DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES Third Edition www.com TEXTBOOKS in MATHEMATICS Series Editors: Al Boggess and Ken Rosen PUBLISHED TITLES ABSTRACT ALGEBRA: AN INTERACTIVE APPROACH, SECOND EDITION William Paulsen ABSTRACT ALGEBRA: AN INQUIRY-BASED APPROACH Jonathan K. Hodge, Steven Schlicker, and Ted Sundstrom ADVANCED LINEAR ALGEBRA Hugo Woerdeman APPLIED ABSTRACT ALGEBRA WITH MAPLE™ AND MATLAB®, THIRD EDITION Richard Klima, Neil Sigmon, and Ernest Stitzinger APPLIED DIFFERENTIAL EQUATIONS: THE PRIMARY COURSE Vladimir Dobrushkin COMPUTATIONAL MATHEMATICS: MODELS, METHODS, AND ANALYSIS WITH MATLAB® AND MPI, SECOND EDITION Robert E. White DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE, SECOND EDITION Steven G. Krantz DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE WITH BOUNDARY VALUE PROBLEMS Steven G.
Krantz DIFFERENTIAL EQUATIONS WITH MATLAB®: EXPLORATION, APPLICATIONS, AND THEORY Mark A. McKibben and Micah D. Webster ELEMENTARY NUMBER THEORY James S. Kraft and Lawrence C.
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Visit the Taylor & Francis Web site at http://www.com and the CRC Press Web site at http://www.com For Hope and Nancy my wife and daughter who still make it all worthwhile www.com Contents Preface to the Third Edition .xi Preface to the Second Edition. xiii Preface to the First Edition.xv Suggestions for the Instructor. xix About the Author. The Nature of Differential Equations.1 2 General Remarks on Solutions .4 3 Families of Curves.
11 4 Growth, Decay, Chemical Reactions, and Mixing. 19 5 Falling Bodies and Other Motion Problems. Fermat and the Bernoullis. 40 Appendix A: Some Ideas From the Theory of Probability: The Normal Distribution Curve (or Bell Curve) and Its Differential Equation.
First Order Equations. 81 11 Reduction of Order .85 12 The Hanging Chain. 88 13 Simple Electric Circuits. Second Order Linear Equations.
107 15 The General Solution of the Homogeneous Equation. 113 16 The Use of a Known Solution to find Another. 119 17 The Homogeneous Equation with Constant Coefficients. 122 18 The Method of Undetermined Coefficients.
127 19 The Method of Variation of Parameters. 133 20 Vibrations in Mechanical and Electrical Systems. 136 21 Newton’s Law of Gravitation and The Motion of the Planets. 146 22 Higher Order Linear Equations.
Coupled Harmonic Oscillators. 155 23 Operator Methods for Finding Particular Solutions.com viii Contents 4. Qualitative Properties of Solutions. 187 24 Oscillations and the Sturm Separation Theorem.
187 25 The Sturm Comparison Theorem. Power Series Solutions and Special Functions. A Review of Power Series. 197 27 Series Solutions of First Order Equations.
206 28 Second Order Linear Equations. 210 29 Regular Singular Points. 219 30 Regular Singular Points (Continued). 229 31 Gauss’s Hypergeometric Equation.
236 32 The Point at Infinity. Two Convergence Proofs. Hermite Polynomials and Quantum Mechanics. Chebyshev Polynomials and the Minimax Property.
Fourier Series and Orthogonal Functions. 289 33 The Fourier Coefficients. 289 34 The Problem of Convergence. 301 35 Even and Odd Functions.
Cosine and Sine Series. 310 36 Extension to Arbitrary Intervals. 325 38 The Mean Convergence of Fourier Series. A Pointwise Convergence Theorem.
Partial Differential Equations and Boundary Value Problems. 351 40 Eigenvalues, Eigenfunctions, and the Vibrating String. 355 41 The Heat Equation. 366 42 The Dirichlet Problem for a Circle.
372 43 Sturm–Liouville Problems. The Existence of Eigenvalues and Eigenfunctions. Some Special Functions of Mathematical Physics. 393 45 Properties of Legendre Polynomials.
The Gamma Function. 407 47 Properties of Bessel Functions. Legendre Polynomials and Potential Theory. Bessel Functions and the Vibrating Membrane.
Additional Properties of Bessel Functions.com Contents ix 9. 447 49 A Few Remarks on the Theory. 452 50 Applications to Differential Equations. 457 51 Derivatives and Integrals of Laplace Transforms .463 52 Convolutions and Abel’s Mechanical Problem.
468 53 More about Convolutions. The Unit Step and Impulse Functions. Systems of First Order Equations. 487 54 General Remarks on Systems.
491 56 Homogeneous Linear Systems with Constant Coefficients. Volterra’s Prey-Predator Equations. The Phase Plane and Its Phenomena. 513 59 Types of Critical Points.
519 60 Critical Points and Stability for Linear Systems. 529 61 Stability By Liapunov’s Direct Method. 541 62 Simple Critical Points of Nonlinear Systems. The Poincaré–Bendixson Theorem.
563 65 More about the van der Pol Equation. Proof of Liénard’s Theorem. The Calculus of Variations. Some Typical Problems of the Subject.
581 67 Euler’s Differential Equation for an Extremal. Hamilton’s Principle and Its Implications. The Existence and Uniqueness of Solutions. 621 69 The Method of Successive Approximations.
The Second Order Linear Equation .com x Contents 73 The Method of Euler .650 75 An Improvement to Euler. 652 76 Higher Order Methods.com Preface to the Third Edition I have taken advantage of this new edition of my book on differential equa- tions to add two batches of new material of independent interest: First, a fairly substantial appendix at the end of Chapter 1 on the famous bell curve. This curve is the graph of the normal distribution func- tion, with many applications in the natural sciences, the social sciences, mathematics—in statistics and probability theory—and engineering. We shall be especially interested how the differential equation for this curve arises from very simple considerations and can be solved to obtain the equa- tion of the curve itself.
And second, a brief section on the van der Pol nonlinear equation and its historical background in World War II that gave it significance in the devel- opment of the theory of radar. This consists, in part, of personal recollections of the eminent physicist Freeman Dyson. Finally, I should add a few words on the meaning of the cover design, for this design amounts to a bit of self-indulgence. The chapter on Fourier series is there mainly to provide machinery needed for the following chapter on partial differential equations.
However, one of the minor offshoots of Fourier series is to find the exact sum of the infinite series formed from the reciprocals of the squares of the positive integers (the first formula on the cover). This sum was discovered by the great Swiss mathematician Euler in 1736, and since his time, several other methods for obtaining this sum, in addition to his own, have been discovered. This is one of the topics dealt with in Sections 34 and 35 and has been one of my own minor hobbies in mathematics for many years. However, from 1736 to the present day, no one has ever been able to find the exact sum of the reciprocals of the cubes of the positive integers (the sec- ond formula on the cover).
Some years ago, I was working with the zeroes of the Bessel functions. I thought for an exciting period of several days that I was on the trail of this unknown sum, but in the end it did not work out. Instead, the trail deviated in an unexpected direction and yielded yet another method for finding the sum in the first formula. These ideas will be found in Section 47.com 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 let- ting 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 prob- lems 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 xix.
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 mat- ters 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 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 University; 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. Simmons xiii www.com 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 experi- ence, 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.