Advanced Engineering Mathematics [5 ed.] 0471862517, 9780471862512

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ADVANCED ENGINEERING MATHEMATICS Erwin Kreyszig

Frfth Edition

ADVANCED ENGINEERING MATHEMATICS

FIFTH EDITION

ADVANCED ENGINEERING MATHEMATICS ERWIN KREYSZIG Professor of Mathematics Ohio State University Columbus, Ohio

JOHN WILEY b SONS New York

Chichester

Brisbane

Toronto

Singapore

Copyright © 1983, by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Sections 107 and 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons. Library of Congress Cataloging in Publication Data Kreyszig, Erwin. Advanced engineering mathematics. Accompanied by instructor’s manual. Includes bibliographical references and index. 1. Mathematical physics. 2. Engineering mathematics. I. Title. QA401.K7 1983 510'.2462 82-15988 ISBN 0-471-86251-7 Printed in the United States of America 10 98765432

PREFACE

Purpose of the book.

This book is intended to introduce students of engi¬ neering and physics to those areas of mathematics which, from a modern point of view, seem to be the most important in connection with practical problems. Topics are chosen according to the frequency of occurrence in applications. New ideas of modern mathematical training, as expressed in recent symposia on engineering education, were taken into account. The book should suit those institutions that have offered extended mathematical training for a long time as well as those that intend to follow the general trend of broadening the program of instruction in mathematics. A course in elementary calculus is the sole prerequisite. The material included in the book has formed the basis of various lecture courses given to undergraduate and graduate students of engineering, physics, and mathematics in this country, in Canada, and in Europe.

Changes in the fifth edition The present edition differs essentially from the first, second, third and fourth editions in the following respects.

Problem sets.

The problems have been changed. More applications are in¬

cluded in the problems.

Modeling is given an even greater weight by further applications in various sections, emphasizing all three phases: modeling—solving—interpreting. Ordinary differential equations. Phase plane methods (discussion of critical points and stability) have been extended (Secs. 3.2, 3.3). Numerical multistep methods have been included (Sec. 21.2).

Partial differential equations.

New sections on numerical methods for ellip¬ tic, parabolic and hyperbolic equations are included in Chap. 21.

Numerical methods have been updated and are now presented in a com¬ puter-oriented algorithmic fashion and, for greater flexibility, arranged in three largely independent chapters: Chap. 19. Numerical Analysis Chap. 20. Numerical Methods in Linear Algebra Chap. 21. Numerical Methods for Differential Equations

Optimization is of increasing importance to the engineer. The new Chap. 22 introduces to basic concepts and methods in this field.

PREFACE

vi

Probability and statistics has been augmented by further applications. References.

The references in Appendix 1 have been updated and their

number has been substantially increased.

Historical notes. These have been very much increased both in number and in the details given. Content and arrangement The arrangement of the subject matter in major parts can be seen from the diagram.

Much space is devoted to ordinary differential equations, linear algebra and vector analysis, complex analysis, and numerical methods, probably the four most important fields for engineers. But also the length of the other chapters on Fourier series, partial differential equations, etc., is such that the presentation may serve as a text for courses of the usual type. To facilitate the use of parts of the book, the chapters are kept as independ¬ ent of one another as possible. The chapters are subdivided into relatively short sections. Each section in¬ cludes typical examples and problems illustrating the concepts, methods, and results and their engineering applications. Historical notes, references to original literature, and over 450 figures are included in the text.

References.

A list of some books for reference and further study can be found at the end of the book, starting on page A-l. Some formulas for special functions are included in Appendix 3, starting on page A-52.

Problems and answers.

The book contains more than 3500 carefully se¬ lected problems, which range from simple routine exercises to practical appli¬ cations of considerable complexity. Answers to odd-numbered problems are included at the end of the book, starting on page A-l.

PREFACE

vii

Tables of functions are included in Appendix 4, starting on page A-60. Suggestions for a sequence of courses.

The material may be taken in sequence and is then suitable for four consecutive semester courses, meeting 3-5 hours a week as follows: First semester.

Ordinary differential equations (Chaps. 1-5).

Second semester.

Linear algebra and vector analysis (Chaps. 6-9).

Third semester.

Fourier series and partial differential equations (Chaps. 10, 11), Numerical methods (Chaps. 19-21).

Fourth semester.

Complex analysis (Chaps. 12-18).

A course on engineering statistics (3-5 hours a week; Chap. 23) may be taught during any of those four semesters or afterwards.

Independent one-semester courses.

The book is also suitable for various independent one-semester courses meeting 3 hours a week; for example: Introduction to ordinary differential equations (Chaps. 1, 2). Laplace transformation (Chap. 5). Vector algebra and calculus (Chaps. 6, 8). Matrices and systems of linear equations (Chap. 7). Fourier series and partial differential equations (Chaps. 10, 11, Secs. 21.4 -21.7). Complex analysis (Chaps. 12-17). Numerical analysis (Chaps. 19-22).

Shorter courses.

Sections that may be omitted in a shorter course are indi¬ cated at the beginning of each chapter.

Principles for selection of topics.

Which topics should be contained in a book of the present type, and how should these topics be arranged and pre¬

sented? To find some answer to these basic questions we may take a look at the historical development of engineering mathematics. This development shows the following two interesting features: 1. Mathematics has become more and more important to engineering science, and it is easy to conjecture that this trend will also continue in the future. In fact, problems in modern engineering are so complex that most of them cannot be solved solely by using physical intuition and past experience. The empirical ap¬ proach has been successful in the solution of many problems in the past, but fails as soon as high speeds, large forces, high temperatures or other abnormal condi¬ tions are involved, and the situation becomes still more critical by the fact that various modern materials (plastics, alloys, etc.) have unusual physical properties. Experimental work has become complicated, time-consuming and expensive. Here mathematics offers aid in planning constructions and experiments, in evalu¬ ating experimental data, and in reducing the work and cost of finding solutions. 2. Mathematical methods which were developed for purely theoretical reasons suddenly became of great importance in engineering mathematics. Examples are the theory of matrices, conformal mapping, and the theory of differential equa¬ tions having periodic solutions. What are the reflections of this situation on the teaching of engineering mathematics? Since the engineer will need more and more mathematics,

viii

PREFACE

should we try to include more and more topics in our courses, devoting less and less time to each topic? Or should we concentrate on a few carefully selected basic ideas of general practical importance which are especially suitable for teaching the student mathematical thinking and developing his own creative ability? Eighty or ninety years ago no one was able to predict that conformal map¬ ping or matrices would ever be of importance in the mathematical part of engineering work. Similarly, it is difficult to conjecture which new mathemati¬ cal theories will have applications to engineering twenty or thirty years from now. But no matter what happens in that respect, if a student has a good training in the fundamentals of mathematics, he will meet the future needs because he will be able to get acquainted with new methods by his own further study. It follows that the most important objective and purpose in engineering mathematics seems to be that the student becomes familiar with mathematical thinking. He should learn to recognize the guiding principles and ideas “be¬ hind the scenes,” which are more important than formal manipulations. He should get the impression that mathematics is not a collection of tricks and recipes but a systematic science of practical importance, resting on a relatively small number of basic concepts and involving powerful unifying methods. He should soon convince himself of the necessity for applying mathematical pro¬ cedures to engineering problems, and he will find that the theory and its appli¬ cations are related to each other like a tree and its fruit. The student will see that the application of mathematics to an engineering problem consists essentially of three phases:

1. Modeling. Translation of the given physical information into a mathe¬ matical form {modeling). In this way we obtain a mathematical model of the physical situation. This model may be a differential equation, a system of linear equations, or some other mathematical expression. 2. Solving.

Treatment of the model by mathematical methods. This will lead to the solution of the given problem in mathematical form.

3. Interpreting.

Interpretation of the mathematical result in physical

terms. All three steps seem to be of equal importance, and the presentation in this book is such that it will help the student to develop skill in carrying out all these steps. In this connection, preference has been given to applications which are of a general nature. In a few considerations it will be unavoidable to rely on results whose proofs are beyond the level of a book of the present type. In any such case these points are marked distinctly, because hiding difficulties and oversimplifying matters is no real help to the student in preparing him for his professional work. Otherwise the book is self-contained: we present matters in detail instead of irritating the reader by frequent references to details in other books. These are some of the guiding principles I used in selecting and presenting the material in this book. I made the choice with greatest care, using past and present teaching and research experience and resisting the temptation to in¬ clude “everything which is important” in engineering mathematics.

PREFACE

ix

Particular efforts have been made in presenting the topics as simply, clearly, and accurately as possible; this also refers to the choice of the notations. In each chapter the level increases gradually, avoiding jumps and accumulations of difficult theoretical considerations. The end of a proof is marked by I. This sign is also used at the end of some of the definitions and at the end of a set of examples when followed by further text. Acknowledgment. I am indebted to many of my former teachers, colleagues, and students for advice and help in preparing this book. Various parts of the manuscript were distributed to my classes in mimeographed form and returned to me with suggestions for improvement. Discussions with engineers and math¬ ematicians (as well as written comments) were of great help to me; I want to mention particularly Professors S. Bergman(t), S. L. Campbell, J. T. Cargo, P. L. Chambrd, A. Cronheim, J. Delany, J. W. Dettman, D. Dicker, W. Fox, R. G. Helsel, W. N. Huff, J. Keener, E. C. Klipple, V. Komkow, H. Kuhn, G. Lamb, H. B. Mann, I. Marx, K. Millet, W. D. Munroe, H.-W. Pu, T. Rado(f) P. V. Reichelderfer, J. T. Scheick, H. A. Smith, J. P. Spencer, J. Todd, H. J. Weiss, A. Wilansky, who are all in this country, Professor H. S. M. Coxeter of Toronto, and Professors H. Florian, H. Graf, F. Hohenberg, K. Klotter, F. Reutter, C. Schmieden, H. Unger, H. Wielandt, all in Europe. I can offer here only an inadequate acknowledgment of my appreciation. Finally, I thank John Wiley and Sons for their effective cooperation and their great care in preparing this edition. Suggestions of many readers were evaluated in preparing the present edition. Any further comment and suggestion for improvement of the book will be grate¬ fully received. ERWIN KREYSZIG

(f

. v

’

_

CONTENTS

CHAPTER 1

Ordinary Differential Equations of the First Order 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12

1

Basic Concepts and Ideas, 1 Geometrical Considerations. Isoclines, 9 Separable Equations, 12 Equations Reducible to Separable Form, 21 Exact Differential Equations, 23 Integrating Factors, 26 Linear First-Order Differential Equations, 28 Variation of Parameters, 33 Electric Circuits, 36 Families of Curves. Orthogonal Trajectories, 42 Picard’s Iteration Method, 47 Existence and Uniqueness of Solutions, 50

CHAPTER 2

Ordinary Linear Differential Equations 2.1 2.2 2.3 2.4

56

Homogeneous Linear Equations of the Second Order, 56 Homogeneous Second-Order Equations with Constant Coefficients, 60 General Solution. Basis. Initial Value Problem, 62 Real Roots, Complex Roots, Double Root of the Characteristic

Equation, 67 2.5 Differential Operators, 72 2.6 Modeling: Free Oscillations, 75 2.7 Cauchy Equation, 84 2.8 Existence and Uniqueness of Solutions, 87 2.9 Homogeneous Linear Equations of Arbitrary Order, 94 2.10 Homogeneous Linear Equations of Any Order with Constant

2.11 2.12 2.13 2.14

Coefficients, 98 Nonhomogeneous Linear Equations, 101 Solving Nonhomogeneous Linear Equations, 103 Modeling: Forced Oscillations. Resonance, 107 Modeling of Electric Circuits, 114 xi

CONTENTS

xii

2.15 Complex Method for Obtaining Particular Solutions, 119 « 2.16 General Method for Solving Nonhomogeneous Equations, 122

CHAPTER 3

Systems of Differential Equations, Phase Plane, Stability 3.1 3.2 3.3

125

Systems of Differential Equations, 125 Phase Plane, 133 Critical Points. Stability, 138

CHAPTER 4

Power Series Solutions of Differential Equations. Orthogonal Functions 4.1 4.2 4.3

4.4 4.5 4.6 4.7 4.8 4.9

147

The Power Series Method, 147 Theoretical Basis of the Power Series Method, 150 Legendre’s Equation. Legendre Polynomials, 156 Extended Power Series Method. Indicial Equation, 161 Bessel’s Equation. Bessel Functions of the First Kind, 173 Bessel Functions of the Second Kind, 179 Orthogonal Sets of Functions, 184 Sturm-Liouville Problem, 188 Orthogonality of Legendre Polynomials and Bessel Functions, 193

CHAPTER 5

Laplace Transformation 5.1 5.2 5.3

5.4 5.5

5.6 5.7 5.8

198

Laplace Transform. Inverse Transform. Linearity, 199 Laplace Transforms of Derivatives and Integrals, 204 Shifting on the 5-Axis, Shifting on the t-Axis, Unit Step Function, 209 Differentiation and Integration of Transforms, 218 Convolution, 221 Partial Fractions, 226 Periodic Functions. Further Applications, 235 Table of Some Laplace Transforms, 245

CHAPTER 6

Linear Algebra Part I: Vectors 6.1 6.2 6.3

6.4 6.5 6.6

6.7 6.8 6.9

Scalars and Vectors, 248 Components of a Vector, 250 Addition of Vectors, Multiplication by Scalars, 252 Vector Spaces. Linear Dependence and Independence, 256 Inner Product (Dot Product), 260 Inner Product Spaces, 266 Vector Product (Cross Product), 268 Vector Products in Terms of Components, 270 Scalar Triple Product. Other Repeated Products, 275

247

CONTENTS

xiii

CHAPTER 7

Linear Algebra Part II: Matrices and Determinants 7.1 7.2 7.3

7.4 7.5

7.6 7.7 7.8 7.9

7.10 7.11 7.12 7.13 7.14 7.15

280

Basic Concepts, 280 Addition of Matrices, Multiplication by Scalars (Numbers), 283 Transpose of a Matrix. Special Matrices, 285 Matrix Multiplication, 289 Systems of Linear Equations. Gauss Elimination, 299 Rank of a Matrix, 307 Systems of Linear Equations: Existence and General Properties of Solutions, 310 The Inverse of a Matrix, 314 Determinants of Second and Third Order, 318 Determinants of Arbitrary Order, 324 Rank in Terms of Determinants. Cramer’s Rule, 336 Bilinear, Quadratic, Hermitian, and Skew-Hermitian Forms, 341 Eigenvalues, Eigenvectors, 345 Eigenvalues of Hermitian, Skew-Hermitian, and Unitary Matrices, 351 Systems of Linear Differential Equations, 355

CHAPTER 8

Vector Differential Calculus. Vector Fields

363

8.1 8.2 8.3 8.4 8.5

Scalar Fields and Vector Fields, 363 Vector Calculus, 366 Curves, 369 Arc Length, 372 Tangent. Curvature and Torsion, 375 8.6 Velocity and Acceleration, 379 8.7 Chain Rule and Mean Value Theorem for Functions of Several Variables, 382 8.8 Directional Derivative. Gradient of a Scalar Field, 386 8.9 Transformation of Coordinate Systems and Vector Components, 393 8.10 Divergence of a Vector Field, 397 8.11 Curl of a Vector Field, 402

CHAPTER 9

Line and Surface Integrals. Integral Theorems 9.1 9.2 9.3

9.4 9.5 9.6 9.7

9.8 9.9

Line Integral, 404 Evaluation of Line Integrals, 407 Double Integrals, 412 Transformation of Double Integrals into Line Integrals, 419 Surfaces, 424 Tangent Plane. First Fundamental Form. Area, 427 Surface Integrals, 434 Triple Integrals. Divergence Theorem of Gauss, 439 Consequences and Applications of the Divergence Theorem, 443

404

CONTENTS

xiv

9.10 Stokes’s Theorem, 448 9.11 Consequences and Applications of Stokes’s Theorem, 452 9.12 Line Integrals Independent of Path, 454

CHAPTER 10

Fourier Series and Integrals 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9

462

Periodic Functions. Trigonometric Series, 462 Fourier Series. Euler Formulas, 465 Functions Having Arbitrary Period, 472 Even and Odd Functions, 475 Half-Range Expansions, 479 Determination of Fourier Coefficients Without Integration, 483 Forced Oscillations, 488 Approximation by Trigonometric Polynomials. Square Error, 492 The Fourier Integral, 494

CHAPTER 1 1

Partial Differential Equations

503

11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10

Basic Concepts, 503 Modeling: Vibrating String. One-Dimensional Wave Equation, 506 Separation of Variables (Product Method), 507 D’Alembert’s Solution of the Wave Equation, 515 One-Dimensional Heat Flow, 520 Heat Flow in an Infinite Bar, 525 Modeling: Vibrating Membrane. Two-Dimensional Wave Equation, 529 Rectangular Membrane, 531 Laplacian in Polar Coordinates, 538 Circular Membrane. Bessel’s Equation, 541 11.11 Laplace’s Equation. Potential, 547 11.12 Laplace’s Equation in Spherical Coordinates. Legendre’s Equation, 550 11.13 Laplace Transformation Applied to Partial Differential Equations, 554

CHAPTER 12

Complex Numbers. Complex Analytic Functions 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9

Complex Numbers, 560 Polar Form of Complex Numbers. Triangle Inequality, 565 Curves and Regions in the Complex Plane, 568 Complex Function. Limit. Derivative. Analytic Function, 570 Cauchy-Riemann Equations. Laplace’s Equation, 575 Rational Functions. Root, 580 Exponential Function, 583 Trigonometric and Hyperbolic Functions, 586 Logarithm. General Power, 589

559

CONTENTS

XV

CHAPTER 13

Conformal Mapping 13.1 13.2 13.3 13.4 13.5 13.6

594

Mapping, 594 Conformal Mapping, 601 Linear Fractional Transformations, 606 Special Linear Fractional Transformations, 608 Mapping by Other Functions, 613 Riemann Surfaces, 620

CHAPTER 14

Complex Integrals 14.1 14.2 14.3 14.4 14.5 14.6

625

Line Integral in the Complex Plane, 625 Basic Properties of the Complex Line Integral, 631 Cauchy’s Integral Theorem, 633 Evaluation of Line Integrals by Indefinite Integration, 641 Cauchy’s Integral Formula, 643 The Derivatives of an Analytic Function, 646

CHAPTER 15

Sequences and Series 15.1 15.2 15.3 15.4 15.5 15.6

650

Sequences, 650 Series, 654 Cauchy Convergence Principle for Sequences and Series, 657 Monotone Real Sequences. Leibniz Test for Real Series, 661 Tests for Convergence and Divergence of Series, 664 Operations on Series, 671

CHAPTER 16

Power Series, Taylor Series, Laurent Series 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8

676

Power Series, 676 Functions Represented by Power Series, 684 Taylor Series, 688 Taylor Series of Elementary Functions, 693 Practical Methods for Obtaining Power Series, 695 Uniform Convergence, 698 Laurent Series, 706 Analyticity at Infinity. Zeros and Singularities, 712

CHAPTER 17

Integration by the Method of Residues 17.1 17.2

Residues, 719 The Residue Theorem, 723

719

CONTENTS

xvi 17.3 17.4

Evaluation of Real Integrals, 725 Further Types of Real Integrals, 729

4

CHAPTER 18

Complex Analytic Functions and Potential Theory 18.1 18.2 18.3 18.4

734

Electrostatic Fields, 734 Two-Dimensional Fluid Flow, 738 General Properties of Harmonic Functions, 745 Poisson’s Integral Formula, 749

CHAPTER 19

Numerical Analysis 19.1 19.2 19.3 19.4 19.5 19.6 19.7

754

Introduction, 755 Solution of Equations by Iteration, 760 Finite Differences, 769 Interpolation, 773 Splines, 781 Numerical Integration and Differentiation, 785 Asymptotic Expansions, 794

CHAPTER 20

Numerical Methods in Linear Algebra 20.1 20.2 20.3 20.4

20.5 20.6

804

Systems of Finear Equations. Gauss Elimination. Matrix Inversion, 804 Systems of Finear Equations: Solution by Iteration, 810 Systems of Linear Equations: Ill-conditioning, 815 Method of Least Squares, 818 Inclusion of Matrix Eigenvalues, 822 Determination of Eigenvalues by Iteration, 826

CHAPTER 21

Numerical Methods for Differential Equations 21.1 21.2 21.3 21.4 21.5 21.6 21.7

830

Numerical Methods for First-Order Differential Equations, 830 Multistep Methods for First-Order Differential Equations, 838 Numerical Methods for Second-Order Differential Equations, 842 Numerical Methods for Elliptic Partial Differential Equations, 847 Neumann and Mixed Boundary Value Problems. Irregular Boundary, 856 Numerical Methods for Parabolic Equations, 861 Numerical Solution of Hyperbolic Equations, 866

CHAPTER 22

Optimization 22.1 22.2

Basic Concepts. Unconstrained Optimization, 870 Linear Programming, 873

870

CONTENTS

22.3 22.4

xvii

Simplex Method, 877 Simplex Method: Degeneracy, Difficulties in Starting, 882

CHAPTER 23

Probability and Statistics 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 23.9 23.10 23.11 23.12 23.13 23.14 23.15 23.16 23.17 23.18 23.19 23.20

887

Nature and Purpose of Mathematical Statistics, 887 Tabular and Graphical Representation of Samples, 889 Sample Mean and Sample Variance, 895 Random Experiments, Outcomes, Events, 898 Probability, 902 Permutations and Combinations, 907 Random Variables. Discrete and Continuous Distributions, 912 Mean and Variance of a Distribution, 918 Binomial, Poisson, and Hypergeometric Distributions, 922 Normal Distribution, 927 Distributions of Several Random Variables, 934 Random Sampling. Random Numbers, 942 Estimation of Parameters, 944 Confidence Intervals, 947 Testing of Hypotheses, Decisions, 956 Quality Control, 966 Acceptance Sampling, 971 Goodness of Fit. x2-Test, 977 Nonparametric Tests, 980 Pairs of Measurements. Fitting Straight Lines, 983

APPENDIX 1

References

A-1

APPENDIX 2

Answers to Odd-Numbered Problems

A-7

APPENDIX 3

Some Formulas for Special Functions

A-52

APPENDIX 4

Tables

INDEX

A-60 1-1

tr-

4

ADVANCED ENGINEERING MATHEMATICS

m ORDINARY I DIFFERENTIAL EQUATIONS I OF THE FIRST ORDER Differential equations are of fundamental importance in engineering math¬ ematics because many physical laws and relations appear mathematically in the form of such equations. We shall consider various physical and geomet¬ rical problems which lead to differential equations, and we shall explain the most important standard methods for solving such equations. These meth¬ ods will in general involve integration. We shall pay particular attention to the derivation of differential equa¬ tions from given physical situations. This transition from the physical prob¬ lem to a corresponding “mathematical model” is called modeling. It is of great practical importance to the engineer and physicist, and will be illus¬ trated by typical examples. The first five chapters of the book are devoted to ordinary differential equations. In the present chapter we shall start with the simplest of these equations, the so-called differential equations of the first order. Differential equations are particularly convenient for modem computers. Corresponding numerical methods for obtaining approximate solutions of first-order differential equations are explained in Secs. 21.1 and 21.2. Prerequisite for this chapter: integral calculus. Sections which may be omitted in a shorter course: 1.8-1.12. References: Appendix 1, Part B. Answers to Problems: Appendix 2.

1.1 Basic Concepts and Ideas In this section we shall define and explain the basic concepts that are essential in connection with differential equations and illustrate these concepts by exam¬ ples. Then we shall consider two simple practical problems taken from physics and geometry. This will give us a first idea of the nature and purpose of differ¬ ential equations and their application. By an ordinary differential equation we mean a relation which involves one or several derivatives of an unspecified function y of x; the relation may also involve y itself, given functions of x, and constants. For example, (1)

y' — cos x,

(2)

y" + 4y = 0,

(3)

x2y"'y' + 2 exy" = (x2 + 2 )y‘

are ordinary differential equations. 1

2

FIRST-ORDER DIFFERENTIAL EQUATIONS

CHAP. 1

The term ordinary distinguishes such an equation from a partial differential equation, which involves partial derivatives of an unspecified function of two or more independent variables. For example,

d2u 32u _ ^ dx2 + 3y2 ~ is a partial differential equation. In the present chapter we shall consider only ordinary differential equations. An ordinary differential equation is said to be of order n if the «th derivative of y with respect to x is the highest derivative of y in that equation. The notion of the order of a differential equation leads to a useful classifica¬ tion of the equations into equations of first order, second order, etc. Thus (1) is a first-order equation, (2) is of the second order, and (3) is of the third order. In the present chapter we shall consider first-order equations. Equations of second and higher order will be discussed in Chaps. 2-5. A function

y = g(x)

(4)

is called a solution of a given first-order differential equation on some interval, say, a < x < b (perhaps infinite) if g(x) is defined and differentiable through¬ out that interval and is such that the equation becomes an identity when y and y' are replaced by g and g', respectively. For example, the function

y

=

g(x)

= e2*

is a solution of the first-order differential equation

for all x, because by differentiation we obtain g' = 2e2*, and by inserting g and g' we see that the equation reduces to the identity 2e2x — 2e2x. Sometimes a solution of a differential equation will appear as an implicit function, that is, implicitly given in the form G(x,y) = 0, and is then called an implicit solution, in contrast to the explicit solution (4). For example,

x2 + y2 — 1=0

O' > 0)

SEC. 1.1

BASIC CONCEPTS AND IDEAS

3

is an implicit solution of the differential equation

77'

=

-x

on the interval —1 < x < 1, as the student may verify. The principal task of the theory of ordinary differential equations is to find all the solutions of a given differential equation and investigate their proper¬ ties. We shall discuss various standard methods which have been developed for that purpose. A differential equation may have many solutions. Let us illustrate this fact by the following examples. EXAMPLE 1 Each of the functions y = sin x,

y = sin x + 3,

y = sin x — f

is a solution of the differential equation (1), y' = cos x, and we know from calculus that every solution of this equation is of the form (5)

y = sin x + c

where c is a constant. If we regard c as arbitrary, then (5) represents the totality of all solutions of the differential equation. (See Fig. 1.)

EXAMPLE 2 The student may verify that each of the functions (6)

y = ex,

y = 2ex,

y = -fe*

is a solution of the differential equation

y' —y for all x. We shall see later that every solution of this equation is of the form

(7)

y

=

ce

4

FIRST-ORDER DIFFERENTIAL EQUATIONS

CHAP. 1

where c is a constant. Formula (7), with arbitrary c, thus represents the totality of all solutions of the differential equation. (See Fig. 2.)

*

Our examples illustrate that a differential equation may (and, in general, will) have more than one solution, even infinitely many solutions, which can be represented by a single formula involving an arbitrary constant c. It is custom¬ ary to call such a function which contains an arbitrary1 constant a general solution of the corresponding differential equation of the first order. If we assign a definite value to that constant, then the solution so obtained is called a particular solution. Thus (7) is a general solution of the equation y' =y, and (6) are particular solutions. In the following sections we shall develop various methods for obtaining general solutions of first-order equations. For a given equation, a general solu¬ tion obtained by such a method is unique, except for notation, and will then be called the general solution of that differential equation. We mention that in some cases there may be further solutions of a given equation which cannot be obtained by assigning a definite value to the arbitrary constant in the general solution; such a solution is then called a singular solution of the equation. For example, the equation (8)

y'2 - xy' + y = 0

has the general solution y = cx — c2 representing a family of straight lines, where each line corresponds to a definite value of c. A further solution is

1The range of the constant may have to be restricted in some cases to avoid imaginary expres¬ sions or other degeneracies.

SEC. 1.1

BASIC CONCEPTS AND IDEAS

5

Fig. 3. Singular solution (representing a parabola) and particular solutions of the equation (8)

as the reader may readily verify by substitution. This is a singular solution of (8), since we cannot obtain it by assigning a definite value to c in the general solution. Obviously, each particular solution represents a tangent to the parabola represented by the singular solution (Fig. 3). Singular solutions will rarely occur in engineering problems. We mention that in some books, a general solution means a formula which includes all solutions of an equation, that is, both the particular and the singular ones. We do not adopt this definition for two reasons. First of all, it is frequently quite difficult to prove that a formula includes all solutions; hence that definition of a general solution is rather useless in practice. Furthermore, we shall see later that a large and very important class of equations (the so-called linear differential equations) does not have singular solutions, and our definition of a general solution can be easily generalized to equations of higher order so that the resulting notion includes all solutions of a differential equation which is linear.

We shall see that the conditions under which a given differential equation has solutions are fairly general. But we should note that there are simple equa¬ tions which do not have solutions at all, and others which do not have a general solution. For example, the equation y'2 = — 1 does not have a solution for real y. (Why?) The equation \y'\ + \y\ — 0 does not have a general solution, be¬ cause its only solution is y = 0.

Modeling Differential equations are of great importance in engineering, because many physical laws and relations appear mathematically in the form of differential equations. Let us consider a simple physical example which will illustrate the typical steps of modeling, that is, the steps which lead from the physical situation {physical system) to a mathematical formulation {mathematical model) and so¬ lution, and to the physical interpretation of the result. This may be the easiest way of obtaining a first idea of the nature and purpose of differential equations and their applications. EXAMPLE 3. Radioactivity, exponential decay Experiments show that a radioactive substance decomposes at a rate proportional to the amount present. Starting with a given amount of substance, say, 2 grams, at a certain time, say, t = 0, what can be said about the amount available at a later time? Solution. 1st Step. Setting up a mathematical model (a differential equation) of the physical process. We denote by y{t) the amount of substance still present at time t. The rate of change is dy/dt.

6

FIRST-ORDER DIFFERENTIAL EQUATIONS

CHAP. 1

According to the physical law governing the process of radiation, dy/dt is proportional to*)’. Thus,

(9) Here A: is a definite physical constant whose numerical value is known for various radioactive substances. (For example, in the case of radium, k s — 1.4 • 10-11 sec-1.) Clearly, since the amount of substance is positive and decreases with time, dy/dt is negative, and so is k. We see that the physical process under consideration is described mathematically by an ordinary differential equation of the first order. Hence this equation is the mathematical model of that physical process. Whenever a physical law involves a rate of change of a function, such as velocity, acceleration, etc., it will lead to a differential equation. For this reason differential equations occur frequently in physics and engineering. 2nd Step. Solving the differential equation. At this early stage of our discussion no systematic method for solving (9) is at our disposal. However, (9) tells us that if there is a solution y(t), its derivative must be proportional toy. From calculus we remember that exponential functions have this prop¬ erty. In fact, the function ekt or more generally (10)

y(t) = cekt

where c is any constant, is a solution of (9) for all t, as can readily be verified by substituting (10) into (9). Since (10) involves an arbitrary constant, it is a general solution of the first order equation (9). [We shall see later (in Sec. 1.3) that (10) includes all solutions of (9), that is, (9) does not have singular solutions.] 3rd Step. Determination of a particular solution. It is clear that our physical process has a unique behavior. Hence we can expect that by using further given information we shall be able to select a definite numerical value of c in (10) so that the resulting particular solution will describe the process properly. The amount of substance y(t) still present at time t will depend on the initial amount of substance given. This amount is 2 grams at t = 0. Hence we have to specify the value of c so that y — 2 when t = 0. This condition is called an initial condition, since it refers to the initial state of the physical system. By inserting this condition (11)

7(0) =2

in (10) we obtain y( 0) = ce° = 2

or

c = 2.

If we use this value of c, then the solution (10) takes the particular form (12)

y(i) = 2ekt.

This particular solution of (9) characterizes the amount of substance still present at any time t ^ 0. The physical constant k is negative, and y(t) decreases, as is shown in Fig. 4. 4th Step. Checking. From (12) we have = 2 kekt = ky

and

j(0) = 2e° = 2.

We see that the function (12) satisfies the equation (9) as well as the initial condition (11). The student should never forget to carry out this important final step which shows him whether the function is (or is not) the solution of the problem.

Fig. 4. Radioactivity (exponential decay)

SEC. 1.1

BASIC CONCEPTS AND IDEAS

7

Let us illustrate that geometrical problems also lead to differential equations. EXAMPLE 4. A geometrical application Find the curve through the point (1, 1) in the xy-plane having at each of its points the slope —y/x. Solution. Clearly the function representing the desired curve must be a solution of the differential equation

03)

>'=-£•

We shall soon learn how to solve such an equation. For the time being the student may verify that

c

is a solution of (13) for every value of the constant c. Some of the corresponding curves are shown in Fig. 5. Since we are looking for the curve which passes through (1, 1), we must have y — 1 when x — 1. This yields c = 1. Flence the solution of our problem is

Problems for Sec. 1.1 In each case state the order of the differential equation and verify that the given function is a solution.

1. y' + 5y = 0, 2. y" + 4y = 0,

y = ce~bx y — A cos 2x + B sin 2x

3. y" - y = 0,

y - cxex + c2fr*

4. y'" =6,

y = x3 + ax2 + bx + c

5. y' 4- y tan x = 0,

y = c cos x

8

CHAP. 1

FIRST-ORDER DIFFERENTIAL EQUATIONS

Solve the following differential equations. 6. y' — sin x

7. y' — ex

9. y" — cos 2x

8. y" — 3

In each case verify that the given function is a solution of the differential equation. Graph the corresponding curves for some values of the constant c. 10. y' +y = 0,

11. y' + y — 2, y — ce 1 + 2

y — ce"x

13. yy' — —x,

12. xy' — 4y = 0, y = cx4

x2 + y2 = c

In each case verify that the given function is a solution of the corresponding differential equation and determine c so that the resulting particular solution satisfies the given condition. Graph this solution.

14. y' = 1,

y — x + c,

15. y' + 2xy = 0, 16. y' + y — 2, 17. y' = 3x2,

y = 0 when x = 7 y = 0.5 when x = 0

y — ce~x2, y = ce~x +2,

y = x3 + c,

y — 3.2 when x = 0 y — —l when x = 1

Find a first-order differential equation involving both y and y' for which the given function is a solution.

18. y — sin 2x

19. y = xe~x

20. y — —e~2x

21. y = x3 — 4

22. j = e-*2

23. y = x3

24. j = tan x

25. x2 + /2 = 9

26. (Falling body) Experiments show that if a body falls in vacuum due to the action of gravity, then its acceleration is constant (equal to g — 9.80 meters/sec2 = 32.1 ft/sec2; this is called the acceleration of gravity). State this law as a differen¬ tial equation for s(t), the distance fallen as a function of t. Show that by solving this equation one can obtain the familiar law

s(t) = y t2. (In practice, this also applies to the free fall in air if we can neglect the air resistance, for instance, if we drop a stone or an iron ball from a position not very high above the ground.) 27. If in Prob. 26 the body starts at / = 0 from initial position s — s0 with initial velocity v = v0, show that then the solution is s(t)

— y t2 +

V01

+

S0.

28. A rocket is shot straight up. During the initial stages of flight it has acceleration It meters/sec2. The engine cuts out at t = 10 sec. How high will the rocket go? (Neglect air resistance.) 29. (Exponential decay; atmospheric pressure) Observations show that the rate of change of the atmospheric pressure p with altitude h is proportional to the pres¬ sure. Assuming that the pressure at 6000 meters (about 18,000 ft) is \ of its value Po at sea level> find the formula for the pressure at any height. Hint. Remember from calculus that if y = ekx, then y' = kekx = ky. 30. (Exponential growth; a population model) If relatively small populations (of humans, animals, bacteria, etc.) are left undisturbed, they often grow according to Malthus’s law 2 which states that the time rate of growth is proportional to the 2THOMAS ROBERT MALTHUS (1766-1834), English social scientist, one of the leaders in classical national economy.

SEC. 1.2

GEOMETRICAL CONSIDERATIONS. ISOCLINES

9

population present. Formulate this as a differential equation. Show that the solution is y{t) = y0ekt. For the United States, observed values, in millions, are as follows. t

0

30

60

90

120

Year Population

1800 5.3

1830 12.9

1860 31.4

1890 62.9

1920 106.5

Use the first two columns for determining^ and k. Then calculate values for 1860, 1890 and 1920 and compare them with the observed values. Comment.

1.2 Geometrical Considerations. Isoclines We shall now start with a systematic study of first-order differential equations. Any such equation may be written in the form (1)

F(x,y,y') = 0.

(This is called the implicit form.) Not always, but in many cases we shall be able to write a first-order differential equation in the explicit form (2)

y' =f(pc,y).

Before we discuss various standard methods for solving equations of the form (2) we shall explain that (2) has a very simple geometric interpretation. This will immediately lead to a useful graphical method for obtaining a rough pic¬ ture of the particular solutions of (2), without actually solving the equation. We assume that the function / is defined in some region of the xy-plane, so that to each point in that region it assigns one (and only one) value. The solutions of (2) can be plotted as curves in the x^-plane. We do not know the solutions, but we see from (2) that a solution passing through a point (x0,j0) must have the slope f(x0,y0) at this point. This suggests the following method. We first graph some of the curves in the xy-plane along which f(x,y) is con¬ stant. These curves are called curves of constant slope, or isoclines of (2). Hence every isocline of (2) is given by f(x,y) — k = const (where the constant k differs from isocline to isocline). Along the isocline f(x,y) = k we draw a number of parallel short line segments (lineal elements) with slope k, which is the slope of solution curves of (2) at any point of that isocline. This we do for all isoclines which we graphed before. In this way we obtain a field of lineal elements, called the direction field of (2). With the help of the lineal elements we can now easily graph approximation curves to the (unknown) solution curves of the given equation (2) and thus obtain a qualita¬ tively correct picture of these solution curves. Let us illustrate the method by a simple example.

10

FIRST-ORDER DIFFERENTIAL EQUATIONS

CHAP. 1

EXAMPLE 1. Isoclines, direction field Graph the direction field of the first-order differential equation (3)

/ = xy

and an approximation to the solution curve through the point (1, 2). Solution. The isoclines are the equilateral hyperbolas xy = k together with the two coordinate axes. We graph some of them. Then we draw lineal elements by sliding a triangle along a fixed ruler. The result is shown in Fig. 6, which also shows an approximation to the solution curve passing through the point (1, 2). In the next section we shall see that (3) can easily be solved, so that our example merely illustrates the technical details of the method of direction field. This method is particularly helpful in engi¬ neering problems involving a first-order differential equation whose solution cannot be expressed in terms of known functions or is a very complicated expression.

Problems for Sec. 1.2 !• Verify that y - cex2/2 is a solution of (3) for any c. For what value of c do we obtain the solution shown in Fig. 6? In each case draw a good direction field. Plot several approximate solution curves. Then solve the equation analytically and compare, to get a feeling for the accuracy of the present method. 2, y' = 2x

3. y' =y

Draw good direction fields for the following differential equations and plot several approximate solution curves. 4. yy' + x = 0

5. y' = x2

6. y' = cosy

l.y'-xyy

8. y' = 2y/x

9. y' = y2

SEC. 1.2

GEOMETRICAL CONSIDERATIONS. ISOCLINES

11

Using direction fields, plot an approximate solution curve of the given differential equation satisfying the given condition.

10. y’ = x, y(-2) = 6

11. y' = 2x, y(2) = 4

12. 4y' + y = 0, 7(0) = -1

13. y' = x + y,

14. y'+y2 = 0, _y(5) = 0.25

15. 977'+ 4x = 0,

y(0) = 0 7(3) = -4

16. A body B moves on a straight line L. Let s(t) be its distance from a fixed point O on L. Suppose that at each instant the velocity of B equals 1 /s(t), and 5 = 1 when t = 0. Find the corresponding differential equation. Graph the direction field and the (approximate) solution curve s(t) of the problem.

17. Graph a direction field of y' — ay — by2 where a — 3 and b — 1, and draw from it the following conclusions. The isoclines 7 = 0 and 7 = 3 are also solution curves. All solution curves in the horizontal strip between7 = 0 and7 = 3 are s-shaped and monotone increasing. A1so7(x) —> 3 as x —» 00 and7(x) 0 as x —> — 00 for any such curve. What about solution curves above the line 7 = 3? In the lower half-plane 7 < 0?

18. (A population model) Malthus’s law states that the time rate of change of a population p(t) is proportional to p(t). This holds for many populations as long as they are not too large. A more refined model is the logistic law given by = ap — bp2.

For the United States, Verhulst predicted in 1845 the values a = 0.03 and b = 1.6 • 10~4, where p(t) is measured in millions. Graph a direction field for 1850 t 5= 1950 and 0 5=7 :£ 200. Plot the solution curve passing through the point (1900, 76) and compare the accuracy with some actual values 1850 23

1870 39

1890 63

1910 92

Show that the direction field indicates that p{t)

1930 123 a/b as t

00.

19. (Approximate integration) Using direction fields, find approximate values of the following integrals (the last two of which are not elementary).

20. (A pursuit problem, tractrix) In a pursuit problem, an object or target moves along a given curve and a second object or pursuer follows or pursues the target, that is, moves in the direction of the target at all times. Assume the target to be a ship S that moves along a straight line and the pursuer to be a destroyer D that

X

of S

Fig. 7. Tractrix and notations in Problem 20

12

FIRST-ORDER DIFFERENTIAL EQUATIONS

CHAP. 1

moves so that the distance a from D to S is constant, say, a = 1 (nautical mile). Show that (Fig. 7) y' — —y/y/a2 — y2. Draw a direction field (using a = 1) and plot an approximate solution curve corresponding to/(0) = a = 1; see Fig. 7. (Such a curve is called a tractrix, from Latin trahere — to pull.)

1.3 Separable Equations Many first-order differential equations can be reduced to the form

g(y)y'

(i)

=/(*)

by algebraic manipulations. Since y' — dy/dx, we find it convenient to write

g(y) dy

(2)

= f(x) dx,

but we keep in mind that this is merely another way of writing (1). Such an equation is called an equation with separable variables, or a separable equation, because in (2) the variables x and j are separated so that jc appears only on the right and y appears only on the left. By integrating on both sides of (2) we obtain

J

(3)

g{y) dy =

f

fix) dx + c.

If we assume that / and g are continuous functions, the integrals in (3) will exist, and by evaluating these integrals we obtain the general solution of (1). EXAMPLE 1 Solve the differential equation 9yy' + 4x = 0. Solution. By separating variables we have

9'y dy — — 4x dx. By integrating on both sides we obtain the general solution 9

2

2 ~ = -lx2 +c

or

jc2

y2

— + — = c

(C

The solution represents a family of ellipses.

EXAMPLE 2 Solve the differential equation

y’ = l + y2Solution. By separating variables and integrating we obtain

dy -- = dx, 1 + y2

arc tany = x + c,

y = tan (x + c).

It is of great importance that the constant of integration is introduced immediately when the integra¬ tion is carried out

SEC. 1.3

13

SEPARABLE EQUATIONS

EXAMPLE 3 Solve the differential equation y' = —2 xy.

Solution. By separating variables we have

— = — 2x dx

(y ^ 0).

7 Integration yields (4)

In Lv| = -x2 + c.

In fact, the result on the left may be verified by differentiation as follows. When y > 0, then (ln>-)' = y'/y. When y < 0, then -y > 0 and (In (-y))' = -■//(-y) = y'/y. Now 7 = \y\ when y > 0 and —y = |_y| when y < 0. Hence we may combine both formulas, finding (In [y|)' = y'/y. Furthermore, from (4) we obtain w = ^x2+c~ Noting that ea+b = eaeb and setting ec — c wheny > 0 and ec = — c wheny < 0, we finally have

This solution represents a family of so-called bell-shaped curves (Fig. 8). We may also admit c = 0, which gives the solution y = 0.

*

. .

Fig 8 Solutions of y‘ = -2xy (“bell-shaped curves” ) in the upper half plane (y > 0)

In many engineering applications we are not interested in the general solu¬ tion of a given differential equation but only in the particular solution y(x) satisfying a given initial condition, say, the condition that at some point x0 the solution y(x) has a prescribed value y0, briefly (5)

y(x0)

^o-

A first-order differential equation together with an initial condition is called an initial value problem. To solve such a problem, we must find the particular solution of the equation which satisfies the given initial condition.

14

CHAP. 1

FIRST-ORDER DIFFERENTIAL EQUATIONS

The term “initial value problem” is suggested by the fact that the independ¬ ent variable often is time, so that (5) prescribes the initial situation at some instant and the solution of the problem shows what happens later. EXAMPLE 4 Solve the initial value problem (x2 + 1)/ + y2 + Solution.

1

=

0,

7(0)

= 1.

1st Step. Solving the differential equation. Separating variables, we find

dy

_

dx

1 + y2

l + x2

By integration we obtain arc tan 7 = —arc tan x + c or arc tan y + arc tan x = c. Taking the tangent on both sides, we have (6)

tan (arc tan y + arc tan x) = tan c.

Now the addition formula for the tangent is tan {a + b) = 1

tan a + tan b — tan a tan b

For a = arc tan y and b = arc tan x this becomes

tan (arc tan y + arc tan x)

y + x 1

— xy'

Consequently, (6) may be written y + x

(7)

1

tan c.

— xy

2nd Step. Use of the initial condition. We determine c from the initial condition. Setting x = 0 and

y = 1 in (7), we have 1 = tan c, so that y + x 1

— xy

=

1

or

y=

1

— x

1

+ JC '

3rd Step. Checking. Check the result.

Applications EXAMPLE 5 (Newton’s law of cooling3) A copper ball is heated to a temperature of 100° C. Then at time r = 0 it is placed in water which is 3 Sir

ISAAC NEWTON (1642-1727), great English physicist and mathematician, became a pro¬

fessor at Cambridge in 1669 and master of the mint in 1699. He and the German mathematician and philosopher GOTTFRIED WILHELM LEIBNIZ (1646-1716) invented (independently) the differential and integral calculus. Newton discovered many basic physical laws and created the method of investigating physical problems by means of calculus. His Philosophiae naturalis principia mathematica (Mathematical Principles of Natural Philosophy, 1687) contain the develop¬ ment of classical mechanics. His work is of greatest importance to both mathematics and physics. Sometimes the Latin spelling LEIBNITZ is used; this spelling appeared in Leibniz’s scientific publications.

SEC. 1.3

15

SEPARABLE EQUATIONS

maintained at a temperature of 30° C. At the end of 3 minutes the temperature of the ball is reduced to 70° C. Find the time at which the temperature of the ball is reduced to 31° C. Physical information. Experiments show that the time rate of change of the temperature T of the ball is proportional to the difference between T and the temperature of the surrounding medium (Newton's law of cooling). Experiments also show that heat flows so rapidly in copper that at any time the temperature is practically the same at all points of the ball. Solution.

1st Step. Setting up the mathematical model In our case, the mathematical formulation of

Newton’s law of cooling is

(8)

^ =

-k(T

- 30),

where we denoted the constant of proportionality by — k in order that

k

> 0.

2nd Step. General solution. The general solution of (8) is obtained by separating variables; we find

T(t) = ce~kt + 30. 3rd Step. Use of initial condition. The given initial condition is 7( 0) = 100. The particular solution

satisfying this condition is 7X0 = 70e~fct + 30. 4th Step. Use of farther information. The constant

k can be determined

from the given information

7(3) = 70. We obtain

7(3) = 700 - c

6. In x sin uy = c

Show that the following differential equations are exact and solve them. 7. y dx + x dy = 0 9. (2x + e*) dx + xev dy = 0 11. sinh x cos y dx = cosh x sin y dy

8. (x dy - y dx)/x2 = 0 10. 2x Iny dx + y~xx2 dy = 0 12. 3re36 d6 + e3e dr — 0

13. (Warning) If a differential equation is exact, we can find u(x,y) = c from (6) by the method explained in the text. However, if the equation is not exact, that method will not work. To see this, consider x dy — y dx = 0. Show that this equation is not exact. Obtain u from (6), calculate du/dy from the result and show that (4b) yields a contradiction. Solve the equation by some other method which has been discussed before. 14. If an equation is separable, show that it is exact. Is the converse true? Are the following equations exact? Find the general solution. 15. (1 + x2) dy + 2xy dx = 0

16. x dy - 4y dx - 0

17. ydx + x( 1 + y) dy = 0

18. (2y dx + dy)e2x = 0

19. (3/ cos 3x dx - sin 3x dy)/y2 = 0

20. sin fry dx + (i cos fiy dy = 0

Under what conditions are the following equations exact? Solve the exact equations. 21. (ax + by) dx

+ (kx + ly) dy = 0

22. (x3 + xy2) dx + (ax2y + bxy2) dy — 0 Under what conditions are the following equations exact? 23. f(x)g(y) dx + h(x,y) dy = 0 24. [/(x) + g(y)) dx + [h(x) + /(/)] dy = 0 It will sometimes be possible to solve a differential equation by several methods. To illustrate this, solve the following equations (a) by the present method, (b) by separat¬ ing variables, (c) by inspection. 26. y sinh x dx + cosh x dy — 0 25. x dx + y dy = 0 27. 4x~5y dx = x~4 dy

28. cos x cosy dx = sin x siny dy

26

CHAP. 1

FIRST-ORDER DIFFERENTIAL EQUATIONS

Find and graph the curve having the given slope y' and passing through the given point (x,y).

29-y' = yr^9-

30-

y' = jzrf-

(0"»

Solve the following initial value problems.

31. 32. 33. 34.

(y - 1) dx + (x - 3) dy = 0, y(0) = 2/3 2(y sin 2x + cos 2x) dx = cos 2x dy, y(ir) = 0 3x2y* dx + 4x3y3 dy = 0, y(\) = 2 cos 2x cosh 2y dx + sin 2x sinh 2y dy — 0, y(it/A)

35. / =

—

0

7(1) = 0

1.6 Integrating Factors Sometimes a given differential equation (1)

P(x,y) dx + Q(x,y) dy = 0

is not exact but can be made exact by multiplying it by a suitable function F(x, y) (^0). This function is then called an integrating factor of (1). With some experience, integrating factors can be found by inspection. For this pur¬ pose the student should keep in mind the differentials listed in Example 2 below. In some important special cases, integrating factors can be determined in a systematic way which will be illustrated in the next section. EXAMPLE 1 Solve x dy — y dx = 0. Solution. The differential equation is not exact. An integrating factor is F = 1/x2, and we obtain F(x){x dy - ydx) = -dy J ** = d

= 0,

y = cx.

Theorem 1 If (1) is not exact and has a general solution u(x,y) = c, then there exists an integrating factor of {1) (even infinitely many such factors). Proof. By differentiating u(x,y) — c we have du ~

dx

dx +

Comparing this and (1), we see that

dy

dy = 0.

SEC. 1.6

27

INTEGRATING FACTORS

must hold identically. Hence there exists a function F(x, y) such that

Using these expressions for du/dx and du/dy, we have from the first formula in this proof (2)

du = FPdx + FQ dy = F(P dx + Q dy).

This shows that F is an integrating factor of (1). Furthermore, multiplication of (2) by a function H{u) yields the expression H(u)F(P dx + Q dy) = H(u) du which is exact. Hence H(u)F(x,y) is another integrating factor, and since H is arbitrary, there are infinitely many integrating factors of (1). I EXAMPLE 2 Find integrating factors of xdy — ydx = 0; cf. Example 1. Solution. Since x dy — y dx

y dx — x dy

B-

’K)=

!

r2

xy

x dy — y dx d( arctan^=^ „2 V x' x2 T y

,

the functions 1 /y2, 1 /xy, and l/(x2 + y2) are such factors. The corresponding solutions x — = c,

y arc tan — = c x

In — = c,

y

X

are essentially the same, because each represents a family of straight lines through the origin.

Problems for Sec. 1.6 Show that the given function is an integrating factor and solve: 1. 2y dx + x dy = 0, 3. xdy -ydx = 0,

2. siny dx + cosy dy = 0,

x

4. y2 dx + (1 + xy) dy

l/x2

5. 2 cos iry dx — tt sin -ny dy,

e21

= 0,

e*>

6. y cos x dx + 3 sin x dy = 0,

y2

In each case show that the given function is an integrating factor and find further integrating factors. 1. y dx + 2x dy = 0,

8. 2y dx + 3x dy = 0,

F =y

F = xy2

In each case find an integrating factor F and solve:

9. 3y dx + 2x dy

=

10. dx + (y/x)2 dy = 0

0

11. 2 dx — ey~x dy = 0 13.

12. x coshy dy — sinhy dx = 0

y cos x dx + 2 sin xdy — 0

14.

(y + 1) dx — (x + 1) dy = 0

Find integrating factors and solve the following initial value problems. 15. 2 dx + sec x cos y dy

- 0,

16. 2x2 dx — 3xy2 dy = 0, 17. 2 sin ydx + cosy dy 18. (2y + xy) dx

+

= 0,

2x dy = 0,

y(0) = 0 y(l) = 0 y(0) = w/2 y(3) = \/2

28

CHAP. 1

FIRST-ORDER DIFFERENTIAL EQUATIONS

Under what conditions are the following functions F integrating factors of (1)? Hint. Assume F(P dx + Q dy) — 0 to be exact and apply (5) in Sec. 1.5. 19. F = xk

20. F — ym

21. F = ex

22. F - xkym

23. Using the result of Prob. 19, solve Probs. 9, 10, and 12. 24. (Checking) Checking of solutions is always important. In connection with the present method it is particularly essential since one may have to exclude the function y(x) given by F(x,y) = 0. To see this, consider (xy)_1 dy — x~2 dx — 0; show that an integrating factor is F — y and leads to d(y/x) = 0, hence y = cx, where c is arbitrary, but F = y — 0 is not a solution of the original equation. 25. It is interesting to note that if F(x,y) is an integrating factor, 1 /F(x,y) = 0 may sometimes yield an additional solution of the corresponding equation. To see this, consider x dy — y dx — 0; show that an integrating factor is F = 1 /x2 and leads toy = cx. Show that \/F — x2 = 0 yields the additional solution x = 0 of the given equation which is not included in y — cx.

1.7 Linear First-Order Differential Equations A first-order differential equation is said to be linear if it can be written

(1)

y' + f(x)y — r(x).

The characteristic feature of this equation is that it is linear iny andywhereas / and r may be any given functions of x. If r{x) = 0, the equation is said to be homogeneous; otherwise, it is said to be nonhomogeneous. (Here, r = 0 means r = 0 for all x in the domain of r.) Let us find a formula for the general solution of (1) in some interval I, assuming that / and r are continuous in I. For the homogeneous equation (2)

y' + f(x)y — 0

this is very simple. Indeed, by separating variables we have

~ - —fix) dx

and thus

In |y| = — Jf(x) dx + c*

or (3)

y(x) = ce~5f(x)dx

(c = ±ec*

when

y ^ 0);

here we may also take c = 0 and obtain the trivial solution y = 0. To solve the nonhomogeneous equation (1), let us write it in the form (fy - r)dx + dy = 0

SEC. 1.7

LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS

29

and show that we can find an integrating factor F(x) depending only on x. If such a factor exists, F(x)(/y — r) dx + F(x) dy = 0 must be exact. Now for this equation, condition (5) of Sec. 1.5 takes the form

f[FUV-r)] = g

or

Ff =

dF dx

By separating variables we have f dx = dF/F. By integration we obtain In \F\ = Jf(x) dx. Hence ^(x) = eUx)

where

h(x) = Jf(x) dx.

This shows that eh(x) is an integrating factor of (1). Let us now multiply (1) by this factor, finding eh(y' + fy) = ehr. Since h' — f this may be written

A (yeh)

= ehr,

dx as the student may verify by differentiating the product yeh. Now we can inte¬ grate on both sides and obtain yeh = j ehr dx + c. Dividing both sides by eh we finally arrive at the desired formula (4)

y>(x) = e~h

J ehr dx + c j,

h = Jf{x) dx,

which represents the general solution of (1) in the form of an integral. (The choice of the values of the constants of integration is immaterial; cf. Prob. 2 below.)

EXAMPLE 1 Solve the linear differential equation y' — y — e2*.

Solution. Here /= -1,

r = e2x,

h=Jfdx = -x

and from (4) we obtain the general solution y(x) = ex

[ f e~xe2x

dx + c] = ex[ex + c] = cex +