Second Course in Ordinary Differential Equations for Scientists and Engineers

450 Pages·1988·17.75 MB·English

by  Mayer Humi

Universitext
Editors
J. Ewing
P.R. Halmos
F.W. Gehring
Universitext
Editors: J. Ewing, F.W. Gehring, and P.R. Halmos
Booss/Bleecker: Topology and Analysis
Charlap: Bieberbach Groups and Flat Manifolds
Chern: Complex Manifolds Without Potential Theory
Chorin/Marsden: A Mathematical Introduction to Fluid Mechanics
Cohn: A Classical Invitation to Algebraic Numbers and Class Fields
Curtis: Matrix Groups, 2nd ed.
van Dalen: Logic and Structure
Devlin: Fundamentals of Contemporary Set Theory
Edwards: A Formal Background to Mathematics I alb
Edwards: A Formal Background to Mathematics II alb
Endler: Valuation Theory
Frauenthal: Mathematical Modeling in Epidemiology
Gardiner: A First Course in Group Theory
Godbillon: Dynamical Systems on Surfaces
Greub: Multilinear Algebra
Hermes: Introduction to Mathematical Logic
Humi/Miller: Second Course in Ordinary Differential Equations
for Scientists and Engineers
Hurwitz/Kritikos: Lectures on Number Theory
Kelly/Matthews: The Non-Euclidean, The Hyperbolic Plane
Kostrikin: Introduction to Algebra
Luecking/Rubel: Complex Analysis: A Functional Analysis Approach
Lu: Singularity Theory and an Introduction to Catastrophe Theory
Marcus: Number Fields
McCarthy: Introduction to Arithmetical Functions
Mines/Richman/Ruitenburg: A Course in Constructive Algebra
Meyer: Essential Mathematics for Applied Fields
Moise: Introductory Problem Course in Analysis and Topology
0ksendal: Stochastic Differential Equations
Rees: Notes on Geometry
Reisel: Elementary Theory of Metric Spaces
Rey: Introduction to Robust and Quasi-Robust Statistical Methods
Rickart: Natural Function Algebras
Smith: Power Series From a Computational Point of View
Smorynski: Self-Reference and Modal Logic
Stanisic: The Mathematical Theory of Turbulence
Stroock: An Introduction to the Theory of Large Deviations
Sunder: An Invitation to von Neumann Algebras
Tolle: Optimization Methods
Mayer Humi
William Miller
Second Course in
Ordinary Differential Equations
for Scientists and Engineers
Springer-Verlag
New York Berlin Heidelberg London Paris Tokyo
Mayer Humi William Miller
Department of Mathematical Sciences Department of Mathematical Sciences
Worcester Polytechnic Institute Worcester Polytechnic Institute
Worcester. MA 01609 Worcester. MA 01609
USA USA
Library of Congress Cataloging-in-Publication Data
Humi, Mayer.
Second course in ordinary differential equations for
scientists and engineers.
(Universitext)
I. Differential equations. I. Miller. William.
II. Title.
QA372.H875 1988 515.3'52 87-32732
© 1988 by Springer-Verlag New York Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer-Verlag. 175 Fifth Avenue. New York. NY 10010. USA). except
for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of
information storage and retrieval. electronic adaptation, computer software. or by similar or dissimilar
methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names. trademarks. etc. in this publication. even if the
former are not especially identified. is not to be taken as a sign that such names. as understood by the
Trade Marks and Merchandise Marks Act. may accordingly be used freely by anyone.
Camera-ready copy provided by the authors.
9 8 7 6 5 4 3 2 I
ISBN-13: 978-0-387-96676-2 e-ISBN-13: 978-1-4612-3832-4
DOl: 10.1007/978-1-4612-3832-4
To Shulamit
To Gerry
PREFACE
The world abounds with introductory texts on
ordinary differential equations and rightly so in view
of the large number of students taking a course in this
subject. However, for some time now there is a growing
need for a junior-senior level book on the more
advanced topics of differential equations. In fact the
number of engineering and science students requiring a
second course in these topics has been increasing.
This book is an outgrowth of such courses taught by us
in the last ten years at Worcester Polytechnic
Institute.
The book attempts to blend mathematical theory
with nontrivial applications from varipus disciplines.
It does not contain lengthy proofs of mathemati~al
theorems as this would be inappropriate for its
intended audience. Nevertheless, in each case we
motivated these theorems and their practical use
through examples and in some cases an "intuitive proof"
is included. In view of this approach the book could
be used also by aspiring mathematicians who wish to
obtain an overview of the more advanced aspects of
differential equations and an insight into some of its
applications. We have included a wide range of topics
in order to afford the instructor the flexibility in
designing such a course according to the needs of the
students. Therefore, this book contains more than
enough material for a one semester course.
The text begins with a review chapter (Chapter 0)
which may be omitted if the students have just recently
completed an introductory course. Chapter 1 sets the
stage for the ·study of Boundary Value Problems, which
are not normally included in elementary texts.
Examples of important boundary value problems are
covered in Chapter 2. Although systems of ordinary
differential equations are contained in beginning
texts, a more detailed approach to such problems is
discussed in Chapter 3. Chapters 4 through 10 examine
specific applications of differential equations such as
Perturbation Theory. Stability. Bifurcations. (See
Table of Contents.)
A word about the numbering system used in this
book. The sections of each chapter are numbered
1.2.3 ... Occasionally there are sub-sections numbered
1.1. 1.2 .... Definitions. Theorems. Lemmas.
Corollaries. Examples start at 1 in each section.
Formulas are tied to sections e.g. (2.1). (2.2) are in
section 2. Exercises start at 1 in each chapter.
Finally. special thanks are due to Mrs. C. M.
Lewis who typed the manuscript. Her efforts went well
beyond the call of duty and she spared no time or
effort to produce an accurate and esthetically
appealing "camera ready" copy of the book. Thanks are
also due to the staff at Springer Verlag for their
encouragement and guidance.
CONTENTS
Page
CHAPTER 0: REVIEW 1
1. Solution of second order ordinary
differential equations by series 1
2. Regular singular points 10
3. Series solutions near a regular
singular point 11
CHAPTER 1: BOUNDARY VALUE PROBLEMS 17
1. Introduction 17
2. Adjoint differential equations and
boundary conditions 25
3. Self -adjoint systems 30
4. A broader approach to self-adjoint
systems 34
5. Sturm-Liouville theory 36
6. Introduction to orthogonality and
completeness 42
CHAPTER 2: SPECIAL FUNCTIONS 60
1. Hypergeometric series 60
2. Bessel functions 89
3. Legendre polynomials 120
4. Gamma function 128
CHAPTER 3: SYSTEMS OF ORDINARY
DIFFERENTIAL EQUATIONS 131
1. Introduction 131
2. Method of elimination 134
3. Some linear algebra 136
4. Linear systems with constant
coefficients 145
5. Linear systems with variable
coefficients 149
6. Elements of linear control theory 153
7. The Laplace transform 161
CHAPTER 4: APPLICATIONS OF SYMMETRY
PRINCIPLES TO DIFFERENTIAL EQUATIONS 169
1. Introduction 169
2. Lie groups 170
3. Lie algebras 174
4. Prolongation of the action 179
5. Invariant differential equations 185
6. The factorization method 193
7. Examples of factorizable equations 198
CHAPTER 5: EQUATIONS WITH PERIODIC
COEFFICIENTS 210
1. Introduction 210
2. Floquet theory for periodic equations 215
3. Hill's and Mathieu equations 222
CHAPTER 6: GREEN'S FUNCTIONS 228
1. Introduction 228
2. General definition of Green's function 231
3. The interpretation of Green's functions 235
4. Generalized functions 241
5. Elementary solutions and Green's
functions 246
6. Eigenfunction representation of Green's
functions 250
7. Integral equations 254
CHAPTER 7: PERTURBATION THEORY 263
1. Pre I iminar ies 263
2. Some basic ideas-regular perturbations 267
3. Singular perturbations 275
4. Boundary layers 280
5. Other perturbation methods 285
*6. Perturbations and partial
differential equations 291
*7. Perturbation of eigenvalue problems 296
*8. The Zeeman and Stark effects 301
CHAPTER 8: PHASE DIAGRAMS AND STABILITY 308
1. General introduction 308
2. Systems of two equations 313
3. Some general theory 328
4. Almost linear systems 334
5. Almost linear systems in R2 338
6. Liapounov direct method 344
7. Periodic solutions (limit cycles) 350
CHAPTER 9: CATASTROPHES AND BIFURCATIONS 356
1. Catastrophes and structural stability 356
2. Classification of catastrophe sets 363
3. Some examples of bifurcations 369
4. Bifurcation of equilibrium states in
one dimension 379
5. Hopf bifurcation 390
6. Bifurcations in Rn 394
CHAPTER 10: STURM IAN THEORY 401
1. Some mathematical preliminaries 401
2. Sturmian theory for first order equations 407
3. Sturmian theory for second order
equations 413
4. Prufer transformations 420