Elementary Differential Equations and Boundary Value Problems, 12th Edition is written from the viewpoint of the applied mathematician, whose interest in differential equations may sometimes be quite theoretical, sometimes intensely practical, and often somewhere in between. In this revision, new author Douglas Meade focuses on developing students conceptual understanding with new concept questions and worksheets for each chapter. Meade builds upon Boyce and DiPrima’s work to combine a sound and accurate (but not abstract) exposition of the elementary theory of differential equations with considerable material on methods of solution, analysis, and approximation that have proved useful in a wide variety of applications. The main prerequisite for engaging with the program is a working knowledge of calculus, gained from a normal two or three semester course sequence or its equivalent. Some familiarity with matrices will also be helpful in the chapters on systems of differential equations.
Table of Contents:
Preface v
1 Introduction 1
1.1 Some Basic Mathematical Models; Direction Fields 1
1.2 Solutions of Some Differential Equations 9
1.3 Classification of Differential Equations 17
2 First-Order Differential Equations 26
2.1 Linear Differential Equations; Method of Integrating Factors 26
2.2 Separable Differential Equations 34
2.3 Modeling with First-Order Differential Equations 41
2.4 Differences Between Linear and Nonlinear Differential Equations 53
2.5 Autonomous Differential Equations and Population Dynamics 61
2.6 Exact Differential Equations and Integrating Factors 72
2.7 Numerical Approximations: Euler’s Method 78
2.8 The Existence and Uniqueness Theorem 86
2.9 First-Order Difference Equations 93
3 Second-Order Linear Differential Equations 106
3.1 Homogeneous Differential Equations with Constant Coefficients 106
3.2 Solutions of Linear Homogeneous Equations; the Wronskian 113
3.3 Complex Roots of the Characteristic Equation 123
3.4 Repeated Roots; Reduction of Order 130
3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients 136
3.6 Variation of Parameters 145
3.7 Mechanical and Electrical Vibrations 150
3.8 Forced Periodic Vibrations 161
4 Higher-Order Linear Differential Equations 173
4.1 General Theory of n;;;; Order Linear Differential Equations 173
4.2 Homogeneous Differential Equations with Constant Coefficients 178
4.3 The Method of Undetermined Coefficients 185
4.4 The Method of Variation of Parameters 189
5 Series Solutions of Second-Order Linear Equations 194
5.1 Review of Power Series 194
5.2 Series Solutions Near an Ordinary Point, Part I 200
5.3 Series Solutions Near an Ordinary Point, Part II 209
5.4 Euler Equations; Regular Singular Points 215
5.5 Series Solutions Near a Regular Singular Point, Part I 224
5.6 Series Solutions Near a Regular Singular Point, Part II 228
5.7 Bessel’s Equation 235
6 The Laplace Transform 247
6.1 Definition of the Laplace Transform 247
6.2 Solution of Initial Value Problems 254
6.3 Step Functions 263
6.4 Differential Equations with Discontinuous Forcing Functions 270
6.5 Impulse Functions 275
6.6 The Convolution Integral 280
7 Systems of First-Order Linear Equations 288
7.1 Introduction 288
7.2 Matrices 293
7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 301
7.4 Basic Theory of Systems of First-Order Linear Equations 311
7.5 Homogeneous Linear Systems with Constant Coefficients 315
7.6 Complex-Valued Eigenvalues 325
7.7 Fundamental Matrices 335
7.8 Repeated Eigenvalues 342
7.9 Nonhomogeneous Linear Systems 351
8 Numerical Methods 363
8.1 The Euler or Tangent Line Method 363
8.2 Improvements on the Euler Method 372
8.3 The Runge-Kutta Method 376
8.4 Multistep Methods 380
8.5 Systems of First-Order Equations 385
8.6 More on Errors; Stability 387
9 Nonlinear Differential Equations and Stability 400
9.1 The Phase Plane: Linear Systems 400
9.2 Autonomous Systems and Stability 410
9.3 Locally Linear Systems 419
9.4 Competing Species 429
9.5 Predator – Prey Equations 439
9.6 Liapunov’s Second Method 446
9.7 Periodic Solutions and Limit Cycles 455
9.8 Chaos and Strange Attractors: The Lorenz Equations 465
10 Partial Differential Equations and Fourier Series 476
10.1 Two-Point Boundary Value Problems 476
10.2 Fourier Series 482
10.3 The Fourier Convergence Theorem 490
10.4 Even and Odd Functions 495
10.5 Separation of Variables; Heat Conduction in a Rod 501
10.6 Other Heat Conduction Problems 508
10.7 The Wave Equation: Vibrations of an Elastic String 516
10.8 Laplace’s Equation 527
A Appendix 537
B Appendix 541
11 Boundary Value Problems and Stur-Liouville Theory 544
11.1 The Occurrence of Two-Point Boundary Value Problems 544
11.2 Sturm-Liouville Boundary Value Problems 550
11.3 Nonhomogeneous Boundary Value Problems 561
11.4 Singular Sturm-Liouville Problems 572
11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion 578
11.6 Series of Orthogonal Functions: Mean Convergence 582
Answers to Problems 591
Index 624
About the Author :
WILLIAM E. BOYCE (deceased) received his B.A. degree in Mathematics from Rhodes College and his M.S. and Ph.D. degrees in Mathematics from Carnegie Mellon University. He was a member of the American Mathematical Society, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics. He was also the Edward P. Hamilton Distinguished Professor Emeritus of Science Education (Department of Mathematical Sciences) at Rensselaer. He authored numerous technical papers in boundary value problems and random differential equations and their applications, as well as several textbooks including two differential equations texts, and was the coauthor (with M.H. Holmes, J.G. Ecker, and W.L. Siegmann) of a text on using Maple to explore Calculus. He was also coauthor (with R.L. Borrelli and C.S. Coleman) of Differential Equations Laboratory Workbook (Wiley 1992), which received the EDUCOM Best Mathematics Curricular Innovation Award in 1993.
RICHARD C. DIPRIMA (deceased) received his B.S., M.S., and Ph.D. degrees in Mathematics from Carnegie Mellon University. He joined the faculty of Rensselaer Polytechnic Institute after holding research positions at MIT, Harvard, and Hughes Aircraft. He held the Eliza Ricketts Foundation Professorship of Mathematics at Rensselaer, was a fellow of the American Society of Mechanical Engineers, the American Academy of Mechanics, and the American Physical Society. He was also a member of the American Mathematical Society, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics. He served as the Chairman of the Department of Mathematical Sciences at Rensselaer, as President of the Society for Industrial and Applied Mathematics, and as Chairman of the Executive Committee of the Applied Mechanics Division of ASME. In 1980, he was the recipient of the William H. Wiley Distinguished Faculty Award given by Rensselaer.
DOUGLAS B. MEADE received B.S. degrees in Mathematics and Computer Science from Bowling Green State University, an M.S. in Applied Mathematics from Carnegie Mellon University, and a Ph.D. in Mathematics from Carnegie Mellon University. After a two-year stint at Purdue University, he joined the mathematics faculty at the University of South Carolina, where he is currently an Associate Professor of Mathematics. He is a member of the American Mathematical Society, Mathematics Association of America, and Society for Industrial and Applied Mathematics; in 2016 he was named an ICTCM Fellow at the International Conference on Technology in Collegiate Mathematics (ICTCM). Prof. Meade currently serves on the AMSASA-MAA-SIAM Data Committee and on the MAA Committee on Articulation and Placement.
