REVIEW OF DIFFERENTIATION - WordPress.com · REVIEW OF DIFFERENTIATION. BRIEF TABLE OF INTEGRALS 1....

Rules

1. Constant: ddx

c = 0 2. Constant Multiple: ddx

cf (x) = c f (x)

. Sum: ddx

[ f (x) g(x)]=

f (x) g (x) 4. Product: ddx

f (x)g(x) = f (x) g (x) + g(x) f (x)

5. Quotient: ddx

f (x)g(x)

= g(x)f (x) f (x)g (x)[g(x)]2

6. Chain: ddx

f (g(x)) = f (g(x)) g (x)

7. Power: ddx

xn = nxn1 8. Power: ddx

[g(x)]n = n[g(x)]n 1 g (x)

Functions

Trigonometric:

9. ddx

sin x = cos x 10. ddx

cos x = sin x 11. ddx

tan x = sec2 x

12.ddx

cot x = csc2 x 13. ddx

sec x = sec x tan x 14. ddx

csc x = csc x cot x

Inverse trigonometric:

15. ddx

sin1 x = 1

1 x216.

ddx

cos1 x = 1

1 x217.

ddx

tan1 x = 11 + x2

18.ddx

cot1 x = 11 + x2

19.ddx

sec1 x = 1

x x2 120.

ddx

csc1 x = 1

x x2 1

Hyperbolic:

21. ddx

sinh x = cosh x 22. ddx

cosh x = sinh x 23. ddx

tanh x = sech2 x

24.ddx

coth x = csch2 x 25. ddx

sech x = sech x tanh x 26. ddx

csch x = csch x coth x

Inverse hyperbolic:

27. ddx

sinh1 x = 1

x2 +128.

ddx

cosh1 x = 1

x2 129.

ddx

tanh1 x = 11 x2

30.ddx

coth1 x = 11 x2

31.ddx

sech1 x = 1

x 1 x232.

ddx

csch1 x = 1

x x2 +1

Exponential:

33. ddx

ex = ex 34. ddx

bx = bx (ln b)

Logarithmic:

35. ddx

ln x = 1x

36.ddx

logb x =1

x(ln b)

3

REVIEW OF DIFFERENTIATION

BRIEF TABLE OF INTEGRALS

1. 1

, 11

nn uu du C n

n

2.

1lndu u C

u

3. u ue du e C 4. 1lnu ua du a C

a

5. sin cosu du u C 6. cos sinu du u C 7. 2sec tanu du u C 8. 2csc cotu du u C 9. sec tan secu u du u C 10. csc cot cscu u du u C

11. tan ln cosu du u C 12. cot ln sinu du u C 13. sec ln sec tanu du u u C 14. csc ln csc cotu du u u C 15. sin sin cosu u du u u u C 16. cos cos sinu u du u u u C 17. 2 1 1

2 4sin sin 2u du u u C 18. 2 1 12 4cos sin 2u du u u C

19. 2tan tanu du u u C 20. 2cot cotu du u u C 21. 3 213sin 2 sin cosu du u u C 22. 3 213cos 2 cos sinu du u u C 23. 3 21

2tan tan ln cosu du u u C 24. 3 212cot cot ln sinu du u u C

25. 3 1 12 2

sec sec tan ln sec tanu du u u u u C 26. 3 1 12 2csc csc cot ln csc cotu du u u u u C 27. sin( ) sin( )sin cos

2( ) 2( )

a b u a b uau bu du Ca b a b

28.

sin( ) sin( )cos cos

2( ) 2( )

a b u a b uau bu du Ca b a b

29. 2 2

sin sin cosau

au ee bu du a bu b bu Ca b

30. 2 2cos cos sin

auau ee bu du a bu b bu C

a b

31. sinh coshu du u C 32. cosh sinhu du u C 33. 2sech tanhu du u C 34. 2csch cothu du u C 35. tanh ln(cosh )u du u C 36. coth ln sinhu du u C 37. ln lnu du u u u C 38. 2 21 12 4ln lnu u du u u u C 39. 1

2 2

1sin

udu Caa u

40.

2 2

2 2

1lndu u a u C

a u

41.

22 2 2 2 1sin

2 2

u a ua u du a u Ca

42. 2

2 2 2 2 2 2ln2 2

u aa u du a u u a u C 43. 1

2 2

1 1tan

udu Ca aa u

44. 2 2

1 1ln

2

a udu Ca a ua u

Note: Some techniques of integration, such as integration by parts and partial fractions, are reviewed in the Student Resource and Solutions Manual that accompanies this text.

TABLE OF LAPLACE TRANSFORMS

f (t)

1. 1

2. t

3. tn n a positive integer

4. t1/2

5. t1/2

6. ta

7. sin kt

8. cos kt

9. sin2 kt

10. cos2 kt

11. eat

12. sinh kt

13. cosh kt

14. sinh2kt

15. cosh2kt

16. teat

17. tn eat n a positive integer

18. eat sin kt

19. eat cos kts a

(s a)2 k2

k

(s a)2 k2

n!

(s a)n1,

1

(s a)2

s2 2k2

s(s2 4k2)

2k2

s(s2 4k2)

s

s2 k2

k

s2 k2

1

s a

s2 2k2

s(s2 4k2)

2k2

s(s2 4k2)

s

s2 k2

k

s2 k2

( 1)

s1, a 1

12s3/2

B

s

n!

sn1,

1

s2

1

s

{ f (t)} F(s) f (t)

20. eat sinh kt

21. eat cosh kt

22. t sin kt

23. t cos kt

24. sin kt kt cos kt

25. sin kt kt cos kt

26. t sinh kt

27. t cosh kt

28.

29.

30. 1 cos kt

31. kt sin kt

32.

33.

34. sin kt sinh kt

35. sin kt cosh kt

36. cos kt sinh kt

37. cos kt cosh kt

38. J0(kt)1

1s2 k2

s3

s4 4k4

k(s2 2k2)

s4 4k4

k(s2 2k2)

s4 4k4

2k2s

s4 4k4

s

(s2 a2)(s2 b2)

cos bt cos at

a2 b2

1

(s2 a2)(s2 b2)

a sin bt b sin at

ab (a2 b2)

k3

s2(s2 k2)

k2

s(s2 k2)

s

(s a)(s b)

aeat bebt

a b

1

(s a)(s b)

eat ebt

a b

s2 k2

(s2 k2)2

2ks

(s2 k2)2

2k3

(s2 k2)2

2ks2

(s2 k2)2

s2 k2

(s2 k2)2

2ks

(s2 k2)2

s a

(s a)2 k2

k

(s a)2 k2

{ f (t)} F(s)

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SEVENTH EDITION

DIFFERENTIALEQUATIONS with Boundary-Value Problems

SEVENTH EDITION

DIFFERENTIALEQUATIONSwith Boundary-Value Problems

DENNIS G. ZILLLoyola Marymount University

MICHAEL R. CULLENLate of Loyola Marymount University

Australia Brazil Japan Korea Mexico Singapore Spain United Kingdom United States

Differential Equations withBoundary-Value Problems,Seventh EditionDennis G. Zill and Michael R. Cullen

Executive Editor: Charlie Van Wagner

Development Editor: Leslie Lahr

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Compositor: ICC Macmillan Inc.

2009, 2005 Brooks/Cole, Cengage Learning

ALL RIGHTS RESERVED. No part of this work covered by thecopyright herein may be reproduced, transmitted, stored, or usedin any form or by any means graphic, electronic, or mechanical,including but not limited to photocopying, recording, scanning,digitizing, taping, Web distribution, information networks, orinformation storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher.

Library of Congress Control Number: 2008924835ISBN-13: 978-0-495-10836-8ISBN-10: 0-495-10836-7

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3

v

CONTENTS

1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1

Preface xi

1.1 Definitions and Terminology 2

1.2 Initial-Value Problems 13

1.3 Differential Equations as Mathematical Models 19

CHAPTER 1 IN REVIEW 32

2 FIRST-ORDER DIFFERENTIAL EQUATIONS 34

2.1 Solution Curves Without a Solution 35

2.1.1 Direction Fields 35

2.1.2 Autonomous First-Order DEs 37

2.2 Separable Variables 44

2.3 Linear Equations 53

2.4 Exact Equations 62

2.5 Solutions by Substitutions 70

2.6 A Numerical Method 75


MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS 82

3.1 Linear Models 83

3.2 Nonlinear Models 94

3.3 Modeling with Systems of First-Order DEs 105


5

4

vi CONTENTS

HIGHER-ORDER DIFFERENTIAL EQUATIONS 117

4.1 Preliminary TheoryLinear Equations 118

4.1.1 Initial-Value and Boundary-Value Problems 118

4.1.2 Homogeneous Equations 120

4.1.3 Nonhomogeneous Equations 125

4.2 Reduction of Order 130

4.3 Homogeneous Linear Equations with Constant Coefficients 133

4.4 Undetermined CoefficientsSuperposition Approach 140

4.5 Undetermined CoefficientsAnnihilator Approach 150

4.6 Variation of Parameters 157

4.7 Cauchy-Euler Equation 162

4.8 Solving Systems of Linear DEs by Elimination 169

4.9 Nonlinear Differential Equations 174


MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS 181

5.1 Linear Models: Initial-Value Problems 182

5.1.1 Spring/Mass Systems: Free Undamped Motion 182

5.1.2 Spring/Mass Systems: Free Damped Motion 186

5.1.3 Spring/Mass Systems: Driven Motion 189

5.1.4 Series Circuit Analogue 192

5.2 Linear Models: Boundary-Value Problems 199

5.3 Nonlinear Models 207


SERIES SOLUTIONS OF LINEAR EQUATIONS 219

6.1 Solutions About Ordinary Points 220

6.1.1 Review of Power Series 220

6.1.2 Power Series Solutions 223

6.2 Solutions About Singular Points 231

6.3 Special Functions 241

6.3.1 Bessels Equation 241

6.3.2 Legendres Equation 248


6

CONTENTS vii

7 THE LAPLACE TRANSFORM 255

7.1 Definition of the Laplace Transform 256

7.2 Inverse Transforms and Transforms of Derivatives 262

7.2.1 Inverse Transforms 262

7.2.2 Transforms of Derivatives 265

7.3 Operational Properties I 270

7.3.1 Translation on the s-Axis 271

7.3.2 Translation on the t-Axis 274

7.4 Operational Properties II 282

7.4.1 Derivatives of a Transform 282

7.4.2 Transforms of Integrals 283

7.4.3 Transform of a Periodic Function 287

7.5 The Dirac Delta Function 292

7.6 Systems of Linear Differential Equations 295


8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS 303

8.1 Preliminary TheoryLinear Systems 304

8.2 Homogeneous Linear Systems 311

8.2.1 Distinct Real Eigenvalues 312

8.2.2 Repeated Eigenvalues 315

8.2.3 Complex Eigenvalues 320

8.3 Nonhomogeneous Linear Systems 326

8.3.1 Undetermined Coefficients 326

8.3.2 Variation of Parameters 329

8.4 Matrix Exponential 334


9 NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 339

9.1 Euler Methods and Error Analysis 340

9.2 Runge-Kutta Methods 345

9.3 Multistep Methods 350

9.4 Higher-Order Equations and Systems 353

9.5 Second-Order Boundary-Value Problems 358


viii CONTENTS

10 PLANE AUTONOMOUS SYSTEMS 363

10.1 Autonomous Systems 364

10.2 Stability of Linear Systems 370

10.3 Linearization and Local Stability 378

10.4 Autonomous Systems as Mathematical Models 388


11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 397

11.1 Orthogonal Functions 398

11.2 Fourier Series 403

11.3 Fourier Cosine and Sine Series 408

11.4 Sturm-Liouville Problem 416

11.5 Bessel and Legendre Series 423

11.5.1 Fourier-Bessel Series 424

11.5.2 Fourier-Legendre Series 427


12 BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES 432

12.1 Separable Partial Differential Equations 433

12.2 Classical PDEs and Boundary-Value Problems 437

12.3 Heat Equation 443

12.4 Wave Equation 445

12.5 Laplaces Equation 450

12.6 Nonhomogeneous Boundary-Value Problems 455

12.7 Orthogonal Series Expansions 461

12.8 Higher-Dimensional Problems 466


CONTENTS ix

13 BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS 471

13.1 Polar Coordinates 472

13.2 Polar and Cylindrical Coordinates 477

13.3 Spherical Coordinates 483


14 INTEGRAL TRANSFORMS 488

14.1 Error Function 489

14.2 Laplace Transform 490

14.3 Fourier Integral 498

14.4 Fourier Transforms 504


15 NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS 511

15.1 Laplaces Equation 512

15.2 Heat Equation 517

15.3 Wave Equation 522


APPENDICES

I Gamma Function APP-1

II Matrices APP-3

III Laplace Transforms APP-21

Answers for Selected Odd-Numbered Problems ANS-1

Index I-1

xi

TO THE STUDENT

Authors of books live with the hope that someone actually reads them. Contrary towhat you might believe, almost everything in a typical college-level mathematics textis written for you and not the instructor. True, the topics covered in the text are cho-sen to appeal to instructors because they make the decision on whether to use it intheir classes, but everything written in it is aimed directly at you the student. So Iwant to encourage youno, actually I want to tell youto read this textbook! Butdo not read this text like you would a novel; you should not read it fast and youshould not skip anything. Think of it as a workbook. By this I mean that mathemat-ics should always be read with pencil and paper at the ready because, most likely, youwill have to work your way through the examples and the discussion. Readoops,workall the examples in a section before attempting any of the exercises; the ex-amples are constructed to illustrate what I consider the most important aspects of thesection, and therefore, reflect the procedures necessary to work most of the problemsin the exercise sets. I tell my students when reading an example, cover up the solu-tion; try working it first, compare your work against the solution given, and thenresolve any differences. I have tried to include most of the important steps in eachexample, but if something is not clear you should always tryand here is wherethe pencil and paper come in againto fill in the details or missing steps. This maynot be easy, but that is part of the learning process. The accumulation of facts fol-lowed by the slow assimilation of understanding simply cannot be achieved withouta struggle.

Specifically for you, a Student Resource and Solutions Manual (SRSM) is avail-able as an optional supplement. In addition to containing solutions of selected prob-lems from the exercises sets, the SRSM has hints for solving problems, extra exam-ples, and a review of those areas of algebra and calculus that I feel are particularlyimportant to the successful study of differential equations. Bear in mind you do nothave to purchase the SRSM; by following my pointers given at the beginning of mostsections, you can review the appropriate mathematics from your old precalculus orcalculus texts.

In conclusion, I wish you good luck and success. I hope you enjoy the text andthe course you are about to embark onas an undergraduate math major it was oneof my favorites because I liked mathematics that connected with the physical world.If you have any comments, or if you find any errors as you read/work your waythrough the text, or if you come up with a good idea for improving either it or theSRSM, please feel free to either contact me or my editor at Brooks/Cole PublishingCompany:

[email protected]

TO THE INSTRUCTOR

WHAT IS NEW IN THIS EDITION?

First, let me say what has not changed. The chapter lineup by topics, the number andorder of sections within a chapter, and the basic underlying philosophy remain thesame as in the previous editions.

PREFACE

In case you are examining this text for the first time, Differential Equations withBoundary-Value Problems, 7th Edition, can be used for either a one-semester coursein ordinary differential equations, or a two-semester course covering ordinary andpartial differential equations. The shorter version of the text, A First Course inDifferential Equations with Modeling Applications, 9th Edition, ends withChapter 9. For a one-semester course, I assume that the students have successfullycompleted at least two-semesters of calculus. Since you are reading this, undoubt-edly you have already examined the table of contents for the topics that are covered.You will not find a suggested syllabus in this preface; I will not pretend to be sowise as to tell other teachers what to teach. I feel that there is plenty of material hereto pick from and to form a course to your liking. The text strikes a reasonable bal-ance between the analytical, qualitative, and quantitative approaches to the study ofdifferential equations. As far as my underlying philosophy it is this: An under-graduate text should be written with the students understanding kept firmly inmind, which means to me that the material should be presented in a straightforward,readable, and helpful manner, while keeping the level of theory consistent with thenotion of a first course.

For those who are familiar with the previous editions, I would like to mention afew of the improvements made in this edition.

Contributed Problems Selected exercise sets conclude with one or two con-tributed problems. These problems were class tested and submitted by in-structors of differential equations courses and reflect how they supplementtheir classroom presentations with additional projects.

Exercises Many exercise sets have been updated by the addition of new prob-lems to better test and challenge the students. In like manner, some exercisesets have been improved by sending some problems into early retirement.

Design This edition has been upgraded to a four-color design, which addsdepth of meaning to all of the graphics and emphasis to highlighted phrases.I oversaw the creation of each piece of art to ensure that it is as mathemati-cally correct as the text.

New Figure Numeration It took many editions to do so, but I finally becameconvinced that the old numeration of figures, theorems, and definitions had tobe changed. In this revision I have utilized a double-decimal numeration sys-tem. By way of illustration, in the last edition Figure 7.52 only indicates thatit is the 52nd figure in Chapter 7. In this edition, the same figure is renumberedas Figure 7.6.5, where

Chapter Section

7.6.5 Fifth figure in the section

I feel that this system provides a clearer indication to where things are, with-out the necessity of adding a cumbersome page number.

Projects from Previous Editions Selected projects and essays from pasteditions of the textbook can now be found on the companion website atacademic.cengage.com/math/zill.

STUDENT RESOURCES

Student Resource and Solutions Manual, by Warren S. Wright, Dennis G. Zill,and Carol D. Wright (ISBN 0495385662 (accompanies A First Course inDifferential Equations with Modeling Applications, 9e), 0495383163 (ac-companies Differential Equations with Boundary-Value Problems, 7e)) pro-vides reviews of important material from algebra and calculus, the solution ofevery third problem in each exercise set (with the exception of the DiscussionProblems and Computer Lab Assignments), relevant command syntax forthe computer algebra systems Mathematica and Maple, lists of importantconcepts, as well as helpful hints on how to start certain problems.

;bb

xii PREFACE

PREFACE xiii

DE Tools is a suite of simulations that provide an interactive, visual explo-ration of the concepts presented in this text. Visit academic.cengage.com/math/zill to find out more or contact your local sales representative to askabout options for bundling DE Tools with this textbook.

INSTRUCTOR RESOURCES

Complete Solutions Manual, by Warren S. Wright and Carol D. Wright (ISBN049538609X), provides worked-out solutions to all problems in the text.

Test Bank, by Gilbert Lewis (ISBN 0495386065) Contains multiple-choiceand short-answer test items that key directly to the text.

ACKNOWLEDGMENTS

Compiling a mathematics textbook such as this and making sure that its thousandsof symbols and hundreds of equations are (mostly) accurate is an enormous task, butsince I am called the author that is my job and responsibility. But many peoplebesides myself have expended enormous amounts of time and energy in workingtowards its eventual publication. So I would like to take this opportunity to express mysincerest appreciation to everyonemost of them unknown to meat Brooks/ColePublishing Company, at Cengage Learning, and at Hearthside Publication Serviceswho were involved in the publication of this new edition. I would, however, like tosingle out a few individuals for special recognition: At Brooks/Cole/Cengage,Cheryll Linthicum, Production Project Manager, for her willingness to listen to anauthors ideas and patiently answering the authors many questions; Larry Didonafor the excellent cover designs; Diane Beasley for the interior design; Vernon Boesfor supervising all the art and design; Charlie Van Wagner, sponsoring editor; StacyGreen for coordinating all the supplements; Leslie Lahr, developmental editor, forher suggestions, support, and for obtaining and organizing the contributed prob-lems; and at Hearthside Production Services, Anne Seitz, production editor, whoonce again put all the pieces of the puzzle together. Special thanks go to JohnSamons for the outstanding job he did reviewing the text and answer manuscriptfor accuracy.

I also extend my heartfelt appreciation to those individuals who took the timeout of their busy academic schedules to submit a contributed problem:

Ben Fitzpatrick, Loyola Marymount UniversityLayachi Hadji, University of AlabamaMichael Prophet, University of Northern IowaDoug Shaw, University of Northern IowaWarren S. Wright, Loyola Marymount UniversityDavid Zeigler, California State UniversitySacramento

Finally, over the years these texts have been improved in a countless number ofways through the suggestions and criticisms of the reviewers. Thus it is fitting to con-clude with an acknowledgement of my debt to the following people for sharing theirexpertise and experience.

REVIEWERS OF PAST EDITIONS

William Atherton, Cleveland State UniversityPhilip Bacon, University of FloridaBruce Bayly, University of ArizonaWilliam H. Beyer, University of AkronR.G. Bradshaw, Clarkson CollegeDean R. Brown, Youngstown State UniversityDavid Buchthal, University of AkronNguyen P. Cac, University of Iowa

T. Chow, California State UniversitySacramentoDominic P. Clemence, North Carolina Agricultural

and Technical State UniversityPasquale Condo, University of MassachusettsLowellVincent Connolly, Worcester Polytechnic InstitutePhilip S. Crooke, Vanderbilt UniversityBruce E. Davis, St. Louis Community College at Florissant ValleyPaul W. Davis, Worcester Polytechnic InstituteRichard A. DiDio, La Salle UniversityJames Draper, University of FloridaJames M. Edmondson, Santa Barbara City CollegeJohn H. Ellison, Grove City CollegeRaymond Fabec, Louisiana State UniversityDonna Farrior, University of TulsaRobert E. Fennell, Clemson UniversityW.E. Fitzgibbon, University of HoustonHarvey J. Fletcher, Brigham Young UniversityPaul J. Gormley, VillanovaTerry Herdman, Virginia Polytechnic Institute and State UniversityZdzislaw Jackiewicz, Arizona State UniversityS.K. Jain, Ohio UniversityAnthony J. John, Southeastern Massachusetts UniversityDavid C. Johnson, University of KentuckyLexingtonHarry L. Johnson, V.P.I & S.U.Kenneth R. Johnson, North Dakota State UniversityJoseph Kazimir, East Los Angeles CollegeJ. Keener, University of ArizonaSteve B. Khlief, Tennessee Technological University (retired)C.J. Knickerbocker, St. Lawrence UniversityCarlon A. Krantz, Kean College of New JerseyThomas G. Kudzma, University of LowellG.E. Latta, University of VirginiaCecelia Laurie, University of AlabamaJames R. McKinney, California Polytechnic State UniversityJames L. Meek, University of ArkansasGary H. Meisters, University of NebraskaLincolnStephen J. Merrill, Marquette UniversityVivien Miller, Mississippi State UniversityGerald Mueller, Columbus State Community CollegePhilip S. Mulry, Colgate UniversityC.J. Neugebauer, Purdue UniversityTyre A. Newton, Washington State UniversityBrian M. OConnor, Tennessee Technological UniversityJ.K. Oddson, University of CaliforniaRiversideCarol S. ODell, Ohio Northern UniversityA. Peressini, University of Illinois, UrbanaChampaignJ. Perryman, University of Texas at ArlingtonJoseph H. Phillips, Sacramento City CollegeJacek Polewczak, California State University NorthridgeNancy J. Poxon, California State UniversitySacramentoRobert Pruitt, San Jose State UniversityK. Rager, Metropolitan State CollegeF.B. Reis, Northeastern UniversityBrian Rodrigues, California State Polytechnic UniversityTom Roe, South Dakota State UniversityKimmo I. Rosenthal, Union CollegeBarbara Shabell, California Polytechnic State University

xiv PREFACE

Seenith Sivasundaram, EmbryRiddle Aeronautical UniversityDon E. Soash, Hillsborough Community CollegeF.W. Stallard, Georgia Institute of TechnologyGregory Stein, The Cooper UnionM.B. Tamburro, Georgia Institute of TechnologyPatrick Ward, Illinois Central CollegeWarren S. Wright, Loyola Marymount UniversityJianping Zhu, University of AkronJan Zijlstra, Middle Tennessee State UniversityJay Zimmerman, Towson University

REVIEWERS OF THE CURRENT EDITIONS

Layachi Hadji, University of AlabamaRuben Hayrapetyan, Kettering UniversityAlexandra Kurepa, North Carolina A&T State University

Dennis G. Zill Los Angeles

PREFACE xv

1

1 INTRODUCTION TO DIFFERENTIALEQUATIONS1.1 Definitions and Terminology1.2 Initial-Value Problems1.3 Differential Equations as Mathematical ModelsCHAPTER 1 IN REVIEW

The words differential and equations certainly suggest solving some kind of

equation that contains derivatives y, y, . . . . Analogous to a course in algebra and

trigonometry, in which a good amount of time is spent solving equations such as

x2 5x 4 0 for the unknown number x, in this course one of our tasks will be

to solve differential equations such as y 2y y 0 for an unknown function

y (x).

The preceding paragraph tells something, but not the complete story, about the

course you are about to begin. As the course unfolds, you will see that there is more

to the study of differential equations than just mastering methods that someone has

devised to solve them.

But first things first. In order to read, study, and be conversant in a specialized

subject, you have to learn the terminology of that discipline. This is the thrust of the

first two sections of this chapter. In the last section we briefly examine the link

between differential equations and the real world. Practical questions such as How

fast does a disease spread? How fast does a population change? involve rates of

change or derivatives. As so the mathematical descriptionor mathematical

modelof experiments, observations, or theories may be a differential equation.

DEFINITIONS AND TERMINOLOGY

REVIEW MATERIAL Definition of the derivative Rules of differentiation Derivative as a rate of change First derivative and increasing/decreasing Second derivative and concavity

INTRODUCTION The derivative dydx of a function y (x) is itself another function (x)found by an appropriate rule. The function is differentiable on the interval (, ), andby the Chain Rule its derivative is . If we replace on the right-hand side ofthe last equation by the symbol y, the derivative becomes

. (1)

Now imagine that a friend of yours simply hands you equation (1)you have no idea how it wasconstructedand asks, What is the function represented by the symbol y? You are now face to facewith one of the basic problems in this course:

How do you solve such an equation for the unknown function y (x)?

dy

dx 0.2xy

e0.1x2

dy>dx 0.2xe0.1x2y e0.1x2

2 CHAPTER 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS

1.1

A DEFINITION The equation that we made up in (1) is called a differentialequation. Before proceeding any further, let us consider a more precise definition ofthis concept.

DEFINITION 1.1.1 Differential Equation

An equation containing the derivatives of one or more dependent variables,with respect to one or more independent variables, is said to be a differentialequation (DE).

To talk about them, we shall classify differential equations by type, order, andlinearity.

CLASSIFICATION BY TYPE If an equation contains only ordinary derivatives ofone or more dependent variables with respect to a single independent variable it issaid to be an ordinary differential equation (ODE). For example,

A DE can contain morethan one dependent variable

(2)

are ordinary differential equations. An equation involving partial derivatives ofone or more dependent variables of two or more independent variables is called a

dy

dx 5y ex,

d 2y

dx2

dy

dx 6y 0, and

dx

dt

dy

dt 2x y

b b

partial differential equation (PDE). For example,

(3)

are partial differential equations.*

Throughout this text ordinary derivatives will be written by using either theLeibniz notation dydx, d2ydx2, d3ydx3, . . . or the prime notation y, y, y, . . . .By using the latter notation, the first two differential equations in (2) can be writtena little more compactly as y 5y ex and y y 6y 0. Actually, the primenotation is used to denote only the first three derivatives; the fourth derivative iswritten y(4) instead of y. In general, the nth derivative of y is written dnydxn or y(n).Although less convenient to write and to typeset, the Leibniz notation has an advan-tage over the prime notation in that it clearly displays both the dependent andindependent variables. For example, in the equation

it is immediately seen that the symbol x now represents a dependent variable,whereas the independent variable is t. You should also be aware that in physicalsciences and engineering, Newtons dot notation (derogatively referred to by someas the flyspeck notation) is sometimes used to denote derivatives with respectto time t. Thus the differential equation d2sdt2 32 becomes s 32. Partialderivatives are often denoted by a subscript notation indicating the indepen-dent variables. For example, with the subscript notation the second equation in(3) becomes uxx utt 2ut.

CLASSIFICATION BY ORDER The order of a differential equation (eitherODE or PDE) is the order of the highest derivative in the equation. For example,

is a second-order ordinary differential equation. First-order ordinary differentialequations are occasionally written in differential form M(x, y) dx N(x, y) dy 0.For example, if we assume that y denotes the dependent variable in (y x) dx 4x dy 0, then y dydx, so by dividing by the differential dx, weget the alternative form 4xy y x. See the Remarks at the end of this section.

In symbols we can express an nth-order ordinary differential equation in onedependent variable by the general form

, (4)

where F is a real-valued function of n 2 variables: x, y, y, . . . , y(n). For both prac-tical and theoretical reasons we shall also make the assumption hereafter that it ispossible to solve an ordinary differential equation in the form (4) uniquely for the

F(x, y, y, . . . , y(n)) 0

first ordersecond order

5( )3 4y exdydxd2y

dx2

d 2xdt2

16x 0

unknown functionor dependent variable

independent variable

2u

x2

2u

y2 0,

2u

x2

2u

t2 2

u

t, and

u

y

v

x

1.1 DEFINITIONS AND TERMINOLOGY 3

*Except for this introductory section, only ordinary differential equations are considered in A FirstCourse in Differential Equations with Modeling Applications, Ninth Edition. In that text theword equation and the abbreviation DE refer only to ODEs. Partial differential equations or PDEsare considered in the expanded volume Differential Equations with Boundary-Value Problems,Seventh Edition.

highest derivative y(n) in terms of the remaining n 1 variables. The differentialequation

, (5)

where f is a real-valued continuous function, is referred to as the normal form of (4).Thus when it suits our purposes, we shall use the normal forms

to represent general first- and second-order ordinary differential equations. For example,the normal form of the first-order equation 4xy y x is y (x y)4x; the normalform of the second-order equation y y 6y 0 is y y 6y. See the Remarks.

CLASSIFICATION BY LINEARITY An nth-order ordinary differential equation (4)is said to be linear if F is linear in y, y, . . . , y(n). This means that an nth-order ODE islinear when (4) is an(x)y(n) an1(x)y(n1) a1(x)y a0(x)y g(x) 0 or

. (6)

Two important special cases of (6) are linear first-order (n 1) and linear second-order (n 2) DEs:

. (7)

In the additive combination on the left-hand side of equation (6) we see that the char-acteristic two properties of a linear ODE are as follows:

The dependent variable y and all its derivatives y, y, . . . , y(n) are of thefirst degree, that is, the power of each term involving y is 1.

The coefficients a0, a1, . . . , an of y, y, . . . , y(n) depend at most on theindependent variable x.

The equations

are, in turn, linear first-, second-, and third-order ordinary differential equations. Wehave just demonstrated that the first equation is linear in the variable y by writing it inthe alternative form 4xy y x. A nonlinear ordinary differential equation is sim-ply one that is not linear. Nonlinear functions of the dependent variable or its deriva-tives, such as sin y or , cannot appear in a linear equation. Therefore

are examples of nonlinear first-, second-, and fourth-order ordinary differential equa-tions, respectively.

SOLUTIONS As was stated before, one of the goals in this course is to solve, orfind solutions of, differential equations. In the next definition we consider the con-cept of a solution of an ordinary differential equation.

nonlinear term:coefficient depends on y

nonlinear term:nonlinear function of y

nonlinear term:power not 1

(1 y)y 2y ex, sin y 0, andd 2ydx2

y 2 0d 4ydx 4

ey

(y x)dx 4x dy 0, y 2y y 0, and d 3y

dx3 x

dy

dx 5y ex

a1(x)dy

dx a0(x)y g(x) and a2(x)

d 2y

dx2 a1(x)

dy

dx a0(x)y g(x)

an(x)dny

dxn an1(x)

dn1y

dxn1 a1(x)

dy

dx a0(x)y g(x)

dy

dx f (x, y) and

d 2y

dx2 f (x, y, y)

dny

dxn f (x, y, y, . . . , y(n1))


right-hand side: 0.left-hand side: y 2y y (xex 2ex) 2(xex ex) xex 0,

116 x

4y1/2 14 x2

right-hand side: xy1/2 x 116 x41/2

x 14 x2 1

4x3,

left-hand side: dy

dx

1

16 (4 x3)

1

4x3,

y 2y y 0; y xexdy>dx xy1/2; y 116 x4

y e0.1x2

F(x, (x), (x), . . . , (n)(x)) 0 for all x in I.


DEFINITION 1.1.2 Solution of an ODE

Any function , defined on an interval I and possessing at least n derivativesthat are continuous on I, which when substituted into an nth-order ordinarydifferential equation reduces the equation to an identity, is said to be asolution of the equation on the interval.

In other words, a solution of an nth-order ordinary differential equation (4) is a func-tion that possesses at least n derivatives and for which

We say that satisfies the differential equation on I. For our purposes we shall alsoassume that a solution is a real-valued function. In our introductory discussion wesaw that is a solution of dydx 0.2xy on the interval (, ).

Occasionally, it will be convenient to denote a solution by the alternativesymbol y(x).

INTERVAL OF DEFINITION You cannot think solution of an ordinary differentialequation without simultaneously thinking interval. The interval I in Definition 1.1.2is variously called the interval of definition, the interval of existence, the intervalof validity, or the domain of the solution and can be an open interval (a, b), a closedinterval [a, b], an infinite interval (a, ), and so on.

EXAMPLE 1 Verification of a Solution

Verify that the indicated function is a solution of the given differential equation onthe interval (, ).

(a) (b)

SOLUTION One way of verifying that the given function is a solution is to see, aftersubstituting, whether each side of the equation is the same for every x in the interval.

(a) From

we see that each side of the equation is the same for every real number x. Notethat is, by definition, the nonnegative square root of .

(b) From the derivatives y xex ex and y xex 2ex we have, for every realnumber x,

Note, too, that in Example 1 each differential equation possesses the constant so-lution y 0, x . A solution of a differential equation that is identicallyzero on an interval I is said to be a trivial solution.

SOLUTION CURVE The graph of a solution of an ODE is called a solutioncurve. Since is a differentiable function, it is continuous on its interval I of defini-tion. Thus there may be a difference between the graph of the function and the

graph of the solution . Put another way, the domain of the function need not bethe same as the interval I of definition (or domain) of the solution . Example 2illustrates the difference.

EXAMPLE 2 Function versus Solution

The domain of y 1x, considered simply as a function, is the set of all real num-bers x except 0. When we graph y 1x, we plot points in the xy-plane corre-sponding to a judicious sampling of numbers taken from its domain. The rationalfunction y 1x is discontinuous at 0, and its graph, in a neighborhood of the ori-gin, is given in Figure 1.1.1(a). The function y 1x is not differentiable at x 0,since the y-axis (whose equation is x 0) is a vertical asymptote of the graph.

Now y 1x is also a solution of the linear first-order differential equationxy y 0. (Verify.) But when we say that y 1x is a solution of this DE, wemean that it is a function defined on an interval I on which it is differentiable andsatisfies the equation. In other words, y 1x is a solution of the DE on any inter-val that does not contain 0, such as (3, 1), , (, 0), or (0, ). Becausethe solution curves defined by y 1x for 3 x 1 and are sim-ply segments, or pieces, of the solution curves defined by y 1x for x 0and 0 x , respectively, it makes sense to take the interval I to be as large aspossible. Thus we take I to be either (, 0) or (0, ). The solution curve on (0, )is shown in Figure 1.1.1(b).

EXPLICIT AND IMPLICIT SOLUTIONS You should be familiar with the termsexplicit functions and implicit functions from your study of calculus. A solution inwhich the dependent variable is expressed solely in terms of the independentvariable and constants is said to be an explicit solution. For our purposes, let usthink of an explicit solution as an explicit formula y (x) that we can manipulate,evaluate, and differentiate using the standard rules. We have just seen in the last twoexamples that , y xex, and y 1x are, in turn, explicit solutionsof dydx xy1/2, y 2y y 0, and xy y 0. Moreover, the trivial solu-tion y 0 is an explicit solution of all three equations. When we get down tothe business of actually solving some ordinary differential equations, you willsee that methods of solution do not always lead directly to an explicit solutiony (x). This is particularly true when we attempt to solve nonlinear first-orderdifferential equations. Often we have to be content with a relation or expressionG(x, y) 0 that defines a solution implicitly.

DEFINITION 1.1.3 Implicit Solution of an ODE

A relation G(x, y) 0 is said to be an implicit solution of an ordinarydifferential equation (4) on an interval I, provided that there exists at leastone function that satisfies the relation as well as the differential equationon I.

It is beyond the scope of this course to investigate the conditions under which arelation G(x, y) 0 defines a differentiable function . So we shall assume that ifthe formal implementation of a method of solution leads to a relation G(x, y) 0,then there exists at least one function that satisfies both the relation (that is,G(x, (x)) 0) and the differential equation on an interval I. If the implicit solutionG(x, y) 0 is fairly simple, we may be able to solve for y in terms of x and obtainone or more explicit solutions. See the Remarks.

y 116 x4

12 x 10

(12, 10)


1

x

y

1

(a) function y 1/x, x 0

(b) solution y 1/x, (0, )

1

x

y

1

FIGURE 1.1.1 The function y 1xis not the same as the solution y 1x

EXAMPLE 3 Verification of an Implicit Solution

The relation x2 y2 25 is an implicit solution of the differential equation

(8)

on the open interval (5, 5). By implicit differentiation we obtain

.

Solving the last equation for the symbol dydx gives (8). Moreover, solvingx2 y2 25 for y in terms of x yields . The two functions

and satisfy the relation (that is,x2 12 25 and x2 22 25) and are explicit solutions defined on the interval(5, 5). The solution curves given in Figures 1.1.2(b) and 1.1.2(c) are segments ofthe graph of the implicit solution in Figure 1.1.2(a).

Any relation of the form x2 y2 c 0 formally satisfies (8) for any constant c.However, it is understood that the relation should always make sense in the real numbersystem; thus, for example, if c 25, we cannot say that x2 y2 25 0 is animplicit solution of the equation. (Why not?)

Because the distinction between an explicit solution and an implicit solutionshould be intuitively clear, we will not belabor the issue by always saying, Here isan explicit (implicit) solution.

FAMILIES OF SOLUTIONS The study of differential equations is similar to that ofintegral calculus. In some texts a solution is sometimes referred to as an integralof the equation, and its graph is called an integral curve. When evaluating an anti-derivative or indefinite integral in calculus, we use a single constant c of integration.Analogously, when solving a first-order differential equation F(x, y, y) 0, weusually obtain a solution containing a single arbitrary constant or parameter c. Asolution containing an arbitrary constant represents a set G(x, y, c) 0 of solutionscalled a one-parameter family of solutions. When solving an nth-order differentialequation F(x, y, y, . . . , y(n)) 0, we seek an n-parameter family of solutionsG(x, y, c1, c2, . . . , cn) 0. This means that a single differential equation can possessan infinite number of solutions corresponding to the unlimited number of choicesfor the parameter(s). A solution of a differential equation that is free of arbitraryparameters is called a particular solution. For example, the one-parameter familyy cx x cos x is an explicit solution of the linear first-order equation xy y x2 sin x on the interval (, ). (Verify.) Figure 1.1.3, obtained by using graphing soft-ware, shows the graphs of some of the solutions in this family. The solution y x cos x, the blue curve in the figure, is a particular solution corresponding to c 0.Similarly, on the interval (, ), y c1ex c2xex is a two-parameter family of solu-tions of the linear second-order equation y 2y y 0 in Example 1. (Verify.)Some particular solutions of the equation are the trivial solution y 0 (c1 c2 0),y xex (c1 0, c2 1), y 5ex 2xex (c1 5, c2 2), and so on.

Sometimes a differential equation possesses a solution that is not a member of afamily of solutions of the equationthat is, a solution that cannot be obtained by spe-cializing any of the parameters in the family of solutions. Such an extra solution is calleda singular solution. For example, we have seen that and y 0 are solutions ofthe differential equation dydx xy1/2 on (, ). In Section 2.2 we shall demonstrate,by actually solving it, that the differential equation dydx xy1/2 possesses the one-parameter family of solutions . When c 0, the resulting particularsolution is . But notice that the trivial solution y 0 is a singular solution, sincey 116 x

4y (14 x2 c)2

y 116 x4

y 2(x) 125 x2y 1(x) 125 x2y 225 x2

d

dxx2

d

dxy2

d

dx 25 or 2x 2y

dy

dx 0

dy

dx

x

y


y

x5

5

y

x5

5

y

x

5

5

5

(a) implicit solution

x2 y2 25

(b) explicit solution

y1 25 x2, 5 x 5

(c) explicit solution

y2 25 x2, 5 x 5

(a)

FIGURE 1.1.2 An implicit solutionand two explicit solutions of y xy

FIGURE 1.1.3 Some solutions of xy y x2 sin x

y

x

c>0

c

it is not a member of the family ; there is no way of assigning a value tothe constant c to obtain y 0.

In all the preceding examples we used x and y to denote the independent anddependent variables, respectively. But you should become accustomed to seeingand working with other symbols to denote these variables. For example, we coulddenote the independent variable by t and the dependent variable by x.

EXAMPLE 4 Using Different Symbols

The functions x c1 cos 4t and x c2 sin 4t, where c1 and c2 are arbitrary constantsor parameters, are both solutions of the linear differential equation

For x c1 cos 4t the first two derivatives with respect to t are x 4c1 sin 4tand x 16c1 cos 4t. Substituting x and x then gives

In like manner, for x c2 sin 4t we have x 16c2 sin 4t, and so

Finally, it is straightforward to verify that the linear combination of solutions, or thetwo-parameter family x c1 cos 4t c2 sin 4t, is also a solution of the differentialequation.

The next example shows that a solution of a differential equation can be apiecewise-defined function.

EXAMPLE 5 A Piecewise-Defined Solution

You should verify that the one-parameter family y cx4 is a one-parameter familyof solutions of the differential equation xy 4y 0 on the inverval (, ). SeeFigure 1.1.4(a). The piecewise-defined differentiable function

is a particular solution of the equation but cannot be obtained from the familyy cx4 by a single choice of c; the solution is constructed from the family by choos-ing c 1 for x 0 and c 1 for x 0. See Figure 1.1.4(b).

SYSTEMS OF DIFFERENTIAL EQUATIONS Up to this point we have beendiscussing single differential equations containing one unknown function. Butoften in theory, as well as in many applications, we must deal with systems ofdifferential equations. A system of ordinary differential equations is two or moreequations involving the derivatives of two or more unknown functions of a singleindependent variable. For example, if x and y denote dependent variables and tdenotes the independent variable, then a system of two first-order differentialequations is given by

(9)dy

dt g(t, x, y).

dx

dt f(t, x, y)

y x4, x 0

x4, x 0

x 16x 16c2 sin 4t 16(c2 sin 4t) 0.

x 16x 16c1 cos 4t 16(c1 cos 4t) 0.

x 16x 0.

y (14 x2 c)2


FIGURE 1.1.4 Some solutions of xy 4y 0

(a) two explicit solutions

(b) piecewise-defined solution

c = 1

c = 1x

y

c = 1,x 0

c = 1,x < 0

x

y

A solution of a system such as (9) is a pair of differentiable functions x 1(t),y 2(t), defined on a common interval I, that satisfy each equation of the systemon this interval.

REMARKS

(i) A few last words about implicit solutions of differential equations are inorder. In Example 3 we were able to solve the relation x2 y2 25 fory in terms of x to get two explicit solutions, and

, of the differential equation (8). But dont read too muchinto this one example. Unless it is easy or important or you are instructed to,there is usually no need to try to solve an implicit solution G(x, y) 0 for yexplicitly in terms of x. Also do not misinterpret the second sentence followingDefinition 1.1.3. An implicit solution G(x, y) 0 can define a perfectly gooddifferentiable function that is a solution of a DE, yet we might not be able tosolve G(x, y) 0 using analytical methods such as algebra. The solution curveof may be a segment or piece of the graph of G(x, y) 0. See Problems 45and 46 in Exercises 1.1. Also, read the discussion following Example 4 inSection 2.2.

(ii) Although the concept of a solution has been emphasized in this section,you should also be aware that a DE does not necessarily have to possessa solution. See Problem 39 in Exercises 1.1. The question of whether asolution exists will be touched on in the next section.

(iii) It might not be apparent whether a first-order ODE written in differentialform M(x, y)dx N(x, y)dy 0 is linear or nonlinear because there is nothingin this form that tells us which symbol denotes the dependent variable. SeeProblems 9 and 10 in Exercises 1.1.

(iv) It might not seem like a big deal to assume that F(x, y, y, . . . , y(n)) 0 canbe solved for y(n), but one should be a little bit careful here. There are exceptions,and there certainly are some problems connected with this assumption. SeeProblems 52 and 53 in Exercises 1.1.

(v) You may run across the term closed form solutions in DE texts or inlectures in courses in differential equations. Translated, this phrase usuallyrefers to explicit solutions that are expressible in terms of elementary (orfamiliar) functions: finite combinations of integer powers of x, roots, exponen-tial and logarithmic functions, and trigonometric and inverse trigonometricfunctions.

(vi) If every solution of an nth-order ODE F(x, y, y, . . . , y(n)) 0 on an inter-val I can be obtained from an n-parameter family G(x, y, c1, c2, . . . , cn) 0 byappropriate choices of the parameters ci, i 1, 2, . . . , n, we then say that thefamily is the general solution of the DE. In solving linear ODEs, we shall im-pose relatively simple restrictions on the coefficients of the equation; with theserestrictions one can be assured that not only does a solution exist on an intervalbut also that a family of solutions yields all possible solutions. Nonlinear ODEs,with the exception of some first-order equations, are usually difficult or impos-sible to solve in terms of elementary functions. Furthermore, if we happen toobtain a family of solutions for a nonlinear equation, it is not obvious whetherthis family contains all solutions. On a practical level, then, the designationgeneral solution is applied only to linear ODEs. Dont be concerned aboutthis concept at this point, but store the words general solution in the back ofyour mindwe will come back to this notion in Section 2.3 and again inChapter 4.

2(x) 125 x21(x) 125 x2



EXERCISES 1.1 Answers to selected odd-numbered problems begin on page ANS-1.

In Problems 18 state the order of the given ordinary differ-ential equation. Determine whether the equation is linear ornonlinear by matching it with (6).

1. (1 x)y 4xy 5y cos x

2.

3. t5y(4) t3y 6y 0

4.

5.

6.

7. (sin )y (cos )y 2

8.

In Problems 9 and 10 determine whether the given first-orderdifferential equation is linear in the indicated dependentvariable by matching it with the first differential equationgiven in (7).

9. (y2 1) dx x dy 0; in y; in x

10. u dv (v uv ueu) du 0; in v; in u

In Problems 1114 verify that the indicated function is anexplicit solution of the given differential equation. Assumean appropriate interval I of definition for each solution.

11. 2y y 0; y ex/2

12.

13. y 6y 13y 0; y e3x cos 2x

14. y y tan x; y (cos x)ln(sec x tan x)

In Problems 1518 verify that the indicated function y (x) is an explicit solution of the given first-orderdifferential equation. Proceed as in Example 2, by consider-ing simply as a function, give its domain. Then by consid-ering as a solution of the differential equation, give at leastone interval I of definition.

15. (y x)y y x 8; y x 42x 2

dy

dt 20y 24; y

6

5

6

5e20t

x 1 x. 2

3 x. x 0

d 2R

dt 2

k

R2

d 2y

dx 2 B1

dy

dx2

d 2u

dr 2

du

dr u cos(r u)

xd3y

dx3 dydx

4

y 0

16. y 25 y2; y 5 tan 5x

17. y 2xy2; y 1(4 x2)

18. 2y y3 cos x; y (1 sin x)1/2

In Problems 19 and 20 verify that the indicated expression isan implicit solution of the given first-order differential equa-tion. Find at least one explicit solution y (x) in each case.Use a graphing utility to obtain the graph of an explicit solu-tion. Give an interval I of definition of each solution .

19.

20. 2xy dx (x2 y) dy 0; 2x2y y2 1

In Problems 2124 verify that the indicated family of func-tions is a solution of the given differential equation. Assumean appropriate interval I of definition for each solution.

21.

22.

23.

24.

25. Verify that the piecewise-defined function

is a solution of the differential equation xy 2y 0on (, ).

26. In Example 3 we saw that y 1(x) andare solutions of dydx

xy on the interval (5, 5). Explain why the piecewise-defined function

is not a solution of the differential equation on theinterval (5, 5).

y 125 x2,

125 x2,5 x 0

0 x 5

y 2(x) 125 x2125 x2

y x2, x 0

x2, x 0

y c1x1 c2x c3x ln x 4x2

x3d 3y

dx3 2x2

d 2y

dx2 x

dy

dx y 12x2;

d 2y

dx2 4

dy

dx 4y 0; y c1e2x c2xe2x

dy

dx 2xy 1; y ex2x

0et

2dt c1e

x2

dP

dt P(1 P); P c1et

1 c1et

dX

dt (X 1)(1 2X); ln2X 1X 1 t

In Problems 2730 find values of m so that the functiony emx is a solution of the given differential equation.

27. y 2y 0 28. 5y 2y

29. y 5y 6y 0 30. 2y 7y 4y 0

In Problems 31 and 32 find values of m so that the functiony xm is a solution of the given differential equation.

31. xy 2y 0

32. x2y 7xy 15y 0

In Problems 3336 use the concept that y c, x ,is a constant function if and only if y 0 to determinewhether the given differential equation possesses constantsolutions.

33. 3xy 5y 10

34. y y2 2y 3

35. (y 1)y 1

36. y 4y 6y 10

In Problems 37 and 38 verify that the indicated pair offunctions is a solution of the given system of differentialequations on the interval (, ).

37. 38.

,

Discussion Problems

39. Make up a differential equation that does not possessany real solutions.

40. Make up a differential equation that you feel confidentpossesses only the trivial solution y 0. Explain yourreasoning.

41. What function do you know from calculus is such thatits first derivative is itself? Its first derivative is aconstant multiple k of itself? Write each answer inthe form of a first-order differential equation with asolution.

42. What function (or functions) do you know from calcu-lus is such that its second derivative is itself? Its secondderivative is the negative of itself? Write each answer inthe form of a second-order differential equation with asolution.

y cos 2t sin 2t 15 ety e2t 5e6t

x cos 2t sin 2t 15 etx e2t 3e6t,

d 2y

dt 2 4x et;

dy

dt 5x 3y;

d 2x

dt 2 4y et

dx

dt x 3y


43. Given that y sin x is an explicit solution of the

first-order differential equation . Find

an interval I of definition. [Hint: I is not the interval(, ).]

44. Discuss why it makes intuitive sense to presume thatthe linear differential equation y 2y 4y 5 sin thas a solution of the form y A sin t B cos t, whereA and B are constants. Then find specific constants Aand B so that y A sin t B cos t is a particular solu-tion of the DE.

In Problems 45 and 46 the given figure represents the graphof an implicit solution G(x, y) 0 of a differential equationdydx f (x, y). In each case the relation G(x, y) 0implicitly defines several solutions of the DE. Carefullyreproduce each figure on a piece of paper. Use differentcolored pencils to mark off segments, or pieces, on eachgraph that correspond to graphs of solutions. Keep in mindthat a solution must be a function and differentiable. Usethe solution curve to estimate an interval I of definition ofeach solution .

45.

dy

dx 11 y2

FIGURE 1.1.5 Graph for Problem 45


y

x

1

1

1 x

1

y46.

47. The graphs of members of the one-parameter familyx3 y3 3cxy are called folia of Descartes. Verifythat this family is an implicit solution of the first-orderdifferential equation

dy

dx

y(y3 2x3)

x(2y3 x3).

48. The graph in Figure 1.1.6 is the member of the family offolia in Problem 47 corresponding to c 1. Discuss:How can the DE in Problem 47 help in finding pointson the graph of x3 y3 3xy where the tangent lineis vertical? How does knowing where a tangent line isvertical help in determining an interval I of definitionof a solution of the DE? Carry out your ideas,and compare with your estimates of the intervals inProblem 46.

49. In Example 3 the largest interval I over which theexplicit solutions y 1(x) and y 2(x) are definedis the open interval (5, 5). Why cant the interval I ofdefinition be the closed interval [5, 5]?

50. In Problem 21 a one-parameter family of solutions ofthe DE P P(1 P) is given. Does any solutioncurve pass through the point (0, 3)? Through thepoint (0, 1)?

51. Discuss, and illustrate with examples, how to solvedifferential equations of the forms dydx f (x) andd2ydx2 f (x).

52. The differential equation x(y)2 4y 12x3 0 hasthe form given in (4). Determine whether the equationcan be put into the normal form dydx f (x, y).

53. The normal form (5) of an nth-order differential equa-tion is equivalent to (4) whenever both forms haveexactly the same solutions. Make up a first-order differ-ential equation for which F(x, y, y) 0 is not equiva-lent to the normal form dydx f (x, y).

54. Find a linear second-order differential equation F(x, y, y, y) 0 for which y c1x c2x2 is a two-parameter family of solutions. Make sure that your equa-tion is free of the arbitrary parameters c1 and c2.

Qualitative information about a solution y (x) of adifferential equation can often be obtained from theequation itself. Before working Problems 5558, recallthe geometric significance of the derivatives dydxand d2ydx2.

55. Consider the differential equation .

(a) Explain why a solution of the DE must be anincreasing function on any interval of the x-axis.

(b) What are What does

this suggest about a solution curve as

(c) Determine an interval over which a solution curve isconcave down and an interval over which the curveis concave up.

(d) Sketch the graph of a solution y (x) of the dif-ferential equation whose shape is suggested byparts (a) (c).

x : ?

limx :

dy>dx and limx :

dy>dx?

dy>dx ex2


56. Consider the differential equation dydx 5 y.(a) Either by inspection or by the method suggested in

Problems 3336, find a constant solution of the DE.

(b) Using only the differential equation, find intervals onthe y-axis on which a nonconstant solution y (x)is increasing. Find intervals on the y-axis on which y (x) is decreasing.

57. Consider the differential equation dydx y(a by),where a and b are positive constants.

(a) Either by inspection or by the method suggestedin Problems 3336, find two constant solutions ofthe DE.

(b) Using only the differential equation, find intervals onthe y-axis on which a nonconstant solution y (x)is increasing. Find intervals on which y (x) isdecreasing.

(c) Using only the differential equation, explain why y a2b is the y-coordinate of a point of inflectionof the graph of a nonconstant solution y (x).

(d) On the same coordinate axes, sketch the graphs ofthe two constant solutions found in part (a). Theseconstant solutions partition the xy-plane into threeregions. In each region, sketch the graph of a non-constant solution y (x) whose shape is sug-gested by the results in parts (b) and (c).

58. Consider the differential equation y y2 4.

(a) Explain why there exist no constant solutions ofthe DE.

(b) Describe the graph of a solution y (x). Forexample, can a solution curve have any relativeextrema?

(c) Explain why y 0 is the y-coordinate of a point ofinflection of a solution curve.

(d) Sketch the graph of a solution y (x) of thedifferential equation whose shape is suggested byparts (a)(c).

Computer Lab Assignments

In Problems 59 and 60 use a CAS to compute all derivativesand to carry out the simplifications needed to verify that theindicated function is a particular solution of the given differ-ential equation.

59. y(4) 20y 158y 580y 841y 0;y xe5x cos 2x

60.

y 20cos(5 ln x)

x 3

sin(5 ln x)

x

x3y 2x2y 20xy 78y 0;

FIRST- AND SECOND-ORDER IVPS The problem given in (1) is also called annth-order initial-value problem. For example,

(2)

and (3)

are first- and second-order initial-value problems, respectively. These two problemsare easy to interpret in geometric terms. For (2) we are seeking a solution y(x) of thedifferential equation y f (x, y) on an interval I containing x0 so that its graph passesthrough the specified point (x0, y0). A solution curve is shown in blue in Figure 1.2.1.For (3) we want to find a solution y(x) of the differential equation y f (x, y, y) onan interval I containing x0 so that its graph not only passes through (x0, y0) but the slopeof the curve at this point is the number y1. A solution curve is shown in blue inFigure 1.2.2. The words initial conditions derive from physical systems where theindependent variable is time t and where y(t0) y0 and y(t0) y1 represent the posi-tion and velocity, respectively, of an object at some beginning, or initial, time t0.

Solving an nth-order initial-value problem such as (1) frequently entails firstfinding an n-parameter family of solutions of the given differential equation and thenusing the n initial conditions at x0 to determine numerical values of the n constants inthe family. The resulting particular solution is defined on some interval I containingthe initial point x0.

EXAMPLE 1 Two First-Order IVPs

In Problem 41 in Exercises 1.1 you were asked to deduce that y cex is a one-parameter family of solutions of the simple first-order equation y y. All thesolutions in this family are defined on the interval (, ). If we impose an initialcondition, say, y(0) 3, then substituting x 0, y 3 in the family determines the

Subject to: y(x0) y0, y(x0) y1Solve: d2y

dx2 f (x, y, y)

Subject to: y(x0) y0Solve: dy

dx f (x, y)

1.2 INITIAL-VALUE PROBLEMS 13

INITIAL-VALUE PROBLEMS

REVIEW MATERIAL Normal form of a DE Solution of a DE Family of solutions

INTRODUCTION We are often interested in problems in which we seek a solution y(x) of adifferential equation so that y(x) satisfies prescribed side conditionsthat is, conditions imposed onthe unknown y(x) or its derivatives. On some interval I containing x0 the problem

(1)

where y0, y1, . . . , yn1 are arbitrarily specified real constants, is called an initial-valueproblem (IVP). The values of y(x) and its first n 1 derivatives at a single point x0, y(x0) y0,y(x0) y1, . . . , y(n1)(x0) yn1, are called initial conditions.

Subject to: y(x0) y0, y(x0) y1, . . . , y(n1)(x0) yn1,Solve: dny

dxn f x, y, y, . . . , y(n1)

1.2

FIGURE 1.2.1 Solution offirst-order IVP

FIGURE 1.2.2 Solution ofsecond-order IVP

xI

solutions of the DE

(x0, y0)

y

m = y1

xI

solutions of the DE

(x0, y0)

y

constant 3 ce0 c. Thus y 3ex is a solution of the IVP

Now if we demand that a solution curve pass through the point (1, 2) rather than(0, 3), then y(1) 2 will yield 2 ce or c 2e1. In this case y 2ex1 isa solution of the IVP

The two solution curves are shown in dark blue and dark red in Figure 1.2.3.

The next example illustrates another first-order initial-value problem. In thisexample notice how the interval I of definition of the solution y(x) depends on theinitial condition y(x0) y0.

EXAMPLE 2 Interval I of Definition of a Solution

In Problem 6 of Exercises 2.2 you will be asked to show that a one-parameter familyof solutions of the first-order differential equation y 2xy2 0 is y 1(x2 c).If we impose the initial condition y(0) 1, then substituting x 0 and y 1into the family of solutions gives 1 1c or c 1. Thus y 1(x2 1). Wenow emphasize the following three distinctions:

Considered as a function, the domain of y 1(x2 1) is the set of realnumbers x for which y(x) is defined; this is the set of all real numbersexcept x 1 and x 1. See Figure 1.2.4(a).

Considered as a solution of the differential equation y 2xy2 0, theinterval I of definition of y 1(x2 1) could be taken to be anyinterval over which y(x) is defined and differentiable. As can be seen inFigure 1.2.4(a), the largest intervals on which y 1(x2 1) is a solutionare (,1), (1, 1), and (1, ).

Considered as a solution of the initial-value problem y 2xy2 0,y(0) 1, the interval I of definition of y 1(x2 1) could be taken tobe any interval over which y(x) is defined, differentiable, and contains theinitial point x 0; the largest interval for which this is true is (1, 1). Seethe red curve in Figure 1.2.4(b).

See Problems 36 in Exercises 1.2 for a continuation of Example 2.

EXAMPLE 3 Second-Order IVP

In Example 4 of Section 1.1 we saw that x c1 cos 4t c2 sin 4t is a two-parameterfamily of solutions of x 16x 0. Find a solution of the initial-value problem

(4)

SOLUTION We first apply x(2) 2 to the given family of solutions: c1 cos 2 c2 sin 2 2. Since cos 2 1 and sin 2 0, we find that c1 2. We next applyx(2) 1 to the one-parameter family x(t) 2 cos 4t c2 sin 4t. Differentiatingand then setting t 2 and x 1 gives 8 sin 2 4c2 cos 2 1, from which wesee that . Hence is a solution of (4).

EXISTENCE AND UNIQUENESS Two fundamental questions arise in consider-ing an initial-value problem:

Does a solution of the problem exist?If a solution exists, is it unique?

x 2 cos 4t 14 sin 4tc2 14

x 16x 0, x2 2, x

2 1.

y y, y(1) 2.

y y, y(0) 3.


FIGURE 1.2.3 Solutions of two IVPs

y

x

(0, 3)

(1, 2)

FIGURE 1.2.4 Graphs of functionand solution of IVP in Example 2

(0, 1)

x

y

11

x

y

11

(a) function defined for all x except x = 1

(b) solution defined on interval containing x = 0

For the first-order initial-value problem (2) we ask:

Existence {Does the differential equation dydx f (x, y) possess solutions?Do any of the solution curves pass through the point (x0, y0)?Uniqueness {When can we be certain that there is precisely one solution curvepassing through the point (x0, y0)?

Note that in Examples 1 and 3 the phrase a solution is used rather than the solu-tion of the problem. The indefinite article a is used deliberately to suggest thepossibility that other solutions may exist. At this point it has not been demonstratedthat there is a single solution of each problem. The next example illustrates an initial-value problem with two solutions.

EXAMPLE 4 An IVP Can Have Several Solutions

Each of the functions y 0 and satisfies the differential equationdydx xy1/2 and the initial condition y(0) 0, so the initial-value problem

has at least two solutions. As illustrated in Figure 1.2.5, the graphs of both functionspass through the same point (0, 0).

Within the safe confines of a formal course in differential equations one can befairly confident that most differential equations will have solutions and that solutions ofinitial-value problems will probably be unique. Real life, however, is not so idyllic.Therefore it is desirable to know in advance of trying to solve an initial-value problemwhether a solution exists and, when it does, whether it is the only solution of the prob-lem. Since we are going to consider first-order differential equations in the next twochapters, we state here without proof a straightforward theorem that gives conditionsthat are sufficient to guarantee the existence and uniqueness of a solution of a first-orderinitial-value problem of the form given in (2). We shall wait until Chapter 4 to addressthe question of existence and uniqueness of a second-order initial-value problem.

THEOREM 1.2.1 Existence of a Unique Solution

Let R be a rectangular region in the xy-plane defined by a x b, c y dthat contains the point (x0, y0) in its interior. If f (x, y) and f y are continuouson R, then there exists some interval I0: (x0 h, x0 h), h 0, contained in[a, b], and a unique function y(x), defined on I0, that is a solution of the initial-value problem (2).

The foregoing result is one of the most popular existence and uniqueness theo-rems for first-order differential equations because the criteria of continuity of f (x, y)and fy are relatively easy to check. The geometry of Theorem 1.2.1 is illustratedin Figure 1.2.6.

EXAMPLE 5 Example 4 Revisited

We saw in Example 4 that the differential equation dydx xy1/2 possesses at leasttwo solutions whose graphs pass through (0, 0). Inspection of the functions

f (x, y) xy1/2 and f

y

x

2y1/2

dy

dx xy1/2, y(0) 0

y 116 x4


y

y = 0

y = x4/16

(0, 0)

1

x

xI0

R

a b

c

d

(x0, y0)

y

FIGURE 1.2.5 Two solutionsof the same IVP

FIGURE 1.2.6 Rectangular region R

shows that they are continuous in the upper half-plane defined by y 0. HenceTheorem 1.2.1 enables us to conclude that through any point (x0, y0), y0 0 in theupper half-plane there is some interval centered at x0 on which the given differentialequation has a unique solution. Thus, for example, even without solving it, we knowthat there exists some interval centered at 2 on which the initial-value problemdydx xy1/2, y(2) 1 has a unique solution.

In Example 1, Theorem 1.2.1 guarantees that there are no other solutions of theinitial-value problems y y, y(0) 3 and y y, y(1) 2 other than y 3ex

and y 2ex1, respectively. This follows from the fact that f (x, y) y andfy 1 are continuous throughout the entire xy-plane. It can be further shown thatthe interval I on which each solution is defined is (, ).

INTERVAL OF EXISTENCE/UNIQUENESS Suppose y(x) represents a solutionof the initial-value problem (2). The following three sets on the real x-axis may notbe the same: the domain of the function y(x), the interval I over which the solutiony(x) is defined or exists, and the interval I0 of existence and uniqueness. Example 2of Section 1.1 illustrated the difference between the domain of a function and theinterval I of definition. Now suppose (x0, y0) is a point in the interior of the rectan-gular region R in Theorem 1.2.1. It turns out that the continuity of the functionf (x, y) on R by itself is sufficient to guarantee the existence of at least one solutionof dydx f (x, y), y(x0) y0, defined on some interval I. The interval I of defini-tion for this initial-value problem is usually taken to be the largest interval contain-ing x0 over which the solution y(x) is defined and differentiable. The interval Idepends on both f (x, y) and the initial condition y(x0) y0. See Problems 3134 inExercises 1.2. The extra condition of continuity of the first partial derivative fyon R enables us to say that not only does a solution exist on some interval I0 con-taining x0, but it is the only solution satisfying y(x0) y0. However, Theorem 1.2.1does not give any indication of the sizes of intervals I and I0; the interval I ofdefinition need not be as wide as the region R, and the interval I0 of existence anduniqueness may not be as large as I. The number h 0 that defines the intervalI0: (x0 h, x0 h) could be very small, so it is best to think that the solution y(x)is unique in a local sense that is, a solution defined near the point (x0, y0). SeeProblem 44 in Exercises 1.2.

REMARKS

(i) The conditions in Theorem 1.2.1 are sufficient but not necessary. This meansthat when f (x, y) and fy are continuous on a rectangular region R, it mustalways follow that a solution of (2) exists and is unique whenever (x0, y0) is apoint interior to R. However, if the conditions stated in the hypothesis ofTheorem 1.2.1 do not hold, then anything could happen: Problem (2) may stillhave a solution and this solution may be unique, or (2) may have several solu-tions, or it may have no solution at all. A rereading of Example 5 reveals that thehypotheses of Theorem 1.2.1 do not hold on the line y 0 for the differentialequation dydx xy1/2, so it is not surprising, as we saw in Example 4 of thissection, that there are two solutions defined on a common interval h x hsatisfying y(0) 0. On the other hand, the hypotheses of Theorem 1.2.1 donot hold on the line y 1 for the differential equation dydx y 1.Nevertheless it can be proved that the solution of the initial-value problemdydx y 1, y(0) 1, is unique. Can you guess this solution?(ii) You are encouraged to read, think about, work, and then keep in mindProblem 43 in Exercises 1.2.



EXERCISES 1.2 Answers to selected odd-numbered problems begin on page ANS-1.

In Problems 1 and 2, y 1(1 c1ex) is a one-parameterfamily of solutions of the first-order DE y y y2. Find asolution of the first-order IVP consisting of this differentialequation and the given initial condition.

1. 2. y(1) 2

In Problems 36, y 1(x2 c) is a one-parameter familyof solutions of the first-order DE y 2xy2 0. Find asolution of the first-order IVP consisting of this differentialequation and the given initial condition. Give the largestinterval I over which the solution is defined.

3. 4.

5. y(0) 1 6.

In Problems 710, x c1 cos t c2 sin t is a two-parameterfamily of solutions of the second-order DE x x 0. Finda solution of the second-order IVP consisting of this differ-ential equation and the given initial conditions.

7. x(0) 1, x(0) 8

8. x(2) 0, x(2) 1

9.

10.

In Problems 1114, y c1ex c2ex is a two-parameterfamily of solutions of the second-order DE y y 0. Finda solution of the second-order IVP consisting of this differ-ential equation and the given initial conditions.

11.

12. y(1) 0, y(1) e

13. y(1) 5, y(1) 5

14. y(0) 0, y(0) 0

In Problems 15 and 16 determine by inspection at least twosolutions of the given first-order IVP.

15. y 3y2/3, y(0) 0

16. xy 2y, y(0) 0

In Problems 1724 determine a region of the xy-plane forwhich the given differential equation would have a uniquesolution whose graph passes through a point (x0, y0) in theregion.

17. 18.dy

dx 1xydy

dx y2/3

y(0) 2y(0) 1,

x(>4) 12, x(>4) 212x(>6) 12, x(>6) 0

y(12) 4y(2) 12y(2)

13

y(0) 13

19. 20.

21. (4 y2)y x2 22. (1 y3)y x2

23. (x2 y2)y y2 24. (y x)y y x

In Problems 2528 determine whether Theorem 1.2.1 guar-antees that the differential equation pos-sesses a unique solution through the given point.

25. (1, 4) 26. (5, 3)

27. (2, 3) 28. (1, 1)

29. (a) By inspection find a one-parameter family of solu-tions of the differential equation xy y. Verify thateach member of the family is a solution of theinitial-value problem xy y, y(0) 0.

(b) Explain part (a) by determining a region R in thexy-plane for which the differential equation xy ywould have a unique solution through a point (x0, y0)in R.

(c) Verify that the piecewise-defined function

satisfies the condition y(0) 0. Determine whetherthis function is also a solution of the initial-valueproblem in part (a).

30. (a) Verify that y tan (x c) is a one-parameter familyof solutions of the differential equation y 1 y2.

(b) Since f (x, y) 1 y2 and fy 2y are continu-ous everywhere, the region R in Theorem 1.2.1 canbe taken to be the entire xy-plane. Use the family ofsolutions in part (a) to find an explicit solution ofthe first-order initial-value problem y 1 y2,y(0) 0. Even though x0 0 is in the interval(2, 2), explain why the solution is not defined onthis interval.

(c) Determine the largest interval I of definition for thesolution of the initial-value problem in part (b).

31. (a) Verify that y 1(x c) is a one-parameterfamily of solutions of the differential equation y y2.

(b) Since f (x, y) y2 and fy 2y are continuouseverywhere, the region R in Theorem 1.2.1 can betaken to be the entire xy-plane. Find a solution fromthe family in part (a) that satisfies y(0) 1. Thenfind a solution from the family in part (a) thatsatisfies y(0) 1. Determine the largest interval Iof definition for the solution of each initial-valueproblem.

y 0, x 0x, x 0

y 1y2 9

dy

dx y xx

dy

dx y

(c) Determine the largest interval I of definition for thesolution of the first-order initial-value problemy y2, y(0) 0. [Hint: The solution is not a mem-ber of the family of solutions in part (a).]

32. (a) Show that a solution from the family in part (a)of Problem 31 that satisfies y y2, y(1) 1, isy 1(2 x).

(b) Then show that a solution from the family in part (a)of Problem 31 that satisfies y y2, y(3) 1, isy 1(2 x).

(c) Are the solutions in parts (a) and (b) the same?

33. (a) Verify that 3x2 y2 c is a one-parameter fam-ily of solutions of the differential equation y dydx 3x.

(b) By hand, sketch the graph of the implicit solution3x2 y2 3. Find all explicit solutions y (x) ofthe DE in part (a) defined by this relation. Give theinterval I of definition of each explicit solution.

(c) The point (2, 3) is on the graph of 3x2 y2 3,but which of the explicit solutions in part (b) satis-fies y(2) 3?

34. (a) Use the family of solutions in part (a) of Problem 33to find an implicit solution of the initial-valueproblem y dydx 3x, y(2) 4. Then, by hand,sketch the graph of the explicit solution of thisproblem and give its interval I of definition.

(b) Are there any explicit solutions of y dydx 3xthat pass through the origin?

In Problems 3538 the graph of a member of a familyof solutions of a second-order differential equation d2ydx2 f (x, y, y) is given. Match the solution curve withat least one pair of the following initial conditions.

(a) y(1) 1, y(1) 2

(b) y(1) 0, y(1) 4

(c) y(1) 1, y(1) 2

(d) y(0) 1, y(0) 2

(e) y(0) 1, y(0) 0

(f ) y(0) 4, y(0) 2

35.


Discussion Problems

In Problems 39 and 40 use Problem 51 in Exercises 1.1 and(2) and (3) of this section.

39. Find a function y f (x) whose graph at each point (x, y)has the slope given by 8e2x 6x and has the y-intercept (0, 9).

40. Find a function y f (x) whose second derivative isy 12x 2 at each point (x, y) on its graph andy x 5 is tangent to the graph at the point corre-sponding to x 1.

41. Consider the initial-value problem y x 2y,. Determine which of the two curves shown

in Figure 1.2.11 is the only plausible solution curve.Explain your reasoning.

y(0) 12


y

x

5

5

5


y

x

5

5

5

36.

37.

38.



y

x

5

5

5

y

x

5

5

5

42. Determine a plausible value of x0 for which thegraph of the solution of the initial-value problem y 2y 3x 6, y(x0) 0 is tangent to the x-axis at(x0, 0). Explain your reasoning.

43. Suppose that the first-order differential equation dydx f (x, y) possesses a one-parameter family ofsolutions and that f (x, y) satisfies the hypotheses ofTheorem 1.2.1 in some rectangular region R of the xy-plane. Explain why two different solution curvescannot intersect or be tangent to each other at a point(x0, y0) in R.

44. The functions and

have the same domain but are clearly different. SeeFigures 1.2.12(a) and 1.2.12(b), respectively. Show thatboth functions are solutions of the initial-value problem

y(x) 0,116 x

4,

x 0 x 0

y(x) 116 x4, x

1.3 DIFFERENTIAL EQUATIONS AS MATHEMATICAL MODELS 19

FIGURE 1.2.11 Graphs for Problem 41

(0, )12

1

1 x

y

dydx xy1/2, y(2) 1 on the interval (, ).Resolve the apparent contradiction between this factand the last sentence in Example 5.

Mathematical Model

45. Population Growth Beginning in the next sectionwe will see that differential equations can be used todescribe or model many different physical systems. Inthis problem suppose that a model of the growing popu-lation of a small community is given by the initial-valueproblem

where P is the number of individuals in the communityand time t is measured in years. How fastthat is, atwhat rateis the population increasing at t 0? Howfast is the population increasing when the populationis 500?

dP

dt 0.15P(t) 20, P(0) 100,

FIGURE 1.2.12 Two solutions of the IVP in Problem 44

(a)

(2, 1)

y

x

(b)

(2, 1)

y

x

DIFFERENTIAL EQUATIONS AS MATHEMATICAL MODELS

REVIEW MATERIAL Units of measurement for weight, mass, and density Newtons second law of motion Hookes law Kirchhoffs laws Archimedes principle

INTRODUCTION In this section we introduce the notion of a differential equation as amathematical model and discuss some specific models in biology, chemistry, and physics. Once wehave studied some methods for solving DEs in Chapters 2 and 4, we return to, and solve, some ofthese models in Chapters 3 and 5.

1.3

MATHEMATICAL MODELS It is often desirable to describe the behavior ofsome real-life system or phenomenon, whether physical, sociological, or even eco-nomic, in mathematical terms. The mathematical description of a system of phenom-enon is called a mathematical model and is constructed with certain goals in mind.For example, we may wish to understand the mechanisms of a certain ecosystem bystudying the growth of animal populations in that system, or we may wish to datefossils by analyzing the decay of a radioactive substance either in the fossil or in thestratum in which it was discovered.

Construction of a mathematical model of a system starts with

(i) identification of the variables that are responsible for changing thesystem. We may choose not to incorporate all these variables into themodel at first. In this step we are specifying the level of resolution ofthe model.

Next

(ii) we make a set of reasonable assumptions, or hypotheses, about thesystem we are trying to describe. These assumptions will also includeany empirical laws that may be applicable to the system.

For some purposes it may be perfectly within reason to be content with low-resolution models. For example, you may already be aware that in beginningphysics courses, the retarding force of air friction is sometimes ignored in modelingthe motion of a body falling near the surface of the Earth, but if you are a scientistwhose job it is to accurately predict the flight path of a long-range projectile,you have to take into account air resistance and other factors such as the curvatureof the Earth.

Since the assumptions made about a system frequently involve a rate of changeof one or more of the variables, the mathematical depiction of all these assumptionsmay be one or more equations involving derivatives. In other words, the mathemat-ical model may be a differential equation or a system of differential equations.

Once we have formulated a mathematical model that is either a differential equa-tion or a system of differential equations, we are faced with the not insignificantproblem of trying to solve it. If we can solve it, then we deem the model to be reason-able if its solution is consistent with either experimental data or known facts aboutthe behavior of the system. But if the predictions produced by the solution are poor,we can either increase the level of resolution of the model or make alternative as-sumptions about the mechanisms for change in the system. The steps of the model-ing process are then repeated, as shown in the following diagram:

Of course, by increasing the resolution, we add to the complexity of the mathemati-cal model and increase the likelihood that we cannot obtain an explicit solution.

A mathematical model of a physical system will often involve the variable time t.A solution of the model then gives the state of the system; in other words, the valuesof the dependent variable (or variables) for appropriate values of t describe the systemin the past, present, and future.

POPULATION DYNAMICS One of the earliest attempts to model human popula-tion growth by means of mathematics was by the English economist Thomas Malthusin 1798. Basically, the idea behind the Malthusian model is the assumption that the rateat which the population of a country grows at a c

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