me/UPSC_Prelims https://t.me/UPSC_Mains https://t.me/UPSC_Prelims https://t.me/UPSC_Mains www.me/UPSC_Prelims https://t.me/UPSC_Mains The sole aim of science is the honor of the human mind, and from this point of view a question about numbers is as important as a question about the system of the world. Jacobi DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES Second Edition George F. Simmons Professor of Mathematics Colorado College with a new chapter on numerical methods by JohnS. Robertson Department of Mathematical Sciences United States Military Academy McGraw-Hill, Inc.
New York St. Louis San Francisco Auckland Bogota Caracas Hamburg Lisbon London Madrid Mexico Milan Montreal New Delhi Paris San Juan Sao Paulo Singapore Sydney Tokyo Toronto https://t.me/UPSC_Prelims https://t.me/UPSC_Mains www.me/UPSC_Prelims https://t.me/UPSC_Mains This book was set in Times Roman. The editors were Richard Wallis and John M. Morriss ; the production supervisor was Louise Karam.
The cover was designed by Carla B auer. Project supervision was done by The Universities Press. Donnelley & Sons Company was printer and binder. DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES Copyright© 1991 , 1 972 by McGraw-Hill , Inc.
All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means , or stored in a data base or retrieval system , w:thout the prior written permission of the publisher. 2 3 4 5 6 7 8 9 0 DOC DOC 9 5 4 3 2 1 ISBN 0-07-057540-1 Library of Congress Cataloging-in-Publication Data Simmons, George Finlay, (date).
Differential equations with applications and historical notes I George F.me/UPSC_Prelims https://t.me/UPSC_Mains www.me/UPSC_Prelims https://t.me/UPSC_Mains ABOUT THE AUTHOR George Simmons has academic degrees from the California Institute of Technology, the University of Chicago , and Yale University. He taught at several colleges and universities before joining the faculty of Colorado College in 1962, where he is Professor of Mathematics. He is also the author of Introduction to Topology and Modern Analysis (McGraw-Hill , 1963), Precalculus Mathematics in a Nutshell (Janson Publications , 198 1 ) , and Calculus with Analytic Geometry (McGraw-Hill , 1985 ). When not working or talking or eating or drinking or cooking, Professor Simmons is likely to be traveling (Western and Southern Europe, Turkey , Israel , Egypt , Russia, China, Southeast Asia) , trout fishing (Rocky Mountain states) , playing pocket billiards , or reading (literature , history , biography and autobiography , science , and enough thrillers to achieve enjoyment without guilt).me/UPSC_Prelims https://t.me/UPSC_Mains www.me/UPSC_Prelims https://t.me/UPSC_Mains https://t.me/UPSC_Prelims https://t.me/UPSC_Mains www.me/UPSC_Prelims https://t.me/UPSC_Mains FOR HOPE AND NANCY my wife and daughter who still make it all worthwhile https://t.me/UPSC_Prelims https://t.me/UPSC_Mains www.me/UPSC_Prelims https://t.me/UPSC_Mains https://t.me/UPSC_Prelims https://t.me/UPSC_Mains www.me/UPSC_Prelims https://t.me/UPSC_Mains CONTENTS Preface to the Second Edition XV Preface to the First Edition xvii Suggestions for the Instructor xxi 1 The Nature of Differential Equations.
General Remarks on Solutions 4 3. Families of Curves. Growth, Decay, Chemical Reactions, and Mixing 17 5. Falling Bodies and Other Motion Problems 29 6.
Fermat and the Bernoullis 35 2 First Order Equations 47 7. The Hanging Chain. Simple Electric Circuits 71 3 Second Order Linear Equations 81 14. The General Solution of the Homogeneous Equation 87 16.
The Use of a Known Solution to Find Another 92 17. The Homogeneous Equation with Constant Coefficients 95 18. The Method of Undetermined Coefficients 99 xi https://t.me/UPSC_Prelims https://t.me/UPSC_Mains www.com xii https://t.me/UPSC_Prelims https://t.me/UPSC_Mains CONTENTS 19. The Method of Variation of Parameters 103 20.
Vibrations in Mechanical and Electrical Systems 106 21 .Newton's Law of Gravitation and the Motion of the Planets 115 22. Higher Order Linear Equations. Coupled Harmonic Oscillators 122 23. Operator Methods for Finding Particular Solutions 128 Appendix A.
Newton 146 4 Qualitative Properties of Solutions 155 24. Oscillations and the Sturm Separation Theorem 155 25. The Sturm Comparison Theorem 161 5 Power Series Solutions and Special Functions 165 26. A Review of Power Series 165 27.
Series Solutions of First Order Equations 172 28. Second Order Linear Equtions. Regular Singular Points 184 30. Regular Singular Points (Continued ) 192 31.
Gauss's Hypergeometric Equation 199 32. The Point at Infinity 204 Appendix A. Two Convergence Proofs 208 Appendix B. Hermite Polynomials and Quantum Mechanics 211 Appendix C.
Chebyshev Polynomials and the Minimax Property 230 Appendix E. Riemann's Equation 237 6 Fourier Series and Orthogonal Functions 246 33. The Fourier Coefficients 246 34. The Problem of Convergence 257 35.
Even and Odd Functions. Cosine and Sine Series 265 36. Extension to Arbitrary Intervals 272 37. The Mean Convergence of Fourier Series 285 Appendix A.
A Pointwise Convergence Theorem 293 7 Partial Differential Equations and Boundary Value Problems 298 39. Eigenvalues, Eigenfunctions, and the Vibrating String 302 41. The Heat Equation 311 42. The Dirichlet Problem for a Circle.
Sturm-Liouville Problems 323 https://t.me/UPSC_Prelims https://t.me/UPSC_Mains www.com :xiii https://t.me/UPSC_Prelims https://t.me/UPSC_Mains CONTENTS Appendix A. The Existence of Eigenvalues and Eigenfunctions 33 1 8 Some Special Functions of Mathematical Physics 335 44. Properties of Legendre Polynomials 342 46. The Gamma Function 348 47.
Properties of Bessel functions 358 Appendix A. Legendre Polynomials and Potential Theory 365 Appendix B. Bessel Functions and the Vibrating Membrane 371 Appendix C. Additional Properties of Bessel Functions 377 9 Laplace Transforms 381 48.
A Few Remarks on the Theory 385 50. Applications to Differential Equations 390 51. Derivatives and Integrals o f Laplace Transforms 394 52. Convolutions and Abel's Mechanical Problem 399 53.
More about Convolutions. The Unit Step and Impulse Functions 405 Appendix A. Abel 413 10 Systems of First Order Equations 417 54. General Remarks on Systems 417 55.
Homogeneous Linear Systems with Constant Coefficients 427 57. Volterra's Prey- Predator Equations 434 11 Nonlinear Equations 440 58. The Phase Plane and Its Phenomena 440 59. Types of Critical Points.
Critical Points and Stability for Linear Systems 455 61. Stability by Liapunov's Direct Method 465 62. Simple Critical Points of Nonlinear Systems 471 63. The Poincare- Bendixson Theorem 486 Appendix A.
Proof of Lienard's Theorem 497 12 The Calculus of Variations 502 65. Some Typical Problems of the Subject 502 66. Euler's Differential Equation for an Extremal 505 https://t.me/UPSC_Prelims https://t.me/UPSC_Mains www.me/UPSC_Prelims https://t.me/UPSC_Mains xiV CONTENTS 67. Isoperimetric problems 515 Appendix A.
Hamilton's Principle and Its Implications 526 13 The Existence and Uniqueness of Solutions 538 68. The Method of Successive Approximations 538 69. The Second Order Linear Equation 552 14 Numerical Methods 556 71. The Method of Euler 559 73.
An Improvement to Euler 565 75. Higher-Order Methods 569 76. Systems 573 Numerical Tables 577 Answers 585 Index 617 https://t.me/UPSC_Prelims https://t.me/UPSC_Mains www.me/UPSC_Prelims https://t.me/UPSC_Mains PREFACE TO THE SECOND EDITION "As correct as a second edition"-so goes the idiom. I certainly hope so , and I also hope that anyone who detects an error will do me the kindness of letting me know , so that repairs can be made.
As Confucius said , "A man who makes a mistake and doesn't correct it is making two mistakes. " I now understand why second editions of textbooks are always longer than first editions: as with governments and their budgets , there is always strong pressure from lobbyists to put things in, but rarely pressure to take things out. The main changes in this new edition are as follows: the number of problems in the first part of the book has been more than doubled; there are two new chapters , on Fourier Series and on Partial Differential Equations ; sections on higher order linear equations and operator methods have been added to Chapter 3; and further material on convolutions and engineering applications has been added to the chapter on Laplace Transforms. Altogether , many different one-semester courses can be built on various parts of this book by using the schematic outline of the chapters given on page xxi.
There is even enough material here for a two semester course , if the appendices are taken into account. Finally , an entirely new chapter on Numerical Methods (Chapter 14) has been written especially for this edition by Major John S. Robertson of the United States Military Academy. Major Robertson's expertise in these matters is much greater than my own, and I am sure that many users of this new edition will appreciate his contribution , as I do.
McGraw-Hill and I would like to thank the following reviewers for their many helpful comments and suggestions: D. Arterburn , New XV https://t.me/UPSC_Prelims https://t.me/UPSC_Mains www.me/UPSC_Prelims https://t.me/UPSC_Mains XVi PREFACE TO THE SECOND EDITION Mexico Tech ; Edward Beckenstein , St. John's University ; Harold Carda, South Dakota School of Mines and Technology ; Wenxiong Chen, University of Arizona; Jerald P. Dauer, University of Tennessee ; Lester B.
Fuller, Rochester Institute of Technology ; Juan Gatica, University of Iowa; Richard H. Herman , The Pennsylvania State Univer sity; Roger H. Marty, Cleveland State University ; Jean-Pierre Meyer, The Johns Hopkins University ; Krzysztof Ostaszewski, University of Louisville ; James L. Rovnyak , University of Virginia; Alan Sharples, New Mexico Tech ; Bernard Shiffman , The Johns Hopkins University ; and Calvin H.
Wilcox , University of Utah .me/UPSC_Prelims https://t.me/UPSC_Mains www.me/UPSC_Prelims https://t.me/UPSC_Mains PREFACE TO THE FIRST EDITION To be worthy of serious attention , a new textbook on an old subject should embody a definite and reasonable point of view which is not represented by books already in print. Such a point of view inevitably reflects the experience , taste , and biases of the author, and should therefore be clearly stated at the beginning so that those who disagree can seek nourishment elsewhere. The structure and contents of this book express my personal opinions in a variety of ways, as follows. The place of dift'erential equations in mathematics.
Analysis has been the dominant branch of mathematics for 300 years , and differential equations are the heart of analysis. This subject is the natural goal of elementary calculus and the most important part of mathematics for understanding the physical sciences. Also , in the deeper questions it generates , it is the source of most of the ideas and theories which constitute higher analysis. Power series, Fourier series, the gamma function and other special functions , integral equations ,.
existence theorems, the need for rigorous justifications of many analytic processes-all these themes arise in our work in their most natural context. And at a later stage they provide the principal motivation behind complex analysis, the theory of Fourier series and more general orthogonal expansions, Lebesgue integration , metric spaces and Hilbert spaces, and a host of other beautiful topics in modern mathematics. I would argue , for example , that one of the main ideas of complex analysis is the liberation of power series from the confining environment of the real number system ; and this motive is most clearly felt by those who have tried to use real power series to solve differential equations. In botany, it is obvious that no one can fully appreciate the blossoms of flowering plants without a reasonable understanding of the roots, stems , and leaves which nourish and support them.
The same principle is true in mathematics , but is often neglected or forgotten.me/UPSC_Prelims https://t.me/UPSC_Mains www.me/UPSC_Prelims https://t.me/UPSC_Mains XViii PREFACE TO THE FI RST EDITION Fads are as common in mathematics as in any other human activity, and it is always difficult to separate the enduring from the ephemeral in the achievements of one's own time. At present there is a strong current of abstraction flowing through our graduate schools of mathematics. This current has scoured away many of the individual features of the landscape and replaced them with the smooth , rounded boulders of general theories.