, however,

let us see what we can learn about solutions without actually finding any of them. Our task

is simplified slightly if we assign numerical values to m and γ , but the procedure is the same regardless of which values we choose. So, let us suppose that m = 10 kg and γ = 2 kg/s. Then equation (4) can be rewritten as

dv

dt = 9.8 −

v

5 . (5)

EXAMPLE 2 | A Falling Object (continued)

Investigate the behavior of solutions of equation (5) without solving the differential equation.

Solution:

First let us consider what information can be obtained directly from the differential equation itself.

Suppose that the velocity v has a certain given value. Then, by evaluating the right-hand side of

differential equation (5), we can find the corresponding value of dv/dt . For instance, if v = 40, then dv/dt = 1.8. This means that the slope of a solution v = v( t) has the value 1.8 at any point where v = 40. We can display this information graphically in the tv-plane by drawing short line segments

with slope 1.8 at several points on the line v = 40. (See Figure 1.1.2(a)). Similarly, when v = 50,

then dv/dt = −0.2, and when v = 60, then dv/dt = −2.2, so we draw line segments with slope −0.2 at several points on the line v = 50 (see Figure 1.1.2(b)) and line segments with slope −2.2 at several points on the line v = 60 (see Figure 1.1.2(c)). Proceeding in the same way with other values

of v we create what is called a direction field, or a slope field. The direction field for differential

equation (5) is shown in Figure 1.1.3.

Remember that a solution of equation (5) is a function v = v( t) whose graph is a curve in

the tv-plane. The importance of Figure 1.1.3 is that each line segment is a tangent line to one

of these solution curves. Thus, even though we have not found any solutions, and no graphs of

solutions appear in the figure, we can nonetheless draw some qualitative conclusions about the

behavior of solutions. For instance, if v is less than a certain critical value, then all the line segments

have positive slopes, and the speed of the falling object increases as it falls. On the other hand, if v

is greater than the critical value, then the line segments have negative slopes, and the falling object

slows down as it falls. What is this critical value of v that separates objects whose speed is increasing

from those whose speed is decreasing? Referring again to equation (5), we ask what value of v will

cause dv/dt to be zero. The answer is v = (5) (9.8) = 49 m/s. In fact, the constant function v( t) = 49 is a solution of equation (5). To verify this statement,

substitute v( t) = 49 into equation (5) and observe that each side of the equation is zero. Because

it does not change with time, the solution v( t) = 49 is called an equilibrium solution. It is

the solution that corresponds to a perfect balance between gravity and drag. In Figure 1.1.3 we show

the equilibrium solution v( t) = 49 superimposed on the direction field. From this figure we can

draw another conclusion, namely, that all other solutions seem to be converging to the equilibrium

solution as t increases. Thus, in this context, the equilibrium solution is often called the terminal

velocity.

t

45

40

50

55

60 υ

2 4 6 8 10

All slopes 1.8 All slopes –0.2 All slopes –2.2

t

45

40

50

55

60 υ

2 4 6 8 10t

45

40

50

55

60 υ

2 4 6 8 10

(a) (b) (c)

FIGURE 1.1.2 Assembling a direction field for equation (5): dv/dt = 9.8−v/5. (a) when v = 40,

dv/dt = 1.8, (b) when v = 50, dv/dt = −0.2, and (c) when v = 60, dv/dt = −2.2.

▼

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4 CHAPTER 1 Introduction

▼

2 4 t6 8 10

50

45

40

55

60

υ

FIGURE 1.1.3 Direction field and equilibrium solution for equation (5):

dv/dt = 9.8 − v/5.

The approach illustrated in Example 2 can be applied equally well to the more general

differential equation (4), where the parameters m and γ are unspecified positive numbers. The results are essentially identical to those of Example 2. The equilibrium solution of equation (4)

is the constant solution v( t) = mg/γ . Solutions below the equilibrium solution increase with time, and those above it decrease with time. As a result, we conclude that all solutions approach

the equilibrium solution as t becomes large.

Direction Fields. Direction fields are valuable tools in studying the solutions of differential

equations of the form

dy

dt = f ( t , y) , (6)

where f is a given function of the two variables t and y, sometimes referred to as the rate

function. A direction field for equations of the form (6) can be constructed by evaluating f

at each point of a rectangular grid. At each point of the grid, a short line segment is drawn

whose slope is the value of f at that point. Thus each line segment is tangent to the graph

of the solution passing through that point. A direction field drawn on a fairly fine grid gives

a good picture of the overall behavior of solutions of a differential equation. Usually a grid

consisting of a few hundred points is sufficient. The construction of a direction field is often

a useful first step in the investigation of a differential equation.

Two observations are worth particular mention. First, in constructing a direction field, we

do not have to solve equation (6); we just have to evaluate the given function f ( t , y) many

times. Thus direction fields can be readily constructed even for equations that may be quite

difficult to solve. Second, repeated evaluation of a given function and drawing a direction field

are tasks for which a computer or other computational or graphical aid are well suited. All the

direction fields shown in this book, such as the one in Figures 1.1.2 and 1.1.3, were computer

generated.

Field Mice and Owls. Now let us look at another, quite different example. Consider a

population of field mice that inhabit a certain rural area. In the absence of predators we assume

that the mouse population increases at a rate proportional to the current population. This

assumption is not a well-established physical law (as Newton’s law ofmotion is in Example 1),

but it is a common initial hypothesis1 in a study of population growth. If we denote time by t

and the mouse population at time t by p( t) , then the assumption about population growth can

be expressed by the equation

dp

dt = r p, (7)

......................................................................................................................................................................... 1A better model of population growth is discussed in Section 2.5.

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1.1 Some Basic Mathematical Models; Direction Fields 5

where the proportionality factor r is called the rate constant or growth rate. To be specific,

suppose that time is measured in months and that the rate constant r has the value 0.5/month.

Then the two terms in equation (7) have the units of mice/month.

Now let us add to the problem by supposing that several owls live in the same

neighborhood and that they kill 15 field mice per day. To incorporate this information into

the model, we must add another term to the differential equation (7), so that it becomes

dp

dt =

p

2 − 450. (8)

Observe that the predation term is −450 rather than −15 because time is measured in months,

so the monthly predation rate is needed.

EXAMPLE 3

Investigate the solutions of differential equation (8) graphically.

Solution:

A direction field for equation (8) is shown in Figure 1.1.4. For sufficiently large values of p it can

be seen from the figure, or directly from equation (8) itself, that dp/dt is positive, so that solutions increase. On the other hand, if p is small, then dp/dt is negative and solutions decrease. Again, the critical value of p that separates solutions that increase from those that decrease is the value of

p for which dp/dt is zero. By setting dp/dt equal to zero in equation (8) and then solving for p, we find the equilibrium solution p( t) = 900, for which the growth term and the predation term in

equation (8) are exactly balanced. The equilibrium solution is also shown in Figure 1.1.4.

1 2 t3 4 5

900

850

800

950

1000

p

FIGURE 1.1.4 Direction field (red) and equilibrium solution (blue) for

equation (8): dp/dt = p/2 − 450.

Comparing Examples 2 and 3, we note that in both cases the equilibrium solution separates

increasing from decreasing solutions. In Example 2 other solutions converge to, or are attracted

by, the equilibrium solution, so that after the object falls long enough, an observer will see

it moving at very nearly the equilibrium velocity. On the other hand, in Example 3 other

solutions diverge from, or are repelled by, the equilibrium solution. Solutions behave very

differently depending on whether they start above or below the equilibrium solution. As

time passes, an observer might see populations either much larger or much smaller than the

equilibrium population, but the equilibrium solution itself will not, in practice, be observed.

In both problems, however, the equilibrium solution is very important in understanding how

solutions of the given differential equation behave.

A more general version of equation (8) is

dp

dt = r p − k, (9)

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6 CHAPTER 1 Introduction

where the growth rate r and the predation rate k are positive constants that are otherwise

unspecified. Solutions of this more general equation are very similar to those of equation (8).

The equilibrium solution of equation (9) is p( t) = k/r . Solutions above the equilibrium solution increase, while those below it decrease.

You should keep in mind that both of the models discussed in this section have their

limitations. The model (5) of the falling object is valid only as long as the object is falling

freely, without encountering any obstacles. If the velocity is large enough, the assumption

that the frictional resistance is linearly proportional to the velocity has to be replaced with

a nonlinear approximation (see Problem 21). The population model (8) eventually predicts

negative numbers of mice (if p < 900) or enormously large numbers (if p > 900). Both of these predictions are unrealistic, so this model becomes unacceptable after a fairly short time

interval.

Constructing Mathematical Models. In applying differential equations to any of the

numerous fields in which they are useful, it is necessary first to formulate the appropriate

differential equation that describes, or models, the problem being investigated. In this section

we have looked at two examples of this modeling process, one drawn from physics and

the other from ecology. In constructing future mathematical models yourself, you should

recognize that each problem is different, and that successful modeling cannot be reduced to the

observance of a set of prescribed rules. Indeed, constructing a satisfactory model is sometimes

the most difficult part of the problem. Nevertheless, it may be helpful to list some steps that

are often part of the process:

1. Identify the independent and dependent variables and assign letters to represent them.

Often the independent variable is time.

2. Choose the units of measurement for each variable. In a sense the choice of units is

arbitrary, but some choices may be much more convenient than others. For example, we

chose to measure time in seconds for the falling-object problem and in months for the

population problem.

3. Articulate the basic principle that underlies or governs the problem you are investigating.

This may be a widely recognized physical law, such as Newton’s law of motion, or it

may be a more speculative assumption that may be based on your own experience or

observations. In any case, this step is likely not to be a purely mathematical one, but will

require you to be familiar with the field in which the problem originates.

4. Express the principle or law in step 3 in terms of the variables you chose in step 1.

This may be easier said than done. It may require the introduction of physical constants

or parameters (such as the drag coefficient in Example 1) and the determination of

appropriate values for them. Or it may involve the use of auxiliary or intermediate

variables that must then be related to the primary variables.

5. If the units agree, then your equation at least is dimensionally consistent, although it may

have other shortcomings that this test does not reveal.

6. In the problems considered here, the result of step 4 is a single differential equation,

which constitutes the desired mathematical model. Keep in mind, though, that in more

complex problems the resulting mathematical model may be much more complicated,

perhaps involving a system of several differential equations, for example.

Historical Background, Part I: Newton, Leibniz, and the Bernoullis. Without knowing

something about differential equations andmethods of solving them, it is difficult to appreciate

the history of this important branch of mathematics. Further, the development of differential

equations is intimately interwoven with the general development of mathematics and cannot

be separated from it. Nevertheless, to provide some historical perspective, we indicate here

some of the major trends in the history of the subject and identify the most prominent early

contributors. The rest of the historical background in this section focuses on the earliest

contributors from the seventeenth century. The story continues at the end of Section 1.2 with

an overview of the contributions of Euler and other eighteenth-century (and early-nineteenth-

century) mathematicians. More recent advances, including the use of computers and other

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1.1 Some Basic Mathematical Models; Direction Fields 7

technologies, are summarized at the end of Section 1.3. Additional historical information is

contained in footnotes scattered throughout the book and in the references listed at the end of

the chapter.

The subject of differential equations originated in the study of calculus by Isaac Newton

(1643--1727) and GottfriedWilhelm Leibniz (1646--1716) in the seventeenth century. Newton

grew up in the English countryside, was educated at Trinity College, Cambridge, and became

Lucasian Professor of Mathematics there in 1669. His epochal discoveries of calculus and of

the fundamental laws of mechanics date to 1665. They were circulated privately among his

friends, but Newtonwas extremely sensitive to criticism and did not begin to publish his results

until 1687 with the appearance of his most famous book Philosophiae Naturalis Principia

Mathematica. Although Newton did relatively little work in differential equations as such, his

development of the calculus and elucidation of the basic principles of mechanics provided a

basis for their applications in the eighteenth century, most notably by Euler (see Historical

Background, Part II in Section 1.2). Newton identified three forms of first-order differential

equations: dy/dx = f ( x) , dy/dx = f ( y) , and dy/dx = f ( x , y) . For the latter equation he developed a method of solution using infinite series when f ( x , y) is a polynomial in x and

y. Newton’s active research in mathematics ended in the early 1690s, except for the solution

of occasional “challenge problems” and the revision and publication of results obtained much

earlier. He was appointed Warden of the British Mint in 1696 and resigned his professorship a

few years later. He was knighted in 1705 and, upon his death in 1727, became the first scientist

buried in Westminster Abbey.

Leibniz was born in Leipzig, Germany, and completed his doctorate in philosophy at the

age of 20 at the University of Altdorf. Throughout his life he engaged in scholarly work in

several different fields. He was mainly self-taught in mathematics, since his interest in this

subject developed when he was in his twenties. Leibniz arrived at the fundamental results of

calculus independently, although a little later than Newton, but was the first to publish them,

in 1684. Leibniz was very conscious of the power of good mathematical notation and was

responsible for the notation dy/dx for the derivative and for the integral sign. He discovered the method of separation of variables (Section 2.2) in 1691, the reduction of homogeneous

equations to separable ones (Section 2.2, Problem 30) in 1691, and the procedure for solving

first-order linear equations (Section 2.1) in 1694. He spent his life as ambassador and adviser

to several German royal families, which permitted him to travel widely and to carry on an

extensive correspondence with other mathematicians, especially the Bernoulli brothers. In the

course of this correspondence many problems in differential equations were solved during the

latter part of the seventeenth century.

The Bernoulli brothers, Jakob (1654--1705) and Johann (1667--1748), of Basel,

Switzerland did much to develop methods of solving differential equations and to extend

the range of their applications. Jakob became professor of mathematics at Basel in 1687,

and Johann was appointed to the same position upon his brother’s death in 1705. Both

men were quarrelsome, jealous, and frequently embroiled in disputes, especially with each

other. Nevertheless, both also made significant contributions to several areas of mathematics.

With the aid of calculus, they solved a number of problems in mechanics by formulating

them as differential equations. For example, Jakob Bernoulli solved the differential equation

y′ = (

a3/(b2y − a3) )1/2

(see Problem 9 in Section 2.2) in 1690 and, in the same paper, first

used the term “integral” in the modern sense. In 1694 Johann Bernoulli was able to solve the

equation dy/dx = y/(ax) (see Problem 10 in Section 2.2). One problem that both brothers solved, and that led to much friction between them, was the brachistochrone problem (see

Problem 24 in Section 2.3). The brachistochrone problemwas also solved by Leibniz, Newton,

and the Marquis de l’Hôpital. It is said, perhaps apocryphally, that Newton learned of the

problem late in the afternoon of a tiring day at the Mint and solved it that evening after dinner.

He published the solution anonymously, but upon seeing it, Johann Bernoulli exclaimed, “Ah,

I know the lion by his paw.”

Daniel Bernoulli (1700--1782), son of Johann, migrated to St. Petersburg, Russia, as a

young man to join the newly established St. Petersburg Academy, but returned to Basel in

1733 as professor of botany and, later, of physics. His interests were primarily in partial

differential equations and their applications. For instance, it is his name that is associated with

the Bernoulli equation in fluid mechanics. He was also the first to encounter the functions that

a century later became known as Bessel functions (Section 5.7).

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8 CHAPTER 1 Introduction

Problems In each of Problems 1 through 4, draw a direction field for the

given differential equation. Based on the direction field, determine the

behavior of y as t → ∞. If this behavior depends on the initial value

of y at t = 0, describe the dependency.

G 1. y′ = 3 − 2y

G 2. y′ = 2y − 3

G 3. y′ = −1 − 2y

G 4. y′ = 1 + 2y

In each of Problems 5 and 6, write down a differential equation of the

form dy/dt = ay + b whose solutions have the required behavior as t → ∞.

5. All solutions approach y = 2/3.

6. All other solutions diverge from y = 2.

In each of Problems 7 through 10, draw a direction field for the

given differential equation. Based on the direction field, determine the

behavior of y as t → ∞. If this behavior depends on the initial value

of y at t = 0, describe this dependency. Note that in these problems

the equations are not of the form y′ = ay + b, and the behavior of

their solutions is somewhat more complicated than for the equations

in the text.

G 7. y′ = y(4 − y)

G 8. y′ = −y(5 − y)

G 9. y′ = y2

G 10. y′ = y( y − 2) 2

Consider the following list of differential equations, some of which

produced the direction fields shown in Figures 1.1.5 through 1.1.10.

In each of Problems 11 through 16, identify the differential equation

that corresponds to the given direction field.

a. y′ = 2y − 1

b. y′ = 2 + y

c. y′ = y − 2

d. y′ = y( y + 3)

e. y′ = y( y − 3)

f. y′ = 1 + 2y

g. y′ = −2 − y

h. y′ = y(3 − y)

i. y′ = 1 − 2y

j. y′ = 2 − y

11. The direction field of Figure 1.1.5.

1

2

3

4

1 2 3 4

y

t

FIGURE 1.1.5 Problem 11.

12. The direction field of Figure 1.1.6.

1

2

3

4

1 2 3 4

y

t

FIGURE 1.1.6 Problem 12.

13. The direction field of Figure 1.1.7.

–4

–3

–2

–1

1 2 3 4y t

FIGURE 1.1.7 Problem 13.

14. The direction field of Figure 1.1.8.

–4

–3

–2

–1

1 2 3 4y t

FIGURE 1.1.8 Problem 14.

15. The direction field of Figure 1.1.9.

–1

1

2

3

4

5

1 2 3 4

y

t

FIGURE 1.1.9 Problem 15.

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1.2 Solutions of Some Differential Equations 9

16. The direction field of Figure 1.1.10.

–1

1

2

3

4

5

1 2 3 4

y

t

FIGURE 1.1.10 Problem 16.

17. Apond initially contains 1,000,000 gal of water and an unknown amount of an undesirable chemical. Water containing 0.01 grams of

this chemical per gallon flows into the pond at a rate of 300 gal/h. The

mixture flows out at the same rate, so the amount of water in the pond

remains constant. Assume that the chemical is uniformly distributed

throughout the pond.

a. Write a differential equation for the amount of chemical in the pond at any time.

b. How much of the chemical will be in the pond after a very long time? Does this limiting amount depend on the amount that

was present initially?

c. Write a differential equation for the concentration of the chemical in the pond at time t . Hint: The concentration is

c = a/v = a( t)/106.

18. A spherical raindrop evaporates at a rate proportional to its surface area. Write a differential equation for the volume of the

raindrop as a function of time.

19. Newton’s law of cooling states that the temperature of an object changes at a rate proportional to the difference between the

temperature of the object itself and the temperature of its surroundings

(the ambient air temperature in most cases). Suppose that the ambient

temperature is 70◦F and that the rate constant is 0.05 (min)−1. Write a

differential equation for the temperature of the object at any time. Note

that the differential equation is the same whether the temperature of

the object is above or below the ambient temperature.

20. A certain drug is being administered intravenously to a hospital patient. Fluid containing 5 mg/cm3 of the drug enters the patient’s

bloodstream at a rate of 100 cm3/h. The drug is absorbed by body

tissues or otherwise leaves the bloodstream at a rate proportional to

the amount present, with a rate constant of 0.4/h.

a. Assuming that the drug is always uniformly distributed throughout the bloodstream, write a differential equation for the

amount of the drug that is present in the bloodstream at any time.

b. How much of the drug is present in the bloodstream after a long time?

N 21. For small, slowly falling objects, the assumption made in the text that the drag force is proportional to the velocity is a good one.

For larger, more rapidly falling objects, it is more accurate to assume

that the drag force is proportional to the square of the velocity.2

a. Write a differential equation for the velocity of a falling object of mass m if the magnitude of the drag force is

proportional to the square of the velocity and its direction is

opposite to that of the velocity.

b. Determine the limiting velocity after a long time. c. If m = 10 kg, find the drag coefficient so that the limiting velocity is 49 m/s. N d. Using the data in part c, draw a direction field and compare it with Figure 1.1.3.

In each of Problems 22 through 25, draw a direction field for the

given differential equation. Based on the direction field, determine the

behavior of y as t → ∞. If this behavior depends on the initial value

of y at t = 0, describe this dependency. Note that the right-hand sides

of these equations depend on t as well as y; therefore, their solutions

can exhibit more complicated behavior than those in the text.

G 22. y′ = −2 + t − y

G 23. y′ = e−t + y

G 24. y′ = 3 sin t + 1 + y

G 25. y′ = − 2t + y

2y

............................................................................................................................. 2See Lyle N. Long and Howard Weiss, “The Velocity Dependence of

Aerodynamic Drag: A Primer for Mathematicians,” American Mathematical

Monthly 106 (1999), 2, pp. 127--135.

1.2 Solutions of Some Differential Equations

In the preceding section we derived the differential equations

m dv

dt = mg − γ v (1)

and

dp

dt = r p − k. (2)

Equation (1) models a falling object, and equation (2) models a population of field mice preyed

on by owls. Both of these equations are of the general form

dy

dt = ay − b, (3)

where a and b are given constants. We were able to draw some important qualitative

conclusions about the behavior of solutions of equations (1) and (2) by considering the

associated direction fields. To answer questions of a quantitative nature, however, we need

to find the solutions themselves, and we now investigate how to do that.

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10 CHAPTER 1 Introduction

EXAMPLE 1 | Field Mice and Owls (continued)

Consider the equation

dp

dt = 0.5p − 450, (4)

which describes the interaction of certain populations of field mice and owls (see equation (8) of

Section 1.1). Find solutions of this equation.

Solution:

To solve equation (4), we need to find functions p( t) that, when substituted into the equation, reduce

it to an obvious identity. Here is one way to proceed. First, rewrite equation (4) in the form

dp

dt =

p − 900

2 , (5)

or, if p �= 900,

dp/dt

p − 900 =

1

2 . (6)

By the chain rule the left-hand side of equation (6) is the derivative of ln |p − 900| with respect to t ,

so we have

d

dt ln |p − 900| =

1

2 . (7)

Then, by integrating both sides of equation (7), we obtain

ln |p − 900| = t

2 + C, (8)

where C is an arbitrary constant of integration. Therefore, by taking the exponential of both sides of

equation (8), we find that

|p − 900| = et/2+C = eC et/2, (9)

or

p − 900 = ±eC et/2, (10)

and finally

p = 900 + cet/2, (11)

where c = ±eC is also an arbitrary (nonzero) constant. Note that the constant function p = 900 is

also a solution of equation (5) and that it is contained in the expression (11) if we allow c to take the

value zero. Graphs of equation (11) for several values of c are shown in Figure 1.2.1.

900

600

t1 2 3 4 5

700

800

1000

1100

1200

p

FIGURE 1.2.1 Graphs of p = 900 + cet/2 for several values of c. Each blue

curve is a solution of dp/dt = 0.5p − 450.

▼

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1.2 Solutions of Some Differential Equations 11

▼ Note that they have the character inferred from the direction field in Figure 1.1.4. For instance,

solutions lying on either side of the equilibrium solution p = 900 tend to diverge from that

solution.

In Example 1 we found infinitely many solutions of the differential equation (4),

corresponding to the infinitely many values that the arbitrary constant c in equation (11) might

have. This is typical of what happens when you solve a differential equation. The solution

process involves an integration, which brings with it an arbitrary constant, whose possible

values generate an infinite family of solutions.

Frequently, we want to focus our attention on a single member of the infinite family of

solutions by specifying the value of the arbitrary constant. Most often, we do this indirectly by

specifying instead a point that must lie on the graph of the solution. For example, to determine

the constant c in equation (11), we could require that the population have a given value at a

certain time, such as the value 850 at time t = 0. In other words, the graph of the solution

must pass through the point (0, 850) . Symbolically, we can express this condition as

p(0) = 850. (12)

Then, substituting t = 0 and p = 850 into equation (11), we obtain

850 = 900 + c.

Hence c = −50, and by inserting this value into equation (11), we obtain the desired solution,

namely,

p = 900 − 50et/2. (13)

The additional condition (12) that we used to determine c is an example of an initial

condition. The differential equation (4) together with the initial condition (12) forms an initial

value problem.

Now consider the more general problem consisting of the differential equation (3)

dy

dt = ay − b

and the initial condition

y(0) = y0, (14)

where y0 is an arbitrary initial value. We can solve this problem by the same method as in

Example 1. If a �= 0 and y �= b/a, then we can rewrite equation (3) as

dy/dt

y − b a

= a. (15)

By integrating both sides, we find that

ln

∣

∣

∣

∣

y( t) − b

a

∣

∣

∣

∣

= at + C, (16)

where C is an arbitrary constant. Then, taking the exponential of both sides of equation (16)

and solving for y, we obtain

y( t) = b

a + ceat , (17)

where c = ±eC is also an arbitrary constant. Observe that c = 0 corresponds to the equilibrium

solution y( t) = b/a. Finally, the initial condition (14) requires that c = y0 − (b/a) , so the solution of the initial value problem (3), (14) is

y( t) = b

a +

(

y0 − b

a

)

eat . (18)

For a �= 0 the expression (17) contains all possible solutions of equation (3) and is called

the general solution.3 The geometric representation of the general solution (17) is an infinite

family of curves called integral curves. Each integral curve is associated with a particular

......................................................................................................................................................................... 3If a = 0, then the solution of equation (3) is not given by equation (17). We leave it to you to find the general solution

in this case.

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12 CHAPTER 1 Introduction

value of c and is the graph of the solution corresponding to that value of c. Satisfying an initial

condition amounts to identifying the integral curve that passes through the given initial point.

To relate the solution (18) to equation (2), which models the field mouse population, we

need only replace a by the growth rate r and replace b by the predation rate k; we assume that

r > 0 and k > 0. Then the solution (18) becomes

p( t) = k

r +

(

p0 − k

r

)

er t , (19)

where p0 is the initial population of field mice. The solution (19) confirms the conclusions

reached on the basis of the direction field and Example 1. If p0 = k/r , then from equation (19) it follows that p( t) = k/r for all t ; this is the constant, or equilibrium, solution. If p0 �= k/r , then the behavior of the solution depends on the sign of the coefficient p0 − k/r of the exponential term in equation (19). If p0 > k/r , then p grows exponentially with time t ; if p0 < k/r , then p decreases and becomes zero (at a finite time), corresponding to extinction of the field mouse population. Negative values of p, while possible for the expression (19),

make no sense in the context of this particular problem.

To put the falling-object equation (1) in the form (3), we must identify a with −γ/m and b with −g. Observe that assuming γ > 0 and m > 0 implies that a < 0 and b < 0. Making these substitutions in the solution (18), we obtain

v( t) = mg

γ +

(

v0 − mg

γ

)

e−γ t/m , (20)

where v0 is the initial velocity. Again, this solution confirms the conclusions reached in

Section 1.1 on the basis of a direction field. There is an equilibrium, or constant, solution

v( t) = mg/γ , and all other solutions tend to approach this equilibrium solution. The speed of convergence to the equilibrium solution is determined by the exponent −γ/m. Thus, for a given mass m, the velocity approaches the equilibrium value more rapidly as the drag

coefficient γ increases.

EXAMPLE 2 | A Falling Object (continued)

Suppose that, as in Example 2 of Section 1.1, we consider a falling object of mass m = 10 kg and

drag coefficient γ = 2 kg/s. Then the equation of motion (1) becomes

dv

dt = 9.8 −

v

5 . (21)

Suppose this object is dropped from a height of 300 m. Find its velocity at any time t . How long will

it take to fall to the ground, and how fast will it be moving at the time of impact?

Solution:

The first step is to state an appropriate initial condition for equation (21). The word “dropped” in the

statement of the problem suggests that the object starts from rest, that is, its initial velocity is zero,

so we will use the initial condition

v(0) = 0. (22)

The solution of equation (21) can be found by substituting the values of the coefficients into the

solution (20), but we will proceed instead to solve equation (21) directly. First, rewrite the equation as

dv/dt

v − 49 = −

1

5 . (23)

By integrating both sides, we obtain

ln ∣

∣v( t) − 49 ∣

∣ = − t

5 + C, (24)

and then the general solution of equation (21) is

v( t) = 49 + ce−t/5, (25)

where the constant c is arbitrary. To determine the particular value of c that corresponds to the initial

condition (22), we substitute t = 0 and v = 0 into equation (25), with the result that c = −49. Then

▼

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1.2 Solutions of Some Differential Equations 13

▼ the solution of the initial value problem (21), (22) is

v( t) = 49 (

1 − e−t/5 )

. (26)

Equation (26) gives the velocity of the falling object at any positive time after being dropped---until

it hits the ground, of course.

Graphs of the solution (25) for several values of c are shown in Figure 1.2.2, with the solution

(26) shown by the green curve. It is evident that, regardless of the initial velocity of the object, all

solutions tend to approach the equilibrium solution v( t) = 49. This confirms the conclusions we

reached in Section 1.1 on the basis of the direction fields in Figures 1.1.2 and 1.1.3.

100

80

60

40

20

2 4 t6 8 1210

v

v = 49 (1 – e–t/5)

(10.51, 43.01)

FIGURE 1.2.2 Graphs of the solution (25), v = 49 + ce−t/5, for several values

of c. The green curve corresponds to the initial condition v(0) = 0. The point

(10.51, 43.01) shows the velocity when the object hits the ground.

To find the velocity of the object when it hits the ground, we need to know the time at which

impact occurs. In other words, we need to determine how long it takes the object to fall 300 m. To

do this, we note that the distance x the object has fallen is related to its velocity v by the differential

equation v = dx/dt , or

dx

dt = 49

(

1 − e−t/5 )

. (27)

Consequently, by integrating both sides of equation (27) with respect to t , we have

x = 49t + 245e−t/5 + k, (28)

where k is an arbitrary constant of integration. The object starts to fall when t = 0, so we know that

x = 0 when t = 0. From equation (28) it follows that k = −245, so the distance the object has fallen

at time t is given by

x = 49t + 245e−t/5 − 245. (29)

Let T be the time at which the object hits the ground; then x = 300 when t = T . By substituting

these values in equation (29), we obtain the equation

49T + 245e−T/5 − 245 = 300. (30)

The value of T satisfying equation (30) can be approximated by a numerical process4 using

a calculator or other computational tool, with the result that T ∼= 10.51 s. At this time, the corresponding velocity vT is found from equation (26) to be vT ∼= 43.01m/s. The point (10.51, 43.01) is also shown in Figure 1.2.2.