Ordinary and Partial Differential Equations

1 Ordinary and Partial Differential EquationsAn Introduction to Dynamical SystemsJohn W. Cain, and Angela M. Reynolds, Textbook Series. Editor: Lon of Proofby Richard Algebraby Jim Algebra: Theory and Applicationsby Thomas and Partial Differential Equationsby John W. Cain and Angela M. ReynoldsDepartment of Mathematics & Applied MathematicsVirginia Commonwealth UniversityRichmond, Virginia,23284 Publication of this edition supported by the Center for Teaching Excellence atvcuOrdinary and Partial Differential Equations : An Introduction to Dynamical 2010by John W.

2 Cain and Angela ReynoldsThis work is licensed under the Creative Commons Attribution-NonCommercial-No and is published with the express permission of the in10pt Palladio L with Pazo Math fonts using PDFLATEXA cknowledgementsJohn W. Cain expresses profound gratitude to his advisor, Dr. David G. Scha-effer, James B. Duke Professor of Mathematics at Duke University. The firstfive chapters are based in part upon Professor Schaeffer s introductory gradu-ate course on Ordinary Differential Equations .

3 The material has been adaptedto accommodate upper-level undergraduate students, essentially by omittingtechnical proofs of the major theorems and including additional examples. Othermajor influences on this book include the excellent texts of Perko [8], Strauss [10],and Strogatz [11]. In particular, the material presented in the last five chapters(including the ordering of the topics) is based heavily on Strauss book. On theother hand, our exposition, examples, and exercises are more user-friendly ,making our text more accessible to readers with less background in Reynolds dedicates her portion of this textbook to her mother, father andsisters, she thanks them for all their support and , Dr.

4 Cain dedicates his portion of this textbook to his parents Jeanetteand Harry, who he loves more than words can and Boundary Value Problems ..42 Linear, Constant-Coefficient Systems .. Matrices.. and Geometric Multiplicities of Eigenvalues.. Eigenvalues.. Eigenvalues and Non-Diagonalizable Matrices.. Portraits and Planar Systems .. , Unstable, and Center Subspaces .. and Determinant .. Systems ..673 Nonlinear Systems: Local Approximations of Functions of Several Variables .. Existence and Uniqueness Theorem.

5 Existence, Dependence on Initial Conditions .. and Linearization .. Hartman-Grobman Theorem .. Stable Manifold Theorem .. Equilibria and Lyapunov Functions ..1054 Periodic, Heteroclinic, and Homoclinic Orbits and the Poincar -Bendixon Theorem .. and Homoclinic Orbits .. Basic Bifurcations .. of solutions on Parameters .. Bifurcations ..1516 Introduction to Delay Differential Value Problems .. Constant-Coefficient Delay Differential Equations .. Equations .. Hutchinson-Wright Equation.

6 1727 Introduction to Difference Notions .. , Constant-Coefficient Difference Equations .. Nonlinear Equations and Stability .. of Nonlinear Equations and Stability .. Bifurcations .. to Control Chaos ..2088 Introduction to Partial Differential Classification of Partial Differential Equations .. of Partial Differential Equations .. Conditions and Boundary Conditions .. solutions of Partial Differential Equations ..2339 Linear, First-Order Partial Differential and Solution of the Transport Equation.

7 Of Characteristics: More Examples ..24110 The Heat and Wave Equations on an Unbounded of the Heat and Wave Equations .. Problem for the Wave Equation .. Problem for the Heat Equation .. and the Heat Equation .. Equations and Duhamel s Principle ..284vi11 Initial-Boundary Value and Wave Equations on a Half-Line .. of Variables .. Equation, Dirichlet Problem.. Equation, Dirichlet Problem.. Equation, Neumann Problem.. Equation, Neumann Problem.. Boundary Conditions: An Example.

8 32412 Introduction to Fourier series .. sine series.. cosine series.. series.. of Fourier Series .. , distances, inner products, and convergence.. theorems..35913 The Laplace and Poisson and Neumann Problems .. and the Maximum Principle .. and Rotation Invariance .. s Equation on Bounded Domains .. problem on a rectangle.. problem on a disc..390 Guide to Commonly Used Notation404 References406 Index407 CHAPTER 1 IntroductionThemathematical sub-discipline ofdifferential Equations and dynamical systemsis foundational in the study of applied mathematics.

9 Differential equationsarise in a variety of contexts, some purely theoretical and some of practicalinterest. As you read this textbook, you will find that the qualitative andquantitative study of Differential Equations incorporates an elegant blend of linearalgebra and advanced calculus. For this reason, it is expected that the reader hasalready completed courses in (i) linear algebra; (ii) multivariable calculus; and(iii) introductory Differential Equations . Familiarity with the following topics isespecially desirable:+From basic Differential Equations : separable Differential Equations and separa-tion of variables; and solving linear, constant-coefficient Differential equationsusing characteristic Equations .

10 +From linear algebra: solving systems ofmalgebraic Equations withnun-knowns; matrix inversion; linear independence; and eigenvalues/eigenvectors.+From multivariable calculus: parametrized curves; Partial derivatives andgradients; and approximating a surface using a tangent of these topics will be reviewed as we encounter them later in thischapter, we will recall a few basic notions from an introductory course indifferential Equations . Readers are encouraged to supplement this book with theexcellent textbooks of Hubbard and West [5], Meiss [7], Perko [8], Strauss [10],and Strogatz [11].