Edwards differential equations and linear algebra 4th ed 2018

C O N T E N T SApplication Modules vi Preface ix CHAPTER 1 First-Order Differential Equations 1 1.1 Differential Equations and Mathematical Models 11.2 Integrals as General and Particula

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Library of Congress Cataloging-in-Publication Data

Names: Edwards, C Henry (Charles Henry), 1937– j Penney, David E j Calvis,

David

Title: Differential equations & linear algebra / C Henry Edwards, David E.

Penney, The University of Georgia; with the assistance of David Calvis,

Baldwin-Wallace College.

Description: Fourth edition j Boston : Pearson, [2018] j Includes

bibliographical references and index.

Identifies: LCCN 2016030491 j ISBN 9780134497181 (hardcover) j ISBN

013449718X (hardcover)

Subjects: LCSH: Differential equations j Algebras, Linear.

Classification: LCC QA372 E34 2018 j DDC 515/.35 dc23

LC record available at https://lccn.loc.gov/2016030491

1 16

ISBN 13: 978-0-13-449718-1 ISBN 10: 0-13-449718-X

C O N T E N T S

Application Modules vi Preface ix

CHAPTER

1

First-Order Differential Equations 1

1.1 Differential Equations and Mathematical Models 11.2 Integrals as General and Particular Solutions 101.3 Slope Fields and Solution Curves 17

1.4 Separable Equations and Applications 301.5 Linear First-Order Equations 46

1.6 Substitution Methods and Exact Equations 58

CHAPTER

2

Mathematical Models and Numerical Methods 75

2.1 Population Models 752.2 Equilibrium Solutions and Stability 872.3 Acceleration-Velocity Models 942.4 Numerical Approximation: Euler’s Method 1062.5 A Closer Look at the Euler Method 117

2.6 The Runge–Kutta Method 127

CHAPTER

3

Linear Systems and Matrices 138

3.1 Introduction to Linear Systems 1473.2 Matrices and Gaussian Elimination 1463.3 Reduced Row-Echelon Matrices 1563.4 Matrix Operations 164

3.5 Inverses of Matrices 1753.6 Determinants 1883.7 Linear Equations and Curve Fitting 203

iii

4.5 Row and Column Spaces 2424.6 Orthogonal Vectors in Rn

2504.7 General Vector Spaces 257

CHAPTER

5

Higher-Order Linear Differential Equations 265

5.1 Introduction: Second-Order Linear Equations 2655.2 General Solutions of Linear Equations 279

5.3 Homogeneous Equations with Constant Coefficients 2915.4 Mechanical Vibrations 302

5.5 Nonhomogeneous Equations and Undetermined Coefficients 3145.6 Forced Oscillations and Resonance 327

CHAPTER

6

Eigenvalues and Eigenvectors 339

6.1 Introduction to Eigenvalues 3396.2 Diagonalization of Matrices 3476.3 Applications Involving Powers of Matrices 354

CHAPTER

7

Linear Systems of Differential Equations 365

7.1 First-Order Systems and Applications 3657.2 Matrices and Linear Systems 375

7.3 The Eigenvalue Method for Linear Systems 3857.4 A Gallery of Solution Curves of Linear Systems 3987.5 Second-Order Systems and Mechanical Applications 4247.6 Multiple Eigenvalue Solutions 437

7.7 Numerical Methods for Systems 454

8

Matrix Exponential Methods 469

8.1 Matrix Exponentials and Linear Systems 4698.2 Nonhomogeneous Linear Systems 4828.3 Spectral Decomposition Methods 490

CHAPTER

9

Nonlinear Systems and Phenomena 503

9.1 Stability and the Phase Plane 5039.2 Linear and Almost Linear Systems 5149.3 Ecological Models: Predators and Competitors 5269.4 Nonlinear Mechanical Systems 539

CHAPTER

10

Laplace Transform Methods 557

10.1 Laplace Transforms and Inverse Transforms 55710.2 Transformation of Initial Value Problems 56710.3 Translation and Partial Fractions 578

10.4 Derivatives, Integrals, and Products of Transforms 58710.5 Periodic and Piecewise Continuous Input Functions 594

CHAPTER

11

Power Series Methods 604

11.1 Introduction and Review of Power Series 60411.2 Power Series Solutions 616

11.3 Frobenius Series Solutions 62711.4 Bessel Functions 642

References for Further Study 652

Appendix A: Existence and Uniqueness of Solutions 654

Appendix B: Theory of Determinants 668

Answers to Selected Problems 677

Index 733

A P P L I C A T I O N M O D U L E S

The modules listed below follow the indicated sections in the text Most provide

computing projects that illustrate the corresponding text sections Many of these

modules are enhanced by the supplementary material found at the new

Expanded Applications website, which can be accessed by visiting

goo.g l /BXB9k4 For more information about the Expanded Applications,

please review the Principal Features of this Revision section of the preface

1.3 Computer-Generated Slope Fields and Solution Curves

1.4 The Logistic Equation

1.5 Indoor Temperature Oscillations

1.6 Computer Algebra Solutions

2.1 Logistic Modeling of Population Data

2.3 Rocket Propulsion

2.4 Implementing Euler’s Method

2.5 Improved Euler Implementation

2.6 Runge-Kutta Implementation

3.2 Automated Row Operations

3.3 Automated Row Reduction

3.5 Automated Solution of Linear Systems

5.1 Plotting Second-Order Solution Families

5.2 Plotting Third-Order Solution Families

5.3 Approximate Solutions of Linear Equations

5.5 Automated Variation of Parameters

5.6 Forced Vibrations

7.1 Gravitation and Kepler’s Laws of Planetary Motion

7.3 Automatic Calculation of Eigenvalues and Eigenvectors

7.4 Dynamic Phase Plane Graphics

7.5 Earthquake-Induced Vibrations of Multistory Buildings

7.6 Defective Eigenvalues and Generalized Eigenvectors

7.7 Comets and Spacecraft

8.1 Automated Matrix Exponential Solutions

8.2 Automated Variation of Parameters

9.1 Phase Plane Portraits and First-Order Equations

9.2 Phase Plane Portraits of Almost Linear Systems

9.3 Your Own Wildlife Conservation Preserve

9.4 The Rayleigh, van der Pol, and FitzHugh-Nagumo Equations

vi

10.1 Computer Algebra Transforms and Inverse Transforms

10.2 Transforms of Initial Value Problems

10.3 Damping and Resonance Investigations

10.5 Engineering Functions

11.2 Automatic Computation of Series Coefficients

11.3 Automating the Frobenius Series Method

P R E F A C E

The evolution of the present text is based on experience teaching introductory ferential equations and linear algebra with an emphasis on conceptual ideas andthe use of applications and projects to involve students in active problem-solving

dif-experiences Technical computing environments like Maple, Mathematica,

MAT-LAB, and Python are widely available and are now used extensively by practicingengineers and scientists This change in professional practice motivates a shift fromthe traditional concentration on manual symbolic methods to coverage also of quali-tative and computer-based methods that employ numerical computation and graphi-cal visualization to develop greater conceptual understanding A bonus of this morecomprehensive approach is accessibility to a wider range of more realistic applica-tions of differential equations

Both the conceptual and the computational aspects of such a course dependheavily on the perspective and techniques of linear algebra Consequently, the study

of differential equations and linear algebra in tandem reinforces the learning of bothsubjects In this book we therefore have combined core topics in elementary differ-ential equations with those concepts and methods of elementary linear algebra thatare needed for a contemporary introduction to differential equations

Principal Features of This Revision

This 4th edition is the most comprehensive and wide-ranging revision in the history

of this text

We have enhanced the exposition, as well as added graphics, in numeroussections throughout the book We have also inserted new applications, includingbiological Moreover we have exploited throughout the new interactive computertechnology that is now available to students on devices ranging from desktop andlaptop computers to smartphones and graphing calculators While the text contin-

ues to use standard computer algebra systems such as Mathematica, Maple, and

MATLAB, we have now added the WolframjAlpha website In addition, this is thefirst edition of this book to feature Python, a computer platform that is freely avail-able on the internet and which is gaining in popularity as an all-purpose scientificcomputing environment

However, with a single exception of a new section inserted in Chapter 7 (notedbelow), the class-tested table of contents of the book remains unchanged Therefore,instructors notes and syllabi will not require revision to continue teaching with thisnew edition

A conspicuous feature of this edition is the insertion of about 80 new generated figures, many of them illustrating interactive computer applications withslider bars or touchpad controls that can be used to change initial values or parame-ters in a differential equation, and immediately see in real time the resulting changes

computer-in the structure of its solutions

ix

Some illustrations of the revisions and updating in this edition:

New Exposition In a number of sections, we have added new text and graphics

to enhance student understanding of the subject matter For instance, see the newintroductory treatments of separable equations in Section 1.4 (page 30), of linearequations in Section 1.5 (page 46), and of isolated critical points in Sections 9.1(page 503) and 9.2 (page 514) Also we have updated the examples and accom-panying graphics in Sections 2.4–2.6, 7.3, and 7.7 to illustrate modern calculatortechnology

New Interactive Technology and Graphics New figures throughout the text lustrate the capability that modern computing technology platforms offer to varyinitial conditions and other parameters interactively These figures are accompanied

il-by detailed instructions that allow students to recreate the figures and make full use

of the interactive features For example, Section 7.4 includes the figure shown, a

Mathematica-drawn phase plane diagram for a linear system of the form x0DAx;

after putting the accompanying code into Mathematica, the user can immediately

see the effect of changing the initial condition

by clicking and dragging the “locator point” tially set at.4; 2/

ini-Similarly, the application module for tion 5.1 now offers MATLAB and TI-Nspiregraphics with interactive slider bars that varythe coefficients of a linear differential equation

Sec-The Section 11.2 application module features

a new MATLABgraphic in which the user canvary the order of a series solution of an ini-tial value problem, again immediately display-ing the resulting graphical change in the corre-sponding approximate solution

– 4 – 2

– 2

– 4

0 0

New Mathematica graphic in Section 7.4

New Content The single entirely new section for this edition is Section 7.4,which is devoted to the construction of a “gallery” of phase plane portraits illus-trating all the possible geometric behaviors of solutions of the 2-dimensional linear

system x0DAx In motivation and preparation for the detailed study of

eigenvalue-eigenvector methods in subsequent sections of Chapter 7 (which then follow in thesame order as in the previous edition), Section 7.4 shows how the particular ar-

rangements of eigenvalues and eigenvectors of the coefficient matrix A correspond

to identifiable patterns—“fingerprints,” so to speak—in the phase plane portrait ofthe system The resulting gallery is shown in the two pages of phase plane portraits

in Figure 7.4.16 (pages 417–418) at the end of the section The new 7.4 cation module (on dynamic phase plane portraits, page 421) shows how studentscan use interactive computer systems to bring to life this gallery by allowing initialconditions, eigenvalues, and even eigenvectors to vary in real time This dynamicapproach is then illustrated with several new graphics inserted in the remainder ofChapter 7

appli-Finally, for a new biological application, see the application module for tion 9.4, which now includes a substantial investigation (page 551) of the nonlinearFitzHugh–Nagumo equations of neuroscience, which were introduced to model thebehavior of neurons in the nervous system

Sec-New Topical Headings Many of the examples and problems are now organizedunder headings that make the topic easy to see at a glance This not only adds tothe readability of the book, but it also makes it easier to choose in-class examplesand homework problems For instance, most of the text examples in Section 1.4 are

now labelled by topic, and the same is true of the wealth of problems following thissection.

New Expanded Applications Website The effectiveness of the application ules located throughout the text is greatly enhanced by the supplementary materialfound at the new Expanded Applications website Nearly all of the application mod-ules in the text are marked with and a unique “tiny URL”—a web address thatleads directly to an Expanded Applications page containing a wealth of electronicresources supporting that module Typical Expanded Applications materials include

mod-an enhmod-anced mod-and expmod-anded PDF version of the text with further discussion or tional applications, together with computer files in a variety of platforms, including

addi-Mathematica, Maple, MATLAB, and in some cases Python and/or TI calculator.

These files provide all code appearing in the text as well as equivalent versions inother platforms, allowing students to immediately use the material in the Applica-tion Module on the computing platform of their choice In addition to the URLsdispersed throughout the text, the Expanded Applications can be accessed by going

to the Expanded Applications home page through this URL:goo.g l /BXB9k4 Notethat when you enter URLs for the Extended Applications, take care to distinguishthe following characters:

lDlowercase L 1Done

IDuppercase I 0Dzero

ODuppercase O

Features of This Text

Computing Features The following features highlight the flavor of computingtechnology that distinguishes much of our exposition

 Almost 600 computer-generated figures show students vivid pictures of

di-rection fields, solution curves, and phase plane portraits that bring symbolicsolutions of differential equations to life

 About three dozen application modules follow key sections throughout the

text Most of these applications outline technology investigations that can becarried out using a variety of popular technical computing systems and whichseek to actively engage students in the application of new technology Thesemodules are accompanied by the new Expanded Applications website previ-ously detailed, which provides explicit code for nearly all of the applications

in a number of popular technology platforms

 The early introduction of numerical solution techniques in Chapter 2 (on ematical models and numerical methods) allows for a fresh numerical empha-sis throughout the text Here and in Chapter 7, where numerical techniquesfor systems are treated, a concrete and tangible flavor is achieved by the inclu-sion of numerical algorithms presented in parallel fashion for systems rangingfrom graphing calculators to MATLABand Python

math-Modeling Features Mathematical modeling is a goal and constant motivation forthe study of differential equations For a small sample of the range of applications

in this text, consider the following questions:

 What explains the commonly observed time lag between indoor and outdoordaily temperature oscillations? (Section 1.5)

 What makes the difference between doomsday and extinction in alligator ulations? (Section 2.1)

pop- How do a unicycle and a car react differently to road bumps? (Sections 5.6and 7.5)

 Why might an earthquake demolish one building and leave standing the onenext door? (Section 7.5)

 How can you predict the time of next perihelion passage of a newly observedcomet? (Section 7.7)

 What determines whether two species will live harmoniously together orwhether competition will result in the extinction of one of them and the sur-vival of the other? (Section 9.3)

Organization and Content This text reshapes the usual sequence of topics toaccommodate new technology and new perspectives For instance:

 After a precis of first-order equations in Chapter 1 (though with the coverage

of certain traditional symbolic methods streamlined a bit), Chapter 2 offers anearly introduction to mathematical modeling, stability and qualitative proper-ties of differential equations, and numerical methods—a combination of topicsthat frequently are dispersed later in an introductory course

 Chapters 3 (Linear Systems and Matrices), 4 (Vector Spaces), and 6 values and Eigenvectors) provide concrete and self-contained coverage of theelementary linear algebra concepts and techniques that are needed for the solu-tion of linear differential equations and systems Chapter 4 includes sections

(Eigen-4.5 (row and column spaces) and 4.6 (orthogonal vectors in Rn

 Chapter 8 is devoted to matrix exponentials with applications to linear systems

of differential equations The spectral decomposition method of Section 8.3offers students an especially concrete approach to the computation of matrixexponentials

 Chapter 9 exploits linear methods for the investigation of nonlinear systemsand phenomena, and ranges from phase plane analysis to applications involv-ing ecological and mechanical systems

 Chapters 10 treats Laplace transform methods for the solution of coefficient linear differential equations with a goal of handling the piecewisecontinuous and periodic forcing functions that are common in physical ap-plications Chapter 11 treats power series methods with a goal of discussingBessel’s equation with sufficient detail for the most common elementary ap-plications

constant-This edition of the text also contains over 1800 end-of-section exercises, cluding a wealth of application problems The Answers to Selected Problems sec-tion (page 677) includes answers to most odd-numbered problems plus a good manyeven-numbered ones, as well as about 175 computer-generated graphics to enhanceits value as a learning aid

Instructor’s Solutions Manual (0-13-449825-9) is available for instructors to

down-load at Pearson’s Instructor Resource Center (pearsonhighered.com/irc) This ual provides worked-out solutions for most of the problems in the book

man-Student’s Solutions Manual (0-13-449814-3) contains solutions for most of the

Grant Gustafson, University of Utah Semion Gutman, University of Oklahoma Richard Jardine, Keene State College Yang Kuang, Arizona State University Dening Li, West Virginia University Carl Lutzer, Rochester Institute of Technology Francisco Sayas-Gonzalez, University of Delaware Morteza Shafii-Mousavi, Indiana University, South Bend Brent Solie, Knox College

Ifran Ul-Haq, University of Wisconsin, Platteville Luther White, University of Oklahoma

Hong-Ming Yin, Washington State University

We are grateful to our editor, Jeff Weidenaar, for advice and numerous gestions that enhanced and shaped this revision; to Jennifer Snyder for her counseland coordination of the editorial process; to Tamela Ambush and Julie Kidd fortheir supervision of the production process; and to Joe Vetere for his assistance withtechnical aspects of production of the supplementary manuals It is a pleasure to(once again) credit Dennis Kletzing and his extraordinary TeXpertise for the attrac-tive presentation of the text and graphics in this book (and its predecessors over thepast decade)

sug-Henry Edwards

h.edwards@mindspring.com

David Calvis

dcalvis@bw.edu

1 First-Order

Differential Equations

1.1 Differential Equations and Mathematical Models

The laws of the universe are written in the language of mathematics Algebra

is sufficient to solve many static problems, but the most interesting naturalphenomena involve change and are described by equations that relate changingquantities

Because the derivativedx=dt D f0.t /of the functionf is the rate at whichthe quantity x D f t /is changing with respect to the independent variable t, it

is natural that equations involving derivatives are frequently used to describe thechanging universe An equation relating an unknown function and one or more of

its derivatives is called a differential equation.

Example 1 The differential equation

The study of differential equations has three principal goals:

1 To discover the differential equation that describes a specified physical

situation

2 To find—either exactly or approximately—the appropriate solution of that

equation

3 To interpret the solution that is found.

In algebra, we typically seek the unknown numbers that satisfy an equation

such asx3

C 7x2

11x C 41 D 0 By contrast, in solving a differential equation, we

1

are challenged to find the unknown functionsy D y.x/for which an identity such

asy0.x/ D 2xy.x/—that is, the differential equation

dy

dx D 2xy

—holds on some interval of real numbers Ordinarily, we will want to find all

solutions of the differential equation, if possible

Example 2 IfC is a constant and

for allx In particular, Eq (1) defines an infinite family of different solutions of this

differen-tial equation, one for each choice of the arbitrary constantC By the method of separation ofvariables (Section 1.4) it can be shown that every solution of the differential equation in (2)

is of the form in Eq (1)

Differential Equations and Mathematical Models

The following three examples illustrate the process of translating scientific laws andprinciples into differential equations In each of these examples the independentvariable is timet, but we will see numerous examples in which some quantity otherthan time is the independent variable

Example 3 Rate of cooling Newton’s law of cooling may be stated in this way: The time rate of change

(the rate of change with respect to timet) of the temperatureT t /of a body is proportional

to the difference betweenT and the temperatureAof the surrounding medium (Fig 1.1.1).That is,

d T

wherekis a positive constant Observe that ifT > A, thend T=dt < 0, so the temperature is

a decreasing function oftand the body is cooling But ifT < A, thend T=dt > 0, so thatT

is increasing

Thus the physical law is translated into a differential equation If we are given thevalues ofkandA, we should be able to find an explicit formula forT t /, and then—with theaid of this formula—we can predict the future temperature of the body

Temperature T

Temperature A

cooling, Eq (3), describes the cooling

of a hot rock in water.

Example 4 Draining tank Torricelli’s law implies that the time rate of change of the volume V of

water in a draining tank (Fig 1.1.2) is proportional to the square root of the depthyof water

draining, Eq (4), describes the

draining of a water tank.

Example 5 Population growth The time rate of change of a populationP t /with constant birth and

death rates is, in many simple cases, proportional to the size of the population That is,

dP

wherekis the constant of proportionality

Let us discuss Example 5 further Note first that each function of the form

dP=dt D kP has infinitely many different solutions of the formP t / D C ek t

, one foreach choice of the “arbitrary” constantC This is typical of differential equations

It is also fortunate, because it may allow us to use additional information to selectfrom among all these solutions a particular one that fits the situation under study

Example 6 Population growth Suppose thatP t / D C e k t

is the population of a colony of bacteria attimet, that the population at timet D 0(hours, h) was 1000, and that the population doubledafter1h This additional information aboutP t /yields the following equations:

1000 D P 0/ D C e 0

D C;

2000 D P 1/ D C e k

:

It follows thatC D 1000and thate k

D 2, sok Dln2  0:693147 With this value ofkthedifferential equation in (6) is

P 1:5/ D 1000
2 3=2

 2828:

The conditionP 0/ D 1000in Example 6 is called an initial condition because

we frequently write differential equations for whicht D 0 is the “starting time.”Figure 1.1.3 shows several different graphs of the formP t / D C ek t

withk Dln2

The graphs of all the infinitely many solutions of dP=dt D kP in fact fill the entiretwo-dimensional plane, and no two intersect Moreover, the selection of any onepointP0 on theP-axis amounts to a determination ofP 0/ Because exactly onesolution passes through each such point, we see in this case that an initial condition

P 0/ D P determines a unique solution agreeing with the given data

Mathematical Models

Our brief discussion of population growth in Examples 5 and 6 illustrates the crucial

process of mathematical modeling (Fig 1.1.4), which involves the following:

1 The formulation of a real-world problem in mathematical terms; that is, the

construction of a mathematical model

2 The analysis or solution of the resulting mathematical problem.

3 The interpretation of the mathematical results in the context of the original

real-world situation—for example, answering the question originally posed

Real-world situation

Mathematical model

Mathematical results

Mathematical analysis Formulation Interpretation

In the population example, the real-world problem is that of determining the

population at some future time A mathematical model consists of a list of

vari-ables (Pandt) that describe the given situation, together with one or more equations

relating these variables (dP=dt D kP,P 0/ D P0) that are known or are assumed tohold The mathematical analysis consists of solving these equations (here, forP as

a function oft) Finally, we apply these mathematical results to attempt to answerthe original real-world question

As an example of this process, think of first formulating the mathematical

model consisting of the equations dP=dt D kP,P 0/ D 1000, describing the teria population of Example 6 Then our mathematical analysis there consisted ofsolving for the solution functionP t / D 1000e ln 2/t D 1000
2t

bac-as our ical result For an interpretation in terms of our real-world situation—the actualbacteria population—we substitutedt D 1:5to obtain the predicted population of

mathemat-P 1:5/  2828bacteria after 1.5 hours If, for instance, the bacteria population isgrowing under ideal conditions of unlimited space and food supply, our predictionmay be quite accurate, in which case we conclude that the mathematical model isadequate for studying this particular population

On the other hand, it may turn out that no solution of the selected differential

equation accurately fits the actual population we’re studying For instance, for no

choice of the constantsC andkdoes the solutionP t / D C ek t

in Eq (7) accuratelydescribe the actual growth of the human population of the world over the past few

centuries We must conclude that the differential equation dP=dt D kP is inadequatefor modeling the world population—which in recent decades has “leveled off” ascompared with the steeply climbing graphs in the upper half (P > 0) of Fig 1.1.3.With sufficient insight, we might formulate a new mathematical model including

a perhaps more complicated differential equation, one that takes into account suchfactors as a limited food supply and the effect of increased population on birth anddeath rates With the formulation of this new mathematical model, we may attempt

to traverse once again the diagram of Fig 1.1.4 in a counterclockwise manner If

we can solve the new differential equation, we get new solution functions to

com-pare with the real-world population Indeed, a successful population analysis mayrequire refining the mathematical model still further as it is repeatedly measuredagainst real-world experience.

But in Example 6 we simply ignored any complicating factors that might fect our bacteria population This made the mathematical analysis quite simple,perhaps unrealistically so A satisfactory mathematical model is subject to two con-tradictory requirements: It must be sufficiently detailed to represent the real-worldsituation with relative accuracy, yet it must be sufficiently simple to make the math-ematical analysis practical If the model is so detailed that it fully represents thephysical situation, then the mathematical analysis may be too difficult to carry out

af-If the model is too simple, the results may be so inaccurate as to be useless Thusthere is an inevitable tradeoff between what is physically realistic and what is math-ematically possible The construction of a model that adequately bridges this gapbetween realism and feasibility is therefore the most crucial and delicate step inthe process Ways must be found to simplify the model mathematically withoutsacrificing essential features of the real-world situation

Mathematical models are discussed throughout this book The remainder ofthis introductory section is devoted to simple examples and to standard terminologyused in discussing differential equations and their solutions

Examples and Terminology

Example 7 IfC is a constant andy.x/ D 1=.C x/, then

dy

dx D

1 C x/ 2 D y 2

on any interval of real numbers not containing the pointx D C Actually, Eq (8) defines a

one-parameter familyof solutions ofdy=dx D y 2

, one for each value of the arbitrary constant

or “parameter”C WithC D 1we get the particular solution

The fact that we can write a differential equation is not enough to guaranteethat it has a solution For example, it is clear that the differential equation

.y0/2

C y2

has no (real-valued) solution, because the sum of nonnegative numbers cannot be

negative For a variation on this theme, note that the equation

The order of a differential equation is the order of the highest derivative that

appears in it The differential equation of Example 8 is of second order, those inExamples 2 through 7 are first-order equations, and

is a fourth-order equation The most general form of an nth-order differential

equation with independent variablex and unknown function or dependent variable

y D y.x/is

Fx; y; y0; y00; : : : ; y.n/

whereF is a specific real-valued function ofn C 2variables

Our use of the word solution has been until now somewhat informal To be

precise, we say that the continuous functionu D u.x/is a solution of the differential equation in (13) on the intervalI provided that the derivativesu0,u00,: : :,u.n/

Remark Recall from elementary calculus that a differentiable function on an open interval

is necessarily continuous there This is why only a continuous function can qualify as a(differentiable) solution of a differential equation on an interval

Example 7 Continued Figure 1.1.5 shows the two “connected” branches of the graphy D 1=.1 x/

The left-hand branch is the graph of a (continuous) solution of the differential equationy0D

y 2

that is defined on the interval 1; 1/ The right-hand branch is the graph of a different

solution of the differential equation that is defined (and continuous) on the different interval

.1; 1/ So the single formulay.x/ D 1=.1 x/actually defines two different solutions (withdifferent domains of definition) of the same differential equationy0D y 2

Example 9 IfAandBare constants and

y.x/ D Acos3x C Bsin3x; (14)then two successive differentiations yield

y0.x/ D 3Asin3x C 3Bcos3x;

y00.x/ D 9Acos3x 9Bsin3x D 9y.x/

for allx Consequently, Eq (14) defines what it is natural to call a two-parameter family of

solutions of the second-order differential equation

on the whole real number line Figure 1.1.6 shows the graphs of several such solutions

Although the differential equations in (11) and (12) are exceptions to the eral rule, we will see that an nth-order differential equation ordinarily has an n-parameter family of solutions—one involvingndifferent arbitrary constants or pa-

and y 3 x/ D 3 cos 3x C 2sin 3x of

the differential equation y 00 C 9y D 0.

rameters

In both Eqs (11) and (12), the appearance ofy0as an implicitly defined tion causes complications For this reason, we will ordinarily assume that any dif-ferential equation under study can be solved explicitly for the highest derivative that

func-appears; that is, that the equation can be written in the so-called normal form

All the differential equations we have mentioned so far are ordinary

differ-ential equations, meaning that the unknown function (dependent variable) depends

on only a single independent variable If the dependent variable is a function of

two or more independent variables, then partial derivatives are likely to be involved;

if they are, the equation is called a partial differential equation For example, the

temperatureu D u.x; t /of a long thin uniform rod at the pointxat timet satisfies(under appropriate simple conditions) the partial differential equation

refer to them simply as differential equations

In this chapter we concentrate on first-order differential equations of the form

dy

We also will sample the wide range of applications of such equations A typical

mathematical model of an applied situation will be an initial value problem, sisting of a differential equation of the form in (17) together with an initial condi-

con-tiony.x0/ D y0 Note that we cally.x0/ D y0 an initial condition whether or not

x0D 0 To solve the initial value problem

dy

means to find a differentiable functiony D y.x/that satisfies both conditions in

Eq (18) on some interval containingx0

Example 10 Given the solutiony.x/ D 1=.C x/of the differential equationdy=dx D y 2

discussed inExample 7, solve the initial value problem

dy

dx D y

2

; y.1/ D 2:

Solution We need only find a value ofC so that the solution y.x/ D 1=.C x/satisfies the initial

conditiony.1/ D 2 Substitution of the valuesx D 1andy D 2in the given solution yields

2 D y.1/ D 1

C 1;

2 ; 1/

of the solution of the different initial value problemy0D y 2

,y.2/ D 2

The central question of greatest immediate interest to us is this: If we are given

a differential equation known to have a solution satisfying a given initial condition,

how do we actually find or compute that solution? And, once found, what can we do

with it? We will see that a relatively few simple techniques—separation of variables(Section 1.4), solution of linear equations (Section 1.5), elementary substitutionmethods (Section 1.6)—are enough to enable us to solve a variety of first-orderequations having impressive applications

1.1 Problems

In Problems 1 through 12, verify by substitution that each

given function is a solution of the given differential equation.

Throughout these problems, primes denote derivatives with

In Problems 13 through 16, substitutey D e r x

into the given differential equation to determine all values of the constantr

for whichy D e r x

is a solution of the equation.

13. 3y0D 2y 14. 4y00D y

15. y00C y0 2y D 0 16. 3y00C 3y0 4y D 0

In Problems 17 through 26, first verify thaty.x/satisfies the

given differential equation Then determine a value of the

con-stantC so thaty.x/satisfies the given initial condition Use a

computer or graphing calculator (if desired) to sketch several

typical solutions of the given differential equation, and

high-light the one that satisfies the given initial condition.

26. y0C ytanx Dcosx;y.x/ D x C C /cosx,y./ D 0

In Problems 27 through 31, a functiony D g.x/is described

by some geometric property of its graph Write a differential equation of the formdy=dx D f x; y/having the functiongas its solution (or as one of its solutions).

27 The slope of the graph ofgat the point.x; y/is the sum

functiongmight look like?

30 The graph of g is normal to every curve of the form

y D x 2

C k(kis a constant) where they meet

31 The line tangent to the graph ofgat.x; y/passes throughthe point y; x/

Differential Equations as Models

In Problems 32 through 36, write—in the manner of Eqs (3) through (6) of this section—a differential equation that is a mathematical model of the situation described.

32 The time rate of change of a populationP is proportional

to the square root ofP

33 The time rate of change of the velocityv of a coastingmotorboat is proportional to the square ofv

34 The accelerationdv=dt of a Lamborghini is proportional

to the difference between250km=h and the velocity of thecar

35 In a city having a fixed population ofP persons, the time

rate of change of the numberN of those persons who have

heard a certain rumor is proportional to the number of

those who have not yet heard the rumor

36 In a city with a fixed population ofPpersons, the time rate

of change of the numberN of those persons infected with

a certain contagious disease is proportional to the product

of the number who have the disease and the number who

do not

In Problems 37 through 42, determine by inspection at least

one solution of the given differential equation That is, use

your knowledge of derivatives to make an intelligent guess.

Then test your hypothesis.

43 (a) Ifkis a constant, show that a general (one-parameter)

solution of the differential equation is given byx.t / D

1=.C k t /, whereC is an arbitrary constant

(b) Determine by inspection a solution of the initial value

Initially, there are P 0/ D

2rodents, and their number is increasing at the rate of

dP=dt D 1rodent per month when there areP D 10dents Based on the result of Problem 43, how long will ittake for this population to grow to a hundred rodents? To

ro-a thousro-and? Whro-at’s hro-appening here?

46 Suppose the velocityvof a motorboat coasting in watersatisfies the differential equationdv=dt D kv 2

The tial speed of the motorboat isv.0/ D 10meters per sec-ond (m=s), andvis decreasing at the rate of 1 m=s2

ini-when

v D 5m=s Based on the result of Problem 43, long does

it take for the velocity of the boat to decrease to 1 m=s?

To 1

10m=s? When does the boat come to a stop?

47 In Example 7 we saw thaty.x/ D 1=.C x/ defines aone-parameter family of solutions of the differential equa-tion dy=dx D y 2

(a) Determine a value of C so that

y.10/ D 10 (b) Is there a value ofC such thaty.0/ D 0?Can you nevertheless find by inspection a solution of

dy=dx D y 2

such thaty.0/ D 0? (c) Figure 1.1.8 shows

typical graphs of solutions of the formy.x/ D 1=.C x/.Does it appear that these solution curves fill the entirexy-plane? Can you conclude that, given any point.a; b/inthe plane, the differential equationdy=dx D y 2

has actly one solutiony.x/satisfying the conditiony.a/ D b?

ex-48 (a) Show thaty.x/ D C x 4

defines a one-parameter ily of differentiable solutions of the differential equation

fam-xy0D 4y(Fig 1.1.9) (b) Show that

(c) Given any two real

num-bersaandb, explain why—in contrast to the situation inpart (c) of Problem 47—there exist infinitely many differ-entiable solutions ofxy0D 4ythat all satisfy the condition

100 80 60 40 20 0

–100 –80 –60 – 40 –20

various values of C

1.2 Integrals as General and Particular Solutions

The first-order equation dy=dx D f x; y/ takes an especially simple form if theright-hand-side functionf does not actually involve the dependent variabley, so

This is a general solution of Eq (1), meaning that it involves an arbitrary constant

C, and for every choice ofC it is a solution of the differential equation in (1) If

G.x/is a particular antiderivative off—that is, ifG0.x/  f x/—then

y D1x 2 C C for various values of C

C2 on the same intervalI are “parallel” in the sense illustrated by Figs 1.2.1 and1.2.2 There we see that the constant C is geometrically the vertical distance be-tween the two curvesy.x/ D G.x/andy.x/ D G.x/ C C

y D sin x C C for various values of C

To satisfy an initial conditiony.x0/ D y0, we need only substitutex D x0and

y D y0 into Eq (3) to obtainy0 D G.x0/ C C, so thatC D y0 G.x0/ With thischoice ofC, we obtain the particular solution of Eq (1) satisfying the initial value

problem

dy

dx D f x/; y.x0/ D y0:

We will see that this is the typical pattern for solutions of first-order differential

equations Ordinarily, we will first find a general solution involving an arbitrary

constantC We can then attempt to obtain, by appropriate choice ofC, a particular solutionsatisfying a given initial conditiony.x0/ D y0

Remark As the term is used in the previous paragraph, a general solution of a first-order

differential equation is simply a one-parameter family of solutions A natural question is

whether a given general solution contains every particular solution of the differential

equa-tion When this is known to be true, we call it the general solution of the differential equaequa-tion.

For example, because any two antiderivatives of the same functionf x/can differ only by aconstant, it follows that every solution of Eq (1) is of the form in (2) Thus Eq (2) serves to

define the general solution of (1).

Example 1 Solve the initial value problem

C 3x C C:

Figure 1.2.3 shows the graphy D x 2

C 3x C Cfor various values ofC The particular solution

we seek corresponds to the curve that passes through the point.1; 2/, thereby satisfying theinitial condition

Second-order equations. The observation that the special first-order equation

the differential equation in Example 1.

dy=dx D f x/is readily solvable (provided that an antiderivative off can be found)extends to second-order differential equations of the special form

where G is an antiderivative of g andC1 is an arbitrary constant Then anotherintegration yields

y.x/ D

Z

y0.x/ dx D

ZŒG.x/ C C1 dx D

ZG.x/ dx C C1x C C2;

where C2 is a second arbitrary constant In effect, the second-order differential

equation in (4) is one that can be solved by solving successively the first-order

Velocity and Acceleration

Direct integration is sufficient to allow us to solve a number of important problems

concerning the motion of a particle (or mass point) in terms of the forces acting

on it The motion of a particle along a straight line (thex-axis) is described by its

Equation (6) is sometimes applied either in the indefinite integral formx.t / D

R v.t/ dtor in the definite integral form

where m is the mass of the particle If the force F is known, then the equation

x00.t / D F t /=mcan be integrated twice to find the position functionx.t /in terms

of two constants of integration These two arbitrary constants are frequently

deter-mined by the initial positionx0 D x.0/and the initial velocityv0 D v.0/of theparticle

accelerationa D F=m, are constant Then we begin with the equation

Z.at C v0/ dt D 1

v0, and its initial positionx0

Example 2 Lunar lander A lunar lander is falling freely toward the surface of the moon at a speed

of 450 meters per second (m=s) Its retrorockets, when fired, provide a constant deceleration

of 2.5 meters per second per second (m=s2

) (the gravitational acceleration produced by themoon is assumed to be included in the given deceleration) At what height above the lunarsurface should the retrorockets be activated to ensure a “soft touchdown” (v D 0at impact)?

Solution We denote byx.t /the height of the lunar lander above the surface, as indicated in Fig 1.2.4

We lett D 0denote the time at which the retrorockets should be fired Thenv 0 D 450

(m=s, negative because the height x.t /is decreasing), anda D C2:5, because an upwardthrust increases the velocityv(although it decreases the speedjvj) Then Eqs (10) and (11)

x.t / D 1:25t 2

450t C x 0 ; (13)wherex 0 is the height of the lander above the lunar surface at the timet D 0when theretrorockets should be activated

From Eq (12) we see thatv D 0(soft touchdown) occurs whent D 450=2:5 D 180s(that is,3minutes); then substitution oft D 180,x D 0into Eq (13) yields

Physical Units

Numerical work requires units for the measurement of physical quantities such asdistance and time We sometimes use ad hoc units—such as distance in miles orkilometers and time in hours—in special situations (such as in a problem involving

an auto trip) However, the foot-pound-second (fps) and meter-kilogram-second(mks) unit systems are used more generally in scientific and engineering problems

In fact, fps units are commonly used only in the United States (and a few othercountries), while mks units constitute the standard international system of scientificunits

fps units mks units

ForceMassDistanceTime

g

pound (lb)slugfoot (ft)second (s)

32 ft=s2

newton (N)kilogram (kg)meter (m)second (s)9.8 m=s2

The last line of this table gives values for the gravitational accelerationg atthe surface of the earth Although these approximate values will suffice for mostexamples and problems, more precise values are9:7805m=s2

and32:088ft=s2

(atsea level at the equator)

Both systems are compatible with Newton’s second lawF D ma Thus 1 N

is (by definition) the force required to impart an acceleration of 1 m=s2

1 in.D2.54 cm (exactly) and 1 lb4.448 N:

For instance,

1 ftD12 in. 2:54cm

in D30.48 cm;and it follows that

1 miD5280 ft 30:48cm

ft D160934.4 cm1.609 km:Thus a posted U.S speed limit of 50 mi=h means that—in international terms—thelegal speed limit is about50  1:609  80:45km=h

Vertical Motion with Gravitational Acceleration

The weightW of a body is the force exerted on the body by gravity Substitution of

a D gandF D W in Newton’s second lawF D magives

for the weightW of the massmat the surface of the earth (whereg  32ft=s2

To discuss vertical motion it is natural to choose they-axis as the coordinatesystem for position, frequently withy D 0corresponding to “ground level.” If we

choose the upward direction as the positive direction, then the effect of gravity on a

vertically moving body is to decrease its height and also to decrease its velocityv Ddy=dt Consequently, if we ignore air resistance, then the accelerationa D dv=dtofthe body is given by

dv

This acceleration equation provides a starting point in many problems involvingvertical motion Successive integrations (as in Eqs (10) and (11)) yield the velocityand height formulas

Here,y0denotes the initial (t D 0) height of the body andv0its initial velocity

Example 3 Projectile motion

(a) Suppose that a ball is thrown straight upward from the ground (y 0 D 0) with initialvelocityv 0 D 96(ft=s, so we useg D 32ft=s2

in fps units) Then it reaches its maximumheight when its velocity (Eq (16)) is zero,

(b) If an arrow is shot straight upward from the ground with initial velocityv 0 D 49(m=s,

Figure 1.2.5 shows a northward-flowing river of widthw D 2a The linesx D ˙a

represent the banks of the river and they-axis its center Suppose that the velocity

vRat which the water flows increases as one approaches the center of the river, andindeed is given in terms of distancexfrom the center by

Suppose that a swimmer starts at the point a; 0/on the west bank and swimsdue east (relative to the water) with constant speedvS As indicated in Fig 1.2.5, hisvelocity vector (relative to the riverbed) has horizontal componentvS and verticalcomponentvR Hence the swimmer’s direction angle˛is given by

Example 4 River crossing Suppose that the river is1mile wide and that its midstream velocity is

v 0 D 9mi=h If the swimmer’s velocity isv S D 3mi=h, then Eq (19) takes the form



D 31 2



41 2

 3

C 1 D 2;

so the swimmer drifts 2 miles downstream while he swims1mile across the river

1.2 Problems

In Problems 1 through 10, find a functiony D f x/

satisfy-ing the given differential equation and the prescribed initial

Velocity Given Graphically

In Problems 19 through 22, a particle starts at the origin and travels along thex-axis with the velocity functionv.t /whose graph is shown in Figs 1.2.6 through 1.2.9 Sketch the graph

of the resulting position functionx.t /for0 5 t 5 10.

velocity function v.t / of Problem 19.

velocity function v.t / of Problem 20.

velocity function v.t / of Problem 21.

velocity function of Problem 22.

Problems 23 through 28 explore the motion of projectiles der constant acceleration or deceleration.

un-23 What is the maximum height attained by the arrow of part

(b) of Example 3?

24 A ball is dropped from the top of a building400ft high.How long does it take to reach the ground? With whatspeed does the ball strike the ground?

25 The brakes of a car are applied when it is moving at

100km=h and provide a constant deceleration of10ters per second per second (m=s2

me-) How far does the cartravel before coming to a stop?

26 A projectile is fired straight upward with an initial

veloc-ity of100m=s from the top of a building20m high and

falls to the ground at the base of the building Find (a) its maximum height above the ground; (b) when it passes the top of the building; (c) its total time in the air.

27 A ball is thrown straight downward from the top of a tall

building The initial speed of the ball is 10 m=s It strikesthe ground with a speed of 60 m=s How tall is the build-ing?

28 A baseball is thrown straight downward with an initial

speed of 40 ft=s from the top of the Washington ment (555 ft high) How long does it take to reach theground, and with what speed does the baseball strike theground?

so that for the first 10 s its acceleration is given by

If the car starts from rest (x 0 D 0,v 0 D 0), find the distance

it has traveled at the end of the first 10 s and its velocity atthat time

Problems 30 through 32 explore the relation between the speed

of an auto and the distance it skids when the brakes are plied.

ap-30 A car traveling at 60 mi=h (88 ft=s) skids 176 ft after itsbrakes are suddenly applied Under the assumption thatthe braking system provides constant deceleration, what

is that deceleration? For how long does the skid continue?

31 The skid marks made by an automobile indicated that its

brakes were fully applied for a distance of 75 m before

it came to a stop The car in question is known to have

a constant deceleration of 20 m=s2

under these tions How fast—in km=h—was the car traveling whenthe brakes were first applied?

condi-32 Suppose that a car skids 15 m if it is moving at 50 km=

when the brakes are applied Assuming that the car hasthe same constant deceleration, how far will it skid if it ismoving at 100 km=h when the brakes are applied?

Problems 33 and 34 explore vertical motion on a planet with gravitational acceleration different than the Earth’s.

33 On the planet Gzyx, a ball dropped from a height of 20 ft

hits the ground in 2 s If a ball is dropped from the top of

a 200-ft-tall building on Gzyx, how long will it take to hit

the ground? With what speed will it hit?

34 A person can throw a ball straight upward from the

sur-face of the earth to a maximum height of 144 ft How

high could this person throw the ball on the planet Gzyx

of Problem 33?

rest at an initial heighthabove the surface of the earth

Show that the speed with which it strikes the ground is

v D p2gh

has enough “spring” in her legs to jump (on earth) from

the ground to a height of 2.25 feet If she jumps straight

upward with the same initial velocity on the moon—where

the surface gravitational acceleration is (approximately)

5.3 ft=s2

—how high above the surface will she rise?

37 At noon a car starts from rest at pointAand proceeds at

constant acceleration along a straight road toward point

B If the car reachesBat 12:50P.M with a velocity of

60 mi=h, what is the distance fromAtoB?

38 At noon a car starts from rest at pointAand proceeds with

constant acceleration along a straight road toward pointC,

35 miles away If the constantly accelerated car arrives at

C with a velocity of60mi=h, at what time does it arrive

atC?

39 River crossing Ifa D 0:5mi andv 0 D 9mi=h as in

Ex-ample 4, what must the swimmer’s speedv S be in order

that he drifts only 1 mile downstream as he crosses the

river?

andv S D 3mi=h as in Example 4, but that the velocity of

the river is given by the fourth-degree function

he-licopter hovering at an altitude of 800 feet above theground From the ground directly beneath the helicopter,

a projectile is fired straight upward toward the bomb, actly 2 seconds after the bomb is released With what ini-tial velocity should the projectile be fired in order to hitthe bomb at an altitude of exactly 400 feet?

ex-42 Lunar lander A spacecraft is in free fall toward the face of the moon at a speed of 1000 mph (mi=h) Itsretrorockets, when fired, provide a constant deceleration

sur-of 20,000 mi= 2

At what height above the lunar surfaceshould the astronauts fire the retrorockets to insure a softtouchdown? (As in Example 2, ignore the moon’s gravi-tational field.)

(1963) describes Diana, a spacecraft propelled by the solarwind Its aluminized sail provides it with a constant accel-eration of0:001g D 0:0098m=s2

Suppose this spacecraftstarts from rest at timet D 0and simultaneously fires aprojectile (straight ahead in the same direction) that trav-els at one-tenth of the speed c D 3  10 8

m=s of light.How long will it take the spacecraft to catch up with theprojectile, and how far will it have traveled by then?

he was going only 25 mph When police tested his car,they found that when its brakes were applied at 25 mph,the car skidded only 45 feet before coming to a stop Butthe driver’s skid marks at the accident scene measured

210 feet Assuming the same (constant) deceleration, termine the speed he was actually traveling just prior tothe accident

v.t / 2

v 2

D 2aŒx.t / x 0 for allt when the tion a D dv=dt is constant Then use this “kinematicformula”—commonly presented in introductory physicscourses—to confirm the result of Example 2

accelera-1.3 Slope Fields and Solution Curves

Consider a differential equation of the form

dy

where the right-hand functionf x; y/involves both the independent variablexandthe dependent variabley We might think of integrating both sides in (1) with re-spect tox, and hence writey.x/ DR f x; y.x// dx C C However, this approachdoes not lead to a solution of the differential equation, because the indicated integral

involves the unknown functiony.x/itself, and therefore cannot be evaluated

explic-itly Actually, there exists no straightforward procedure by which a general

differen-tial equation can be solved explicitly Indeed, the solutions of such a simple-lookingdifferential equation asy0D x2

C y2

cannot be expressed in terms of the ordinary

elementary functions studied in calculus textbooks Nevertheless, the graphical and

numerical methods of this and later sections can be used to construct approximate

solutions of differential equations that suffice for many practical purposes

Slope Fields and Graphical Solutions

There is a simple geometric way to think about solutions of a given differentialequationy0 D f x; y/ At each point.x; y/ of the xy-plane, the value off x; y/

determines a slopem D f x; y/ A solution of the differential equation is simply

a differentiable function whose graph y D y.x/has this “correct slope” at eachpoint x; y.x//through which it passes—that is, y0.x/ D f x; y.x// Thus a so-

lution curve of the differential equationy0 D f x; y/—the graph of a solution of

the equation—is simply a curve in the xy-plane whose tangent line at each point

.x; y/has slope m D f x; y/ For instance, Fig 1.3.1 shows a solution curve ofthe differential equationy0 D x y together with its tangent lines at three typicalpoints

y0D x ytogether with tangent lines having

 slopem 1 D x 1 y 1at the point.x 1 ; y 1 /;

 slopem 2 D x 2 y 2at the point.x 2 ; y 2 /; and

 slopem 3 D x 3 y 3at the point.x 3 ; y 3 /

This geometric viewpoint suggests a graphical method for constructing proximatesolutions of the differential equation y0 D f x; y/ Through each of arepresentative collection of points.x; y/in the plane we draw a short line segmenthaving the proper slopem D f x; y/ All these line segments constitute a slope

ap-field (or a direction ap-field) for the equationy0D f x; y/

Example 1 Figures 1.3.2 (a)–(d) show slope fields and solution curves for the differential equation

dy

with the valuesk D 2,0:5, 1, and 3of the parameterkin Eq (2) Note that each slopefield yields important qualitative information about the set of all solutions of the differentialequation For instance, Figs 1.3.2(a) and (b) suggest that each solutiony.x/approaches˙1

asx ! C1ifk > 0, whereas Figs 1.3.2(c) and (d) suggest that y.x/ ! 0asx ! C1

ifk < 0 Moreover, although the sign ofkdetermines the direction of increase or decrease

ofy.x/, its absolute valuejkjappears to determine the rate of change ofy.x/ All this isapparent from slope fields like those in Fig 1.3.2, even without knowing that the generalsolution of Eq (2) is given explicitly byy.x/ D C e k x

A slope field suggests visually the general shapes of solution curves of thedifferential equation Through each point a solution curve should proceed in such

solution curves for y 0 D 2y.

0 1 2 3 4

x

–2 –1 – 4 –3

4 3 2 1 0 –1 –2 –3 – 4

solution curves for y 0 D 0:5/y.

0 1 2 3 4

x

–2 –1 – 4 –3

4 3 2 1 0 –1 –2 –3 – 4

solution curves for y 0 D y

0 1 2 3 4

x

–2 –1 – 4 –3

FIGURE 1.3.2(d) Slope field

and solution curves for y 0 D 3y

FIGURE 1.3.3. Values of the slope y 0 D x y for 4  x; y  4.

a direction that its tangent line is nearly parallel to the nearby line segments of theslope field Starting at any initial point.a; b/, we can attempt to sketch freehand anapproximate solution curve that threads its way through the slope field, followingthe visible line segments as closely as possible

Example 2 Construct a slope field for the differential equationy0D x yand use it to sketch an

approx-imate solution curve that passes through the point 4; 4/

Solution Figure 1.3.3 shows a table of slopes for the given equation The numerical slopem D x y

appears at the intersection of the horizontalx-row and the verticaly-column of the table Ifyou inspect the pattern of upper-left to lower-right diagonals in this table, you can see that itwas easily and quickly constructed (Of course, a more complicated functionf x; y/on theright-hand side of the differential equation would necessitate more complicated calculations.)Figure 1.3.4 shows the corresponding slope field, and Fig 1.3.5 shows an approximate so-lution curve sketched through the point 4; 4/so as to follow this slope field as closely aspossible At each point it appears to proceed in the direction indicated by the nearby linesegments of the slope field

Although a spreadsheet program (for instance) readily constructs a table ofslopes as in Fig 1.3.3, it can be quite tedious to plot by hand a sufficient number

of slope segments as in Fig 1.3.4 However, most computer algebra systems clude commands for quick and ready construction of slope fields with as many linesegments as desired; such commands are illustrated in the application material forthis section The more line segments are constructed, the more accurately solutioncurves can be visualized and sketched Figure 1.3.6 shows a “finer” slope field for

in-0 5 0

5

x

–5 –5

corresponding to the table of slopes in Fig 1.3.3.

0 1 2 3 4 5

x

–1 –2 –3 – 4 –5 –5 (– 4, 4)

through 4; 4/

the differential equationy0 D x y of Example 2, together with typical solutioncurves treading through this slope field

If you look closely at Fig 1.3.6, you may spot a solution curve that appears

to be a straight line! Indeed, you can verify that the linear functiony D x 1 is

a solution of the equationy0 D x y, and it appears likely that the other solutioncurves approach this straight line as an asymptote as x ! C1 This inferenceillustrates the fact that a slope field can suggest tangible information about solutionsthat is not at all evident from the differential equation itself Can you, by tracing theappropriate solution curve in this figure, infer thaty.3/  2for the solutiony.x/ofthe initial value problemy0D x y,y 4/ D 4?

typical solution curves for y 0 D x y Applications of Slope Fields

The next two examples illustrate the use of slope fields to glean useful information

in physical situations that are modeled by differential equations Example 3 is based

on the fact that a baseball moving through the air at a moderate speedv(less thanabout 300 ft=s) encounters air resistance that is approximately proportional tov Ifthe baseball is thrown straight downward from the top of a tall building or from ahovering helicopter, then it experiences both the downward acceleration of gravityand an upward acceleration of air resistance If they-axis is directed downward,

then the ball’s velocityv D dy=dtand its gravitational accelerationg D 32ft=s2

areboth positive, while its acceleration due to air resistance is negative Hence its totalacceleration is of the form

dv

A typical value of the air resistance proportionality constant might bek D 0:16

Example 3 Falling baseball Suppose you throw a baseball straight downward from a helicopter

hov-ering at an altitude of 3000 feet You wonder whether someone standing on the ground below

typical solution curves for

v 0 D 32 0:16v

could conceivably catch it In order to estimate the speed with which the ball will land, youcan use your laptop’s computer algebra system to construct a slope field for the differentialequation

dv

The result is shown in Fig 1.3.7, together with a number of solution curves ing to different values of the initial velocityv.0/with which you might throw the baseballdownward Note that all these solution curves appear to approach the horizontal linev D 200

correspond-as an correspond-asymptote This implies that—however you throw it—the bcorrespond-aseball should approach the

limiting velocityv D 200ft=s instead of accelerating indefinitely (as it would in the absence

of any air resistance) The handy fact that 60 mi= D88 ft=s yields

Comment If the ball’s initial velocity is v.0/ D 200, then Eq (4) gives v0.0/ D 32 0:16/.200/ D 0, so the ball experiences no initial acceleration Its velocity therefore remains

unchanged, and hence v.t /  200is a constant “equilibrium solution” of the differentialequation If the initial velocity is greater than 200, then the initial acceleration given by

Eq (4) is negative, so the ball slows down as it falls But if the initial velocity is less than

200, then the initial acceleration given by (4) is positive, so the ball speeds up as it falls Ittherefore seems quite reasonable that, because of air resistance, the baseball will approach alimiting velocity of 200 ft=s—whatever initial velocity it starts with You might like to verifythat—in the absence of air resistance—this ball would hit the ground at over 300 mi=h

In Section 2.1 we will discuss in detail the logistic differential equation

dP

that often is used to model a population P t / that inhabits an environment with

carrying capacityM This means thatM is the maximum population that this ronment can sustain on a long-term basis (in terms of the maximum available food,for instance)

envi-Example 4 Limiting population If we takek D 0:0004andM D 150, then the logistic equation in (5)

takes the form

typical solution curves for

P 0 D 0:06P 0:0004P 2

(with timet measured in years) The negative term 0:0004P 2

represents the inhibition ofgrowth due to limited resources in the environment

Figure 1.3.8 shows a slope field for Eq (6), together with a number of solution curvescorresponding to possible different values of the initial populationP 0/ Note that all thesesolution curves appear to approach the horizontal lineP D 150as an asymptote This impliesthat—whatever the initial population—the populationP t /approaches the limiting popula-

tionP D 150ast ! 1

Comment If the initial population isP 0/ D 150, then Eq (6) gives

P0.0/ D 0:0004.150/.150 150/ D 0;

so the population experiences no initial (instantaneous) change It therefore remains

un-changed, and henceP t /  150is a constant “equilibrium solution” of the differential tion If the initial population is greater than 150, then the initial rate of change given by (6)

equa-is negative, so the population immediately begins to decrease But if the initial population equa-isless than 150, then the initial rate of change given by (6) is positive, so the population imme-diately begins to increase It therefore seems quite reasonable to conclude that the populationwill approach a limiting value of 150—whatever the (positive) initial population

Existence and Uniqueness of Solutions

Before one spends much time attempting to solve a given differential equation, it

is wise to know that solutions actually exist We may also want to know whether

there is only one solution of the equation satisfying a given initial condition—that

is, whether its solutions are unique.

Example 5 (a) Failure of existence The initial value problem

y0D 1

has no solution, because no solutiony.x/ D R 1=x/ dx Dlnjxj C Cof the differential equation

is defined atx D 0 We see this graphically in Fig 1.3.9, which shows a direction field andsome typical solution curves for the equation y0 D 1=x It is apparent that the indicateddirection field “forces” all solution curves near they-axis to plunge downward so that nonecan pass through the point.0; 0/

–1

2

0

1 0

x

(0, 0)

solution curves for the equation y 0 D 1=x.

1 1

0 0

different solution curves for the initial value problem y 0 D 2py , y 0/ D 0.

(b) Failure of uniqueness On the other hand, you can readily verify that the initial valueproblem

y0D 2py; y.0/ D 0 (8)

has the two different solutionsy 1 x/ D x 2

andy 2 x/  0(see Problem 27) Figure 1.3.10shows a direction field and these two different solution curves for the initial value problem in(8) We see that the curvey 1 x/ D x 2

threads its way through the indicated direction field,whereas the differential equationy0D 2pyspecifies slopey0D 0along thex-axisy 2 x/ D 0

Example 5 illustrates the fact that, before we can speak of “the” solution of

an initial value problem, we need to know that it has one and only one solution.

Questions of existence and uniqueness of solutions also bear on the process ofmathematical modeling Suppose that we are studying a physical system whose be-havior is completely determined by certain initial conditions, but that our proposed

mathematical model involves a differential equation not having a unique solution

satisfying those conditions This raises an immediate question as to whether themathematical model adequately represents the physical system

The theorem stated below implies that the initial value problemy0D f x; y/,

y.a/ D bhas one and only one solution defined near the pointx D aon thex-axis,provided that both the functionf and its partial derivative@f =@y are continuousnear the point.a; b/in the xy-plane Methods of proving existence and uniqueness

theorems are discussed in the Appendix

THEOREM 1 Existence and Uniqueness of Solutions

Suppose that both the function f x; y/and its partial derivative Dyf x; y/arecontinuous on some rectangle R in thexy-plane that contains the point a; b/

in its interior Then, for some open intervalI containing the pointa, the initialvalue problem

dy

has one and only one solution that is defined on the intervalI (As illustrated inFig 1.3.11, the solution intervalImay not be as “wide” as the original rectangle

Rof continuity; see Remark 3 below.)

Remark 1 In the case of the differential equation dy=dx D y of Example 1 and

and x -interval I of Theorem 1, and the

solution curve y D y.x/ through the

point a; b/

Fig 1.3.2(c), both the functionf x; y/ D yand the partial derivative@f =@y D 1are tinuous everywhere, so Theorem 1 implies the existence of a unique solution for any initialdata.a; b/ Although the theorem ensures existence only on some open interval containing

con-x D a, each solutiony.x/ D C e x

actually is defined for allx

Remark 2 In the case of the differential equationdy=dx D 2py of Example 5(b) and

Eq (8), the functionf x; y/ D 2pyis continuous wherevery > 0, but the partial derivative

@f =@y D 1=pyis discontinuous wheny D 0, and hence at the point.0; 0/ This is why it ispossible for there to exist two different solutionsy 1 x/ D x 2

andy 2 x/  0, each of whichsatisfies the initial conditiony.0/ D 0

Remark 3 In Example 7 of Section 1.1 we examined the especially simple differentialequationdy=dx D y 2

Here we havef x; y/ D y 2

and@f =@y D 2y Both of these functionsare continuous everywhere in thexy-plane, and in particular on the rectangle 2 < x < 2,

0 < y < 2 Because the point.0; 1/lies in the interior of this rectangle, Theorem 1 guarantees

a unique solution—necessarily a continuous function—of the initial value problem

through the initial point 0; 1/ leaves

the rectangle R before it reaches the

Applying Theorem 1 withf x; y/ D 2y=xand@f =@y D 2=x, we conclude that Eq (11) must

have a unique solution near any point in the xy-plane wherex 6D 0 Indeed, we see ately by substitution in (11) that

immedi-y.x/ D C x 2

(12)

satisfies Eq (11) for any value of the constant C and for all values of the variablex Inparticular, the initial value problem

many solution curves through the point

.0; 0/ , but no solution curves through

the point 0; b/ if b 6D 0.

xdy

dx D 2y; y.0/ D 0 (13)

has infinitely many different solutions, whose solution curves are the parabolasy D C x 2

illustrated in Fig 1.3.13 (In caseC D 0the “parabola” is actually thex-axisy D 0.)Observe that all these parabolas pass through the origin.0; 0/, but none of them passesthrough any other point on they-axis It follows that the initial value problem in (13) hasinfinitely many solutions, but the initial value problem

xdy

dx D 2y; y.0/ D b (14)

has no solution ifb 6D 0.Finally, note that through any point off they-axis there passes only one of the parabolas

y D C x 2

Hence, ifa 6D 0, then the initial value problem

xdy

dx D 2y; y.a/ D b (15)

has a unique solution on any interval that contains the pointx D abut not the originx D 0

In summary, the initial value problem in (15) has

 a unique solution near.a; b/ifa 6D 0;

 no solution ifa D 0butb 6D 0;

 infinitely many solutions ifa D b D 0

Still more can be said about the initial value problem in (15) Consider a cal initial point off they-axis—for instance the point 1; 1/indicated in Fig 1.3.14.Then for any value of the constantC the function defined by

many solution curves through the point

.1; 1/

half of the parabolay D x2

and the right half of the parabolay D C x2

Thus theunique solution curve near 1; 1/branches at the origin into the infinitely manysolution curves illustrated in Fig 1.3.14

We therefore see that Theorem 1 (if its hypotheses are satisfied) guaranteesuniqueness of the solution near the initial point.a; b/, but a solution curve through

.a; b/may eventually branch elsewhere so that uniqueness is lost Thus a solutionmay exist on a larger interval than one on which the solution is unique For instance,the solutiony.x/ D x2

of the initial value problem in (17) exists on the wholex-axis,but this solution is unique only on the negativex-axis 1 < x < 0

1.3 Problems

In Problems 1 through 10, we have provided the slope field of

the indicated differential equation, together with one or more

solution curves Sketch likely solution curves through the

ad-ditional points marked in each slope field.

FIGURE 1.3.20.