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 0070721807, 9780070721807

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ADVANCED ENGNEERING MATHEMAT CS

ADVANCED ENGINEERING MATHEMATICS FOURTH EDITION

C. RAY WYLIE

William R. Kenan, Jr., Professor of Mathematics Chairman, Department of Mathematics Furman University

McGRAW-HILL BOOK COMPANY New York St. Louis San Francisco Dusseldorf Johannesburg Kuala Lumpur London Mexico Montreal New Delhi Panama Paris Sao Paulo Singapore Sydney Tokyo Toronto

ADVANCED ENGINEERING MATHEMATICS Copyright © 1960, 1966, 1975 by McGraw-Hill, Inc. All rights reserved. Copyright 1951 by McGraw-Hill, Inc. All rights reserved. Printed in the United States of America. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. 1234567890SGDO 798765 This book was set in Times New Roman. The editors were A. Anthony Arthur and Michael LaBarbera; the designer was Jo Jones; the production supervisor was Sam Ratkewitch. New drawings were done by Eric G. Hieber Associates Inc. The printer was Segerdahl Corporation; the binder, R. R. Donnelley & Sons Company. Library of Congress Cataloging in Publication Data Wylie, Clarence Raymond, date Advanced engineering mathematics. 1. Mathematics—1961QA401.W9 1975 510 ISBN 0-07-072180-7

I. Title. 74-14523

CONTENTS

Preface

ix

To the Student

x[

1 Ordinary Differential Equations of the First Order

1

1.1 Introduction

1

1.2 Fundamental Definitions

2

1.3 Separable First-Order Equations

8

1.4 Homogeneous First-Order Equations

12

1.5 Exact First-Order Equations

15

1.6 Linear First-Order Equations

20

1.7 Applications of First-Order Differential Equations

23

2 Linear Differential Equations with Constant Coefficients

39

2.1 The General Linear Second-Order Equation

39

2.2 The Homogeneous Linear Equation with Constant Coefficients

46

2.3 The Nonhomogeneous Equation

53

2.4 Particular Integrals by the Method of Variation of Parameters

60

2.5 Equations of Higher Order

63

2.6 Applications

67

2.7 Green’s Functions

78

3 Simultaneous Linear Differential Equations

89

3.1 Introduction

89

3.2 The Reduction of a System to a Single Equation

89

3.3 Complementary Functions and Particular Integrals for Systems of Equations

98

Contents

VI

4 Finite Differences

104

4.1 The Differences of a Function

104

4.2 Interpolation Formulas

116

4.3 Numerical Differentiation and Integration

124

4.4 The Numerical Solution of Differential Equations

133

4.5 Difference Equations

141

4.6 The Method of Least Squares

153

5 Mechanical and Electric Circuits

171

5.1 Introduction

171

5.2 Systems with One Degree of Freedom

171

5.3 The Translational Mechanical System

179

5.4 The Series Electric Circuit

194

5.5 Systems with Several Degrees of Freedom

201

6 Fourier Series and Integrals

214

6.1 Introduction

214

6.2 The Euler Coefficients

215

6.3 Half-Range Expansions

221

6.4 Alternative Forms of Fourier Series

229

6.5 Applications

233

6.6 The Fourier Integral as the Limit of a Fourier Series

240

6.7 From the Fourier Integral to the Laplace Transform

252

7 The Laplace Transformation

257

7.1 Theoretical Preliminaries

257

7.2 The General Method

263

7.3 The Transforms of Special Functions

268

7.4 Further General Theorems

275

7.5 The Heaviside Expansion Theorems

289

7.6 The Transforms of Periodic Functions

295

7.7 Convolution and the Duhamel Formulas

309

8 Partial Differential Equations

321

8.1 Introduction

321

8.2 The Derivation of Equations

321

8.3 The d’Alembert Solution of the Wave Equation

334

8.4 Separation of Variables

342

8.5 Orthogonal Functions and the General Expansion Problem

351

8.6 Further Applications

369

8.7 Laplace Transform Methods

381

Contents

9 Bessel Functions and Legendre Polynomials

vii 388

9.1 Theoretical Preliminaries

388

9.2 The Series Solution of Bessel’s Equation

394

9.3 Modified Bessel Functions

402

9.4 Equations Solvable in Terms of Bessel Functions

408

9.5 Identities for the Bessel Functions

410

9.6 The Orthogonality of the Bessel Functions

419

9.7 Applications of Bessel Functions

425

9.8 Legendre Polynomials

440

10 Determinants and Matrices

454

10.1 Determinants

454

10.2 Elementary Properties of Matrices

469

10.3 Adjoints and Inverses

483

10.4 Rank and the Equivalence of Matrices

‘491

10.5 Systems of Linear Equations

498

10.6 Matric Differential Equations

516

11 Further Properties of Matrices

524

11.1 Quadratic Forms

524

11.2 The Characteristic Equation of a Matrix

532

11.3 The Transformation of Matrices

549

11.4 Functions of a Square Matrix

564

11.5 The Cayley-Hamilton Theorem

575

11.6 Infinite Series of Matrices

583

12 The Calculus of Variations

592

12.1 Introduction

592

12.2 Extrema of Functions of Several Variables

592

12.3 Lagrange’s Multipliers

595

12.4 Extremal Properties of the Characteristic Values of (A — XB)X = 0

600

12.5 The Euler Equation for J* f(x,y,y') dx

607

12.6 Variations

613

12.7 The Extrema of Integrals under Constraints

616

12.8 Sturm-Liouville Problems

621

12.9 Hamilton’s Principle and Lagrange’s Equation

626

13 Vector Analysis

631

13.1 The Algebra of Vectors

631

13.2 Vector Functions of One Variable

644

13.3 The Operator V

650

Contents

Viii

13.4 Line, Surface, and Volume Integrals

659

13.5 Integral Theorems

672

13.6 Further Applications

686

14 Tensor Analysis

696

14.1 Introduction

696

14.2 Oblique Coordinates

696

14.3 Generalized Coordinates

706

14.4 Tensors

719

14.5 Divergence and Curl

724

14.6 Covariant Differentiation

728

15 Analytic Functions of a Complex Variable

733

15.1 Introduction

733

15.2 Algebraic Preliminaries

733

15.3 The Geometric Representation of Complex Numbers

736

15.4 Absolute Values

741

15.5 Functions of a Complex Variable

745

15.6 Analytic Functions

v—^

756

15.7 The Elementary Functions of z

757

15.8 Integration in the Complex Plane

765

16 Infinite Series in the Complex Plane

778

16.1 Series of Complex Terms

778

16.2 Taylor’s Expansion

788

16.3 Laurent’s Expansion

795

17 The Theory of Residues

803

17.1 The Residue Theorem

803

17.2 The Evaluation of Real Definite Integrals

810

17.3 The Complex Inversion Integral

818

17.4 Stability Criteria

824

18 Conformal Mapping

837

18.1 The Geometrical Representation of Functions of z

837

18.2 Conformal Mapping

840

18.3 The Bilinear Transformation

845

18.4 The Schwarz-Christoffel Transformation

856

Answers to Odd-numbered Exercises

866

Index

977

PREFACE

The first edition of this book was written to provide an introduction to those branches of postcalculus mathematics with which the average analytical engineer or physicist needs to be reasonably familiar in order to carry on his own work effectively and keep abreast of current developments in his field. In the present edition, as in the second and third, although the material has been largely rewritten, the various additions, deletions, and refinements have been made only because they seemed to contribute to the achievement of this goal. Because ordinary differential equations are probably the most immediately useful part of postcalculus mathematics for the student of applied science, and because the techniques for solving simple ordinary differential equations stem naturally from the techniques of calculus, this book begins with a chapter on ordinary differential equations of the first order and their applications. This is followed by two other chapters on ordinary differential equations, which develop the theory and applications of linear equations and systems of linear equations with constant coefficients. In particular, a new section on Green’s function and its interpretation as an influence function has been added to the first of these two chapters. Following these is a chapter on finite differences containing the usual applications to interpolation, numerical differentiation and integration, and the step-by-step solution of differential equations using both Milne’s method and the Runge-Kutta method. There is also a section on linear difference equations with constant coefficients closely paralleling the preceding development for differential equations and a section on the method of least squares and the related topic of orthogonal polynomials. It is hoped that the material in this chapter will provide a useful background in classical finite differences, on which a more extensive course in computer-oriented numerical analysis can be based. Chapter 5 is devoted to the application of the preceding ideas to mechanical and electrical systems, and, as in the first three editions, the mathematical identity of these fields is emphasized. The next two chapters deal first with partial differential equations and boundary-value problems and second with Bessel functions and Legendre poly¬ nomials, very much as in the third edition though with a number of new examples and exercises. Chapters 10 and 11 deal with determinants and matrices as far as the CayleyHamilton theorem, Sylvester’s identity, and infinite series of matrices and their use in solving matric differential equations. Chapter 12 is a new chapter on the calculus of variations, covering such topics as the maxima and minima of functions of several variables, Lagrange’s multipliers, the extremal properties of the eigenvalues of matric equations, Euler’s equation, Hamilton’s principle, and Lagrange’s equation. Chapters

X

Preface

13 and 14 deal with vector and tensor analysis. The last four chapters provide an introduction to the theory of functions of a complex variable, with applications to the evaluation of real definite integrals, the complex inversion integral, stability criteria, conformal mapping, and the Schwarz-Christoffel transformation. This book falls naturally into three major subdivisions. The first nine chapters constitute a reasonably self-contained treatment of ordinary and partial differential equations and their applications. The next five chapters cover the related areas of linear algebra, the calculus of variations, and vector and tensor analysis. The last four chapters cover the elementary theory and applications of functions of a complex variable. With this organization, the book, which contains enough material for a 2-year postcalculus course in applied mathematics, is well adapted for use as a text for any of several shorter courses. In this edition, as in the first three, every effort has been made to keep the presenta¬ tion detailed and clear while at the same time maintaining acceptable standards of precision and accuracy. To achieve this, more than the usual number of worked examples and carefully drawn figures have been included, and in every development there has been a conscious attempt to make the transitions from step to step so clear that a student with no more than a good background in calculus, working with paper and pencil, should seldom be held up more than momentarily. Over 750 new exercises of varying degrees of difficulty have been added to the 1,387 problems which appeared in the third edition. Many of these involve extensions of topics presented in the text or related topics which could not be treated because of limitations of space. Hints are included in many of the exercises, and answers to the odd-numbered ones are given at the end of the book. As in the first three editions, words and phrases defined informally in the body of the text are set in boldface. Illustrative examples are set.in type of a different size from that used for the main body of the text. The indebtedness of the author to his colleagues, students, and former teachers is too great to catalog, and to all who have given help and encouragement in the prepara¬ tion of this book, I can offer here only a most inadequate acknowledgment of my appreciation. In particular, I am deeply grateful to those users of this book, both teachers and students, who have been kind enough to write me their impressions and criticisms of the first three editions and their suggestions for an improved fourth edition. Finally, I must express my gratitude to my wife Ellen and my student Moffie Hills, who have shared with me the task of proofreading the manuscript in all its stages. C. Ray Wylie

TO THE STUDENT

This book has been written to help you in your development as an applied scientist, whether engineer, physicist, chemist, or mathematician. It contains material which you will find of great use, not only in the technical courses you have yet to take, but also in your profession after graduation as long as you deal with the analytical aspects of your field. I have tried to write a book which you will find not only useful but also easy to study from, at least as easy as a book on advanced mathematics can be. There is a good deal of theory in it, for it is the theoretical portion of a subject which is the basis for the nonroutine applications of tomorrow. But nowhere will you find theory for its own sake, interesting and legitimate as this may be to a pure mathematician. Our theoretical discussions are designed to illuminate principles, to indicate generaliza¬ tions, to establish limits within which a given technique may or may not safely be used, or to point out pitfalls into which one might otherwise stumble. On the other hand, there are many applications illustrating, with the material at hand, the usual steps in the solution of a physical problem: formulation, manipulation, and inter¬ pretation. These examples are, without exception, carefully set up and completely worked, with all but the simplest steps included. Study them carefully, with paper and pencil at hand, for they are an integral part of the text. If you do this, you should find the exercises, though challenging, still within your ability to work. Terms defined informally in the body of the text are always indicated by the use of boldface type. Italic type is used for emphasis. It is suggested that you read each section through for the main ideas before you concentrate on filling in any of the details. You will probably be surprised at how many times a detail which seems to hold you up in one paragraph is explained in the next as the discussion unfolds. Because this book is long and contains material suitable for various courses, your instructor may begin with any of a number of chapters. However, the overall structure of the book is the following: The first nine chapters are devoted to the general theme of ordinary and partial differential equations and related topics. Here you will find basic analytical techniques for solving the equations in which physical problems must be formulated when continuously changing quantities are involved. Chapters 10 through 14 deal with the somewhat related topics of matrix theory and linear algebra, the calculus of variations, vector analysis, and an introduction to generalized co¬ ordinates and tensor analysis. Finally, Chapters 15 to 18 provide an introduction to the theory and applications of functions of a complex variable. (Chapter 4, in par¬ ticular, is worthy of note because it provides an introduction to numerical analysis,

XI

To the Student

the modern field which deals with techniques for obtaining numerical answers to problems too complicated to be solved by exact analytic methods.) It has been gratifying to receive letters from students who have used this book, giving me their reactions to it, pointing out errors and misprints in it, and offering suggestions for its improvement. Should you be inclined to do so, I should be happy to hear from you also. And now good luck and every success. C. Ray Wylie

ADVANCED ENGINEERING MATHEMATICS

CHAPTER 1 Ordinary Differential Equations of the First Order

1.1 Introduction An equation involving one or more derivatives of a function is called a differential equation. By a solution of a differential equation is meant a relation between the dependent and independent variables which is free of derivatives and which, when substituted into the given equation, reduces it to an identity. The study of the exist¬ ence, nature, and determination of solutions of differential equations is of fundamental importance not only to the pure mathematician but also to anyone engaged in the mathematical analysis of natural phenomena. In general, a mathematician considers it a triumph if he is able to prove that a given differential equation possesses a solution and if he can deduce a few of the more important properties of that solution. A physicist or engineer, on the other hand, is usually greatly disappointed if a specific expression for the solution cannot be ex¬ hibited. The usual compromise is to find some practical procedure by means of which the required solution can be approximated with satisfactory accuracy. Not all differential equations are difficult enough to make this necessary, however, and there are several large and very important classes of equations for which solutions can readily be found. For instance, an equation such as

ax is really a differential equation, and the integral y ~

f(x) dx + c

is a solution. More generally, the equation

TT dxn = *(*> is a differential equation whose solution can be found by n successive integrations. Except in name, the process of integration is actually an example of a process for solving differential equations. In this and the following two chapters we shall consider differential equations which are next in difficulty after those which can be solved by direct integration. These equations form only a very small part of the class of all differential equations, and yet with a knowledge of them a scientist is equipped to handle a great variety of applications. To get so much for so little is indeed remarkable.

Ordinary Differential Equations of the First Order

2

Chapter 1

1.2 Fundamental Definitions If the derivatives which appear in a differential equation are total derivatives, the equation is called an ordinary differential equation; if partial derivatives occur, the equation is called a partial differential equation. By the order of a differential equation is meant the order of the highest derivative which appears in the equation. EXAMPLE 1

The equation x2y" + xy' + (x2 — 4)y = 0 is an ordinary differential equation of the second order connecting the dependent variable y with its first and second derivatives and with the independent variable x.

EXAMPLE 2

The equation

’Z)

Show that not all the constants which appear in the following expressions are essential, and in each case rearrange the expression so that all the constants which remain are essential: 11

Aex+k

12

a + In bx

13

a In xb

14

aX + b cx + d

15 17

A sin (x + b) + C sin (x + d) a cosh2 0 + b sinh2 6 + c cosh 19

16 18

/f[cos (x + a) + cos (x — a)] a sin 3x + b sin x + c sin3 x

A

20

B

C

.* + 1 x + 2 x2 + 3x + 2 a(x — 6y — 1) + b(3x + 4y + 5) + c(5x + 3y + 4)

Verify that each of the following equations has the indicated solution for all values of the constants a and b: 21 y" + 4y = 0 22 y" - 4y = 0 23 y" + 3y + ly = lle2x 24 y" - 6/ + 9y = 0 25 (cos lx)y' + (2 sin lx)y = 2 26 Ixy dy = (y2 — x) dx 27 °y* + (y')2 + 1=0 3 2u du 28 — = — 3x2 dt d2u 32u 29 4 — = — 3x2 dt2

y = a cos lx + b sin lx y = ae2x + be~2x y — ae~x + be~2x + e2x y = ae2x + bxe3x y = a cos 2x + sin 2x y2 = ax — x In x y = In cos (x — a) + b u = ae~9‘ cos (3x + b) u = af(x + It) + bg(x — It)

If a and b are arbitrary constants, find a differential equation of minimum order of which each of the following expressions is a general solution: 30 32 34 36

37

38

39

y = ae~2x + bex y = ae~‘ + be' + ce2t y = a cosh 2x + b sinh lx

31 y = ae~2t + bte~2' 33 y = lax + bx2 35 y = sin (ax + b) Find a differential equation which has as a general solution the expression which defines the family of all parabolas which touch the x axis and have their axes vertical. Find a differential equation which has as a general solution the expression which defines the family of all lines which touch the parabola ly = x2. Verify that the equation of the given parabola defines a function which is a singular solution of the required differential equation. Verify that for all values of m the function y = mx + f(m) is a solution of the differential equation y = xy' + f(y'). [Thi3 differential equation is called Clairaut’s equation, after the French mathematician A. C. Clairaut (1713-1765).] Verify that for all values of the arbitrary constants a and b both jq = a and y2 = bx2 satisfy each of the differential equations xy" = y'

40

and

lyy" = (y')2

but that y = a + bx2 will satisfy only the first of these equations. Explain. Verify that for all values of the arbitrary constants a and b both _>q = a and y2 = by/x satisfy each of the differential equations

2 xy" + y = 0

and

8 x3(y")2 — yy' = 0

but that y = a + by/x will satisfy only the first of these equations. Explain.

8

Chapter 1

Ordinary Differential Equations of the First Order

41

Verify that for all values of the arbitrary constants a and b both yx = a(x - l)2 and y2 = b(x + l)2 satisfy each of the differential equations (x2 - \)y" - 2xy' + 2y = 0

42

2yy" - (/)2 = 0

but that y = a(x — l)2 + b{x + l)2 will satisfy only the first of these equations. Explain. Verify that for all values of the arbitrary constants cv and c2 the differential equation xy' = 2y + x is satisfied by the function

y

43

and

f ctx2 — x

x < 0

| c2x2 — x

x > 0

Explain. Verify that for all values of the arbitrary constants cu c2, and c3 the differential equation (x2 - l)y' = 4xy is satisfied by the function y =

x < —1 -1 < x < 1

fcqfx2 — l)2 c2(x2 - l)2 (c3(x2 - l)2

44

X > 1

Explain. Verify that for all values of the arbitrary constants {c„} (« = ..., —2, —1, 0, 1, 2,...) the differential equation (1 — cos x)y' = (sin x)y

is satisfied by the function y = c„( 1 — cos x)

Inn < x < 2(n + 1)^

Explain.

1.3 Separable First-Order Equations In many cases a first-order differential equation can be reduced by algebraic manipula¬ tions to the form (1)

f{x) dx = g(y) dy

Such an equation is said to be separable because the variables x and y can be separated from each other in such a way that x appears only in the coefficient of dx and y appears only in the coefficient of dy. An equation of this type can be solved at once by integra¬ tion, and we have the general solution

(2)

/(x) dx — |g(y) dy + c

where c is an arbitrary constant of integration. It must be borne in mind, however, that the integrals which appear in (2) may be impossible to evaluate in terms of elementary functions, and numerical or graphical integration may be required before this solution can be put to practical use. Other forms which should be recognized as being separable are

(3)

/(x)G(y) dx = F{x)g{y) dy x = M(x)N(y) dx

(4) V

Section 1.3

Separable First-Order Equations

9

A general solution of Eq. (3) can be found by first dividing by the product F(x)G(y) to separate the variables and then integrating:

7(x)

'

dx =

F(x)

g(y)

dy + c

G(y)

Similarly, a general solution of Eq. (4) can be found by first multiplying by dx and dividing by N(y) and then integrating:

dy

M (x) dx + c

N(y) Clearly, the process of solving a separable equation will often involve division by one or more expressions. In such cases the results are valid where the divisors are not equal to zero but may or may not be meaningful for values of the variables for which the division is impossible. Such values require special consideration and, as we shall see in the next example, may lead us to singular solutions. EXAMPLE 1

Solve the differential equation dx + xy dy = y2 dx + y dy. It is not immediately evident that this equation is separable. In any case, however, the best first step in solving an equation of this sort is to collect terms on dx and dy. This gives (1 - y2) dx = y{ 1 — x) dy which is of the form (3). Hence, division by the product (1 — x)(l - y2) will separate the variables and reduce the equation to the standard form (1): dx

y dy

1 — x

1 — y2

Now, multiplying by -2 and integrating, we obtain the following equation defining y as an implicit function of x: 2 In |1 — x\ = In |1 — y2\ + c In this case, as in many problems of this sort, it is possible to write the solution in a more convenient form by first combining the logarithmic terms and then taking antilogarithms:

where k2 = ec is necessarily positive. Finally, clearing of fractions and eliminating the absolute values, we have (1 - x)2 = ±k2( 1 — y2)

k ^ 0

The two possibilities here can, of course, be combined into one by writing (1 - x)2 = 2(1 - y2) where now A can take on any real value, positive or negative, except 0. The solution of the differential equation thus defines the family of conics (x — 1)^ (5)

+y2 = 1

A * 0

typical members of which are shown in Fig. 1.2. If X > 0, the solution curves are all ellipses; if X < 0, the solution curves are all hyperbolas.

10

Ordinary Differential Equations of the First Order

Chapter 1

Figure 1.2 Typical members of the solution family [(x — 1)2/A] + y2 = 1 of the differential equation (1 — y2) dx = y(l — x) dy.

In most practical problems a general solution of a differential equation is required to satisfy specific conditions which permit its arbitrary constants to be uniquely determined. For instance, in the present problem we might ask for the particular solution curve which passes through the point (—yrrO- Substituting these values of x and y, we then have

from which we find the value A = — 1 and thence the specific solution

(6)

y2 = 1 + (x - l)2

Equation (6) defines the unique member of the family of curves (5) which passes through the point ( — j,^). However, over any interval which contains both x = — yand x = 1, there are many functions which satisfy the given differential equation and are such that y = ^ when x = — y. In fact, the upper branch of any curve of the family (5) for x > 1 can be associated with the upper branch of the curve (6) for x < 1 to give a function which satisfies the given equation and fulfills the condition that y = ^ when x = — j. This is, of course, consistent with the fact that according to Theorem 1, Sec. 1.2, the uniqueness of the solution for which y = when x = —y can be guaranteed only over an interval around x = — y which does not contain x = 1, since y' is undefined at x = 1. It should be noted that in separating variables in the given differential equation it was necessary to divide by 1 — x and by 1 — y2; hence, the possibility that x = 1 and the possibility that y = ± 1 were implicitly ruled out. Therefore, had we desired the

Section 1.3

Separable First-Order Equations

11

particular solution curve which passed through any point with coordinates of the form C*o,l), or (.Vo, —1), we could not have found that curve, even if it existed, by starting with the general solution (5) and particularizing the arbitrary constant 2. Instead, it would have been necessary to return to the differential equation and search for the required solution by some method other than separation of variables. In this case it is obvious that the linear equations x = 1, y = 1, and y — —1 all define solutions of the given differential equation and, moreover, satisfy, respectively, the conditions (l,y0), (a:o,1), and (.v0,— 1). None of these can be obtained from our general solution, although * = 1 can be included in the first form of it by permitting X to take on the (previously excluded) value zero. In this case, then, only y — 1 and y = — 1 appear as singular solutions of the given equation. EXERCISES

Find a general solution of each of the following equations: 1 3 5

7 9

11

y' = —2 xy y = 3x2(l + y2) 2(xy + x)y' = y (y + x2y) dy = (xy2 yy' = 2 (xy + x) yex+y dy = dx

at) dx

2 4 6 8 10 12

(sin x) dy = 2y (cos x) dx x dy = 3y dx x dy = (y2 - 3y + 2) dx y dx — x dy = x(dy — y dx) dx + y dy = x2y dy xex2+y dx = y dy

, = 2(y2 + y - 2)

14 y

15 16

y" + (y')2 + 1=0 xy" = /-

x2 + 4x + 3

Hint: Observe that y" = dy'/dx. 17 yy" = (y')2

Find the particular solution of each of the following equations which satisfies the indicated conditions: 18 19 20 21 22

2-vy' + y = 0 y' + 2y = 0

x = 4, y = 1

* = 0, y = 100 x = — 3, y = 1 x = 1, y = 4 Is there a solution of the equation x dy = 3(y — 1) dx satisfying the two con¬ ditions y = 3 when x = 1 and y = 9 when x = 22 Is there a solution of this equation which satisfies the two conditions y = 3 when x = — 1 and y — 9 when x = 2? Explain. 23 Find a solution of the equation (1 — x2) dy + 4xy dx = 0 with the property that y = 9 when x = — 2, y = 2 when x = 0, and y = 0 when x = 2. 24 Show that every solution of the equation y' = ky is of the form y = Aekx. Hint: Let y be any solution of the given equation, and consider the derivative of the fraction y/ekx. 25 A critical student watching his professor integrate the separable equation f(x) dx = g(y) dy objected that the procedure was incorrect, since one side was integrated with repect to x while the other side was integrated with respect to y. How would you answer the student’s objection? 26 Show that there is no loss of generality if the arbitrary constant added when a separable equation is integrated is written in the form In c rather than just c. Do you think this would ever be a convenient thing to do? Is there any loss of generality if the integration constant is written in the form c2? tan c2 sin c? ec2 sinh c? cosh cl 27 Show that the change of dependent variable from y to t; defined by the substitution v = ax + by + c will always transform the equation y' = f(ax + by + c) into a separable equation. 2x dx — dy = x(x dy — 2y dx) dy = x(2y dx — x dy)

Using the substitution described in Exercise 27, find a general solution of each of the following equations: 28 30

y' = (x — y)2 y' = (x + y - 3)2 - 2(x + y - 3)

29 31

y' = e2x+y~l - 2 y' = (x - y + l)2 + x - y

Ordinary Differential Equations of the First Order

12

Chapter 1

1.4 Homogeneous First-Order Equations If all terms in the coefficient functions M(x,y) and N(x,y) in the general first-order differential equation (1)

M(x,y) dx = N(x,y) dy

are of the same total degree in the variables x and y, then either of the substitutions y = ux and x = vy will reduce the equation to one which is separable. More generally, if M(x,y) and N(x,y) have the property that for all positive values of A the substitution of Ax for x and Ay for y converts them, respectively, into the expressions AnM(x,y)

and

AnN(x,y)

then Eq. (1) can always be reduced to a separable equation by either of the substitu¬ tions y — ux and x — vy. Functions with the property that the substitutions x -*■ Ax

and

y -*■ Ay

A > 0

merely reproduce the original forms multiplied by An are called homogeneous functions of degree n. As a direct extension of this terminology, the differential equation (1) is said to be homogeneous when M(x,y) and N(x,y) are homogeneous functions of the same degree.

EXAMPLE 1

Is the function F(x,y) = x(ln Vx2 + y2 - In y) + yex/y

homogeneous? To decide this question, we replace x by lx and y by Xy, getting F{Xx,Xy) = .U'Qn VX2x2 + X2y2 — In Xy) + XyeXxlXy

= Ax[(ln Vx2 + y2 + In X) — (In y + In A)] + Xyexly = X[x(ln Vx2 + y2 — In y) + yex,y] =

mx,y)

The given function is therefore homogeneous of degree 1.

If Eq. (1), assumed now to be homogeneous, is written in the form dy _ M(x,y) dx

N(x,y)

it is evident that the fraction on the right is a homogeneous function of degree 0, since the same power of A will multiply both numerator and denominator when the test substitutions x -> Ax and y -> Ay are made. But if M (Ax,Ay) _ M(x,y) N (Ax,Ay)

V

N(x,y)

Section 1.4

Homogeneous First-Order Equations

13

it follows, by assigning to the arbitrary symbol X the value l/x if x is positive and the value — l/x if * is negative, that M(l,y/x) M(x,y) _ M(Xx,Xy)

N(l,y/x)

N(x,y)

M(— l, —y/x)

N(Xx,Xy)

x > 0 x < 0

N (—1,—y/x) In either case it is clear that the result is a function of the fractional argument y/x. Thus, an alternative standard form for a homogeneous first-order differential equation is

(2) Although in practice it is not necessary to reduce a homogeneous equation to the form (2) in order to solve it, the theory of the substitution y = ux, or u = y/x, is most easily developed when the equation is written in this form. Now if y = ux, then dy/dx = u + x du/dx (or, equivalently, dy = u dx + x du). Hence under this substitution, Eq. (2) becomes du r,/ x u + x — - R(u) dx or x du = [i?(u) — u] dx

(3)

If R(u) = u, then Eq. (2) is simply dy = y dx

x

and this is separable at the outset. If R(u) ^ u, then we can divide (3) by the product x[i?(w) — u], getting du

_ dx

R(u) — u

x

The variables have now been separated, and the equation can be integrated at once. Finally, by replacing u by its value y/x, we can obtain the equation defining the original dependent variable y as a function of x.

EXAMPLE 2

Solve the equation (x2 + 3y2) dx — 2xy dy = 0 By inspection, this equation is homogeneous, since all terms in the coefficient of each differential are of the second degree. Hence we substitute y = ux and dy = u dx + x du, getting (x2 + 3u2x2) dx - 2x2u(u dx + x du) = 0

or, dividing by x2 and collecting terms, (1 + u2) dx — 2xu du = 0

14

Chapter 1

Ordinary Differential Equations of the First Order

Separating variables, we obtain dx

2u du

x

1 + u2

0

=

and then, by integrating, we find In |jc| — In 11 -h u21 = c This can be written as In

1

+

= In ec = In k

where k = ec > 0

u2

Hence |x/(l + u2)| = k\ or, replacing u by y/x and dropping absolute values, ---- = +k 1 + (y/x)2

-

Finally, clearing of fractions, we have x3 = K(x2 + y2)

K = ±k

From the preceding steps, it appears that K must be different from zero. However, it is easy to verify by direct substitution that the function corresponding to K = 0, namely, x = 0, is also a solution of the given equation. Hence, in the general solution we have just obtained, K is actually unrestricted.

EXERCISES

Determine which, if any, of the following functions are homogeneous: sin

1 + y2

+

+ y2

- In (x + y)

+ y2 tan

x + 2y

r

+ y2 + 1

3x - y xy + 2 Prove that the substitution x = vy will also transform any homogeneous first-order differential equation into one which is separable. Under what conditions, if any, do you think that the substitution x = vy would be more convenient than the substitution y = «x? Show that the product of a homogeneous function of degree m and a homogeneous function of degree n is a homogeneous function of degree m + n. Show that the quotient of a homogeneous function of degree m by a homogeneous function of degree n is a homogeneous function of degree m — n. If f(x,y,c0 = 0 and f(x,y,c2) = 0 are two solution curves of a homogeneous first-order differential equation, and if Pi and P2 are, respectively, the points of intersection of these curves and an arbitrary line, y = mx, through the origin, prove that the slopes of these two curves at Pt and P2 are equal. Find a general solution of each of the following differential equations: 10

(x2 + y2) dx = 2xy dy

11

2x/ = y — x

12

xy' — y = Vx2 — y2

13

x2 dy = (xy — y2) dx

Solve each of the following equations and discuss the family of solution curves: dy _ 2 x - y x — 2y dx dy _ x + 2y dx

2x + y

V

15 17

dy _

x - y

x + 3y dy _ x + y

dx

dx

x - y

Section 1.5

Exact First-Order Equations

15

Find the particular solution of each of the following equations which satisfies the given conditions: 18

x2y dx = {x3 — y3) dy

x = 1, y = 1

19 20 21

xy' = y + y/x2 + y2 (3y3 — a:3) dx = 3*y2 dy

* = 4, y = 3

22 23 24

04 + y4) dx = 2x3y dy

, y , y y = sec - -I— x x {x3 + y3) dx = 2xy2 dy

x — 1, y = 2 x = 1, y = 0 x = 2 ,y — n

x = 1, y = 0 If aB ^ bA, show that by choosing d and D suitably the equation dy

ax + by + c

dx

Ax + By + C

can be reduced to a homogeneous equation in the new variables t and z by the substitutions x = t + d

and

y = z + D

Using the substitutions described in Exercise 24, find a general solution of each of the following equations: x — y + 5 ^ , 2x + 2y + 1 26 y 25 y' = x + y — 1 " 3x + y — 2 27 Discuss Exercise 24 in the case when aB = bA. Hint: Recall Exercise 27, Sec. 1.3. 28 Prove that rM- c = 0 is a sufficient condition for all solutions of the equation y' = {ax + by)/(cx + ey) to be conics. Prove further that when this is the case, the conics are all ellipses if c2 + ae < 0 and are all hyperbolas if c2 + ae > 0. 29 Extend the result of Exercise 28 by showing that b -I- c = 0 is also a necessary condition for all solutions of the equation y' = {ax + by)/{cx + ey) to be conics. 30 If M{x,y) dx = N{x,y) dy is a homogeneous equation, prove that if it is expressed in terms of the polar coordinates r and 9 by means of the substitutions x = r cos 9 and y = r sin 6, it becomes separable. Solve each of the following equations, using the method described in Exercise 30: x + 2y

31

/ = '--

33

Give an example of a function which is homogeneous according to our definition but is not homogeneous if the condition f{Xx,Xy) = X"f{x,y) is required to hold for all real values of A. If f{x,y) is a homogeneous function of degree «, show that

34

32

x — y

/

9/3/

2x — y

,

x — + y — = nf dx dy

What is the generalization of this result to functions of more than two variables? (This result is commonly referred to as Euler’s theorem for homogeneous functions.)

1.5 Exact First-Order Equations Associated with each suitably differentiable function of two variables f(x,y) is an expression called its total differential, namely, df=5-fdx ox

dy dy

16

Chapter 1

Ordinary Differential Equations of the First Order

Conversely, if the differential equation M(x,y) dx + N(x,y) dy — 0 has the property that M(x,y) = %■ dx

and

N(x,y) = ~dy

then it can be written in the form — dx + — dy — df = 0 8x dy from which it follows that f(x,y) = k is a solution for all values of the constant k. An equation of this sort is said to be exact since, as it stands, its left member is an exact differential. When M(x,y) and N(x,y) are sufficiently simple, it is possible to tell by inspection whether or not there exists a function / with the property that yf- = M(x,y) dx

and

= N{x,y) dy

In general, however, this cannot be done, and it is desirable to have a straightforward test to determine when a given first-order equation is exact. Such a criterion is provided by the following theorem.

THEOREM 1 If dM/dy and dNjdx are continuous, then the differential equation M(x,y) dx + N(x,y) dy — 0 is exact if and only if dM _ dN dy

dx

Proof To prove the theorem, let us assume first that the given equation is exact. Under this assumption there exists a function / such that and dM =

Hence,

dy

d2f

efiV =

and

dy dx

dx

d2f dx dy

Moreover, d2f/(dy dx) and d2f/(dx dy) are continuous since we have just found them to be equal, respectively, to dM/dy and dN/dx, which are continuous by hypothesis. Therefore it follows, from the familiar properties of partial derivatives, that the order of differentiation is immaterial and d2f dy dx

=

d2f dx dy

Hence dM/dy = dN/dx, and the “only if” part of the theorem is established. To complete the proof we must now show that if dM/dy = dN/dx, then there is a function / such that df/dx = M and df/dy = N. To do this, let us first integrate

V

Section 1.5

Exact First-Order Equations

17

M(x,y) with respect to x, holding y fixed. This gives us the expression (1)

f(x,y) =

M(x,y) dx + c(y)

a arbitrary

in which, since the integration is done with respect to x while y is held constant, the integration “constant” is actually a function of y to be determined. Clearly, df/dx = M(pc,y); and our proof will be complete if we can determine c(y) so that df/dy — N(x,y). Now, observing that under the hypothesis that dMjdy is continuous the operations of integrating with respect to x and differentiating with respect to y can legitimately be interchanged, and recalling our current assumption that dM/dy — dN/dx, we have, from (1),

w dy

M{x,y) dx + c'(y) dy

( * dM

~T-dx + c'(y)

«

dy dx + c'(y)

= N(x,y) - N(a,y) + c\y) Thus, df/dy will equal N(x,y), as required, if c(y) is determined so that c'(y) = N(a,y), that is, if c(y) = f N(a,y) dy

& arbitrary

We have thus shown that if dM/dy = dN/dx, then (2)

f(x,y) = | M(x,y)dx+ N(a,y)dy Ja Jb

is a function such that df = — dx + — dy = M{x,y) dx + N(x,y) dy dx dy This establishes the “if” assertion of the theorem, and our proof is complete. Since the proof of the preceding theorem tells us that when the equation M(x,y) dx + N(x,y) dy = 0 is exact, its left member is, in fact, the total differential of the function/defined by (2), it follows that the solution in this case can be found at once by integration. Thus we have the following corollary.

COROLLARY 1

If the differential equation M(x,y) dx + N(x,y) dy = 0 is exact,

then, for all values of the constant k,

j*

M(x,y) dx +

is a solution of the equation.

j*

N(a,y) dy = k

18

Chapter 1

Ordinary Differential Equations of the First Order

EXAMPLE 1

Show that the equation (2x + 3y — 2) dx + (3x — 4y + 1) dy = 0 is exact, and find a general solution. Applying the test provided by Theorem 1, we find 8M

d(2x + 3y - 2)

8y

8y



J

3N

d(3x - 4y + 1)

ox

dx

Since the two partial derivatives are equal, the equation is exact. Its solution can therefore be found by means of Corollary 1, Theorem 1:

r

i

(2x + 3y — 2) dx +

(x2 + 3xy - 2x)

(3a — 4y + 1) dy = k

+ (3 ay — 2y2 + y)

= k

k2 + 3 xy — 2y2 — 2 x + y = k + a2 + 3 ab — 2 b2 — 2 a + b = K

Occasionally a differential equation which is not exact can be made exact by multiplying it by some simple expression. For example, if the (exact) equation 2xy3 dx + 3x2yz dy = 0 is simplified by the natural process of dividing out the common factor xy2, the resulting equation, namely, 2y dx + 3x dy = 0, is not exact. Conversely, however, the last equation can be restored to its original exact form by multiplying it through by xy2. This illustrates the general resultf that every first-order equation which possesses a family of solutions can be made exact by multiplying it by a suitable factor, called an integrating factor. In general, the determination of an integrating factor for a given equation is difficult. However, as the following examples show, in particular cases an integrating factor can often be found by inspection. EXAMPLE 2

Show that l/(x2 + y2) is an integrating factor for the equation (x2 + y2 — x) dx — y dy = 0, and then solve the equation. The test provided by Theorem 1 shows that in its present form the given equation is not exact. However, if it is multiplied by the indicated factor, it can be rewritten in the form (1--——- | dx--—- dy = 0

\

* +

yJ

x

+

Testing again, we find that the last equation is exact, and we can now use Corollary 1 to find a general solution of it. However, it is simpler to observe that it can also be written , x dx y y dy dx----— = 0 or dx - \ r/fln (a2 + y2)] = 0 x2 + y2

Hence, integrating, we have x — In yjx2 + y2 = k EXAMPLE 3

Find an integrating factor for the equation y dx + (x2y3 + x) dy = 0, and solve the equation. Since this equation can be rewritten in the form (y dx + x dy) + x2y3 dy = 0

f See, for instance, Golomb and Shanks, op. cit., pp. 52-53.

V.

Section 1.5

Exact First-Order Equations

19

and since y dx + x dy = d(xy), it is natural to multiply the equation by llx2y2, getting

dtoO , —— +

.

n

ydy = 0

(xy)2

This equation can now be integrated by inspection, and we have 1

y2

xy

2

=k

EXAMPLE 4

Find an integrating factor for the equation x dy - y dx = (4x2 + y2) dy, and solve the equation. In this equation, the terms on the left seem related equally well to dx

, (x\

W -

y dx — x dy

J'2

If we pursue the first suggestion and multiply the equation by llx2, we obtain

This equation is still not exact, but it is separable, and division by 4 + y2/x2 gives us =

d(y/x)

4 + (y/x)2

y

Integrating this, we have finally j Tan-1

= y + k

The results of the last three examples suggest the following observations, which are often helpful: a. If a first-order differential equation contains the combination x dx + y dy = \ d(x2 + y2), try some function of x2 + y2 as a multiplier to make the equation integrable. b. If a first-order differential equation contains the combination x dy + y dx = d(xy), try some function of xy as a multiplier to make the equation integrable. c. If a first-order differential equation contains the combination x dy — y dx, try l/x2 or l/y2 as a multiplier to make the equation integrable. EXERCISES

Show that the following equations are exact and integrate each one: 1 2 3

(y2 — 1) dx + (2xy — sin y) dy = 0 (2xy + x3) dx + (a2 + y2) dy = 0 (3a2 — 6xy) dx — (3a2 + 2y) dy — 0

4 5

(aVa2 + y2 + y) dx + (yVa2 + y2 + a) dy = 0 (2ay* + sin y) dx + (4x2y3 + a cos y) dy = 0

Find a general solution of each of the following equations by first multiplying by a suitable factor and then integrating: 6 8

10

y( 1 + xy) dx + (2y - x) dy = 0 (xy2 + y) dx + (x - x2y) dy = 0

a dy + 3 y dx — xy dy

7 9

3(y4 + 1) dx + 4ay3 dy = 0 (x2 + y2 + 2a) dy = 2y dx

20

Chapter 1

Ordinary Differential Equations of the First Order

Solve each of the following equations by two methods : 11 13 15 16

17 18

19

2y dx + (3y - 2x) dy = 0 dx dy x dy + y dx --—

_

y

12

(x2 - y2) dy = 2xy dx 1+x2 2x In y dx + -dy = 0

14

x

y

\lx2 + y2 dx = x dy — y dx Solve the equation (xy2 - y) dx + (x2y - x) dy = 0 first by integrating it as an exact equation and then by multiplying it by 1 /x2y2 before integrating it. Reconcile

your results. Show that if the equation of Exercise 16 is multiplied by any differentiable function of the product xy, it is still exact. If (x,y) is an integrating factor of the differential equation M(x,y) dx + N(x,y) dy = 0, show that satisfies the partial differential equation

Show that f(,x,y) = k is a general solution of the differential equation M(x,y) dx + N(x,y) dy = 0 if and only if 0/

0/

M— - N— = 0 dy dx

20

21

22

23 24

Using the result of the preceding exercise, show that if the equation M(x,y) dx + N(x,y) dy = 0 is both homogeneous and exact, its solution is xM(x,y) + yN(x,y) = k. Hint: Recall the result of Exercise 34, Sec. 1.4. Show that if {x,y) is an integrating factor leading to the solution f(x,y) = k for the differential equation M{x,y) dx + N(x,y) dy = 0, then $F(f) is also an integrating factor, where F is an arbitrary differentiable function. If the equation M(x,y) dx + N(x,y) dy = 0 is homogeneous, show that 1 /(xM + yN) is an integrating factor. Hint: Observe that M dx + N dy

dx

(x dy — y dx)N

xM + yN

x

x(xM + yN)

dx

(x dy — y dx)/x2

x

M/N + y/x

Prove the “if” part of Theorem 1 by first integrating N(x,y) with respect to y. Show that the arbitrary constants a and b which appear in the formula of Corollary 1, Theorem 1, add no generality to the solution. Hint: Consider the partial deriv¬ atives with respect to a and b of the left-hand side of the formula.

1.6 Linear First-Order Equations First-order equations which are linear form an important class of differential equations which can always be routinely solved by the use of an integrating factor. By definition, a linear first-order differential equation cannot contain products, powers, or other nonlinear combinations of y or y'. Hence its most general form is F(x) ^ + G(x)y = H(x) dx If we divide this equation by F(x) and rename the coefficients, it appears in the more usual form

(1)

ydx

V

+

my

= CM

Section 1.6

21

Linear First-Order Equations

The presence of two terms on the left side of (1) involving, respectively, dy/dx and y suggests strongly that this expression is in some way related to the derivative of a product, say (j)(x)y, having y as one factor. Now the derivative of (p(x)y is

= m ii + WMy

(2) dx

dx

dx

and the left member of (1) can be made identically equal to this if we first multiply Eq. (1) by (p(x), getting (3)

0(x) ^ + 0(x)P(x)y =