Differential Equations Theory, Technique, and Practice

Differential equations is one of the oldest subjects in modern mathematics. It was not long after Newton and Leibniz invented the calculus that Bernoulli and Euler and others began to consider the heat equation and the wave equation of mathematical physics. Newton himself solved differential equations both in the study of planetary motion and also in his consideration of optics. Today differential equations is the centerpiece of much of engineering, of physics, of significant parts of the life sciences, and in many areas of mathematical modeling. This text describes classical ideas and provides an entree to the newer ones. The author pays careful attention to advanced topics like the Laplace transform, Sturm–Liouville theory, and boundary value problems (on the traditional side) but also pays due homage to nonlinear theory, to modeling, and to computing (on the modern side). This book began as a modernization of George Simmons’ classic, Differential Equations with Applications and Historical Notes. Prof. Simmons invited the author to update his book. Now in the third edition, this text has become the author’s own and a unique blend of the traditional and the modern. The text describes classical ideas and provides an entree to newer ones. Modeling brings the subject to life and makes the ideas real. Differential equations can model real life questions, and computer calculations and graphics can then provide real life answers. The symbiosis of the synthetic and the calculational provides a rich experience for students, and prepares them for more concrete, applied work in future courses. Additional Features Anatomy of an Application sections. Historical notes continue to be a unique feature of this text. Math Nuggets are brief perspectives on mathematical lives or other features of the discipline that will enhance the reading experience. Problems for Review and Discovery give students some open-ended material for exploration and further learning. They are an important means of extending the reach of the text, and for anticipating future work. This new edition is re-organized to make it more useful and more accessible. The most frequently taught topics are now up front. And the major applications are isolated in their own chapters. This makes this edition the most useable and flexible of any previous editions.

Preface 1. What Is a Differential Equation? 1.1 Introductory Remarks 1.2 A Taste of Ordinary Differential Equations 1.3 The Nature of Solutions 2. Solving First-Order Equations 2.1 Separable Equations 2.2 First-Order Linear Equations 2.3 Exact Equations 2.4 Orthogonal Trajectories and Curves 2.5 Homogeneous Equations 2.6 Integrating Factors 2.7 Reduction of Order 2.7.1 Dependent Variable Missing 2.7.2 Independent Variable Missing 3. Some Applications of the First-Order Theory 3.1 The Hanging Chain and Pursuit Curves 3.1.1 The Hanging Chain 3.1.2 Pursuit Curves 3.2 Electrical Circuits Anatomy of an Application Problems for Review and Discovery 4. Second-Order Linear Equations 4.1 Second-Order Linear Equations with Constant Coefficients 4.2 The Method of Undetermined Coefficients 4.3 The Method of Variation of Parameters 4.4 The Use of a Known Solution to Find Another 4.5 Higher-Order Equations 5. Applications of the Second-Order Theory 5.1 Vibrations and Oscillations 5.1.1 Undamped Simple Harmonic Motion 5.1.2 Damped Vibrations 5.1.3 Forced Vibrations 5.1.4 A Few Remarks About Electricity 5.2 Newton’s Law of Gravitation and Kepler’s Laws 5.2.1 Kepler’s Second Law 5.2.2 Kepler’s First Law 5.2.3 Kepler’s Third Law Historical Note Anatomy of an Application Problems for Review and Discovery 6. Power Series Solutions and Special Functions6.1 Introduction and Review of Power Series 6.1.1 Review of Power Series 6.2 Series Solutions of First-Order Equations 6.3 Ordinary Points 6.4 Regular Singular Points 6.5 More on Regular Singular Points Historical Note Historical Note Anatomy of an Application Problems for Review and Discovery 7. Fourier Series: Basic Concepts7.1 Fourier Coefficients 7.2 Some Remarks about Convergence 7.3 Even and Odd Functions: Cosine and Sine Series 7.4 Fourier Series on Arbitrary Intervals 7.5 Orthogonal Functions Historical Note Anatomy of an Application Problems for Review and Discovery 8. Laplace Transforms8.0 Introduction 8.1 Applications to Differential Equations 8.2 Derivatives and Integrals 8.3 Convolutions 8.3.1 Abel’s Mechanics Problem 8.4 The Unit Step and Impulse Functions Historical Note Anatomy of an Application Problems for Review and Discovery 9. The Calculus of Variations 9.1 Introductory Remarks 9.2 Euler’s Equation 9.3 Isoperimetric Problems and the Like 9.3.1 Lagrange Multipliers 9.3.2 Integral Side Conditions 9.3.3 Finite Side Conditions Historical Note Anatomy of an Application Problems for Review and Discovery 10. Systems of First-Order Equations 10.1 Introductory Remarks 10.2 Linear Systems 10.3 Systems with Constant Coefficients 10.4 Nonlinear Systems Anatomy of an Application Problems for Review and Discovery 11. Partial Differential Equations and Boundary Value Problems 11.1 Introduction and Historical Remarks 11.2 Eigenvalues and the Vibrating String 11.2.1 Boundary Value Problems 11.2.2 Derivation of the Wave Equation 11.2.3 Solution of the Wave Equation 11.3 The Heat Equation 11.4 The Dirichlet Problem for a Disc 11.4.1 The Poisson Integral 11.5 Sturm—Liouville Problems Historical Note Historical Note Anatomy of an Application Problems for Review and Discovery 12. The Nonlinear Theory 12.1 Some Motivating Examples 12.2 Specializing Down 12.3 Types of Critical Points: Stability 12.4 Critical Points and Stability 12.5 Stability by Liapunov’s Direct Method 12.6 Simple Critical Points of Nonlinear Systems 12.7 Nonlinear Mechanics: Conservative Systems 12.8 Periodic Solutions Historical Note Anatomy of an Application Problems for Review and Discovery 13. Qualitative Properties and Theoretical Aspects13.1 A Bit of Theory 13.2 Picard’s Existence and Uniqueness Theorem 13.2.1 The Form of a Differential Equation 13.2.2 Picard’s Iteration Technique 13.2.3 Some Illustrative Examples 13.2.4 Estimation of the Picard Iterates 13.3 Oscillations and the Sturm Separation Theorem 13.4 The Sturm Comparison Theorem Anatomy of an Application Problems for Review and Discovery Appendix: Review of Linear Algebra BibliographyIndex

Steven G. Krantz is a professor of mathematics at Washington University in St. Louis. He has previously taught at UCLA, Princeton University, and Pennsylvania State University. He has written more than 130 books and more than 250 scholarly papers and is the founding editor of the Journal of Geometric Analysis. An AMS Fellow, Dr. Krantz has been a recipient of the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize. He received a Ph.D. from Princeton University.