The new 9th edition of Elementary Differential Equations and Boundary Value Problems, like its predecessors, 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. The authors have sought 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 9th edition includes new problems and examples, as well as expanded explanations to help motivate students. The book is written primarily for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study. The main prerequisite for reading the book 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:
Chapter 1 Introduction 1
1.1 Some Basic Mathematical Models; Direction Fields 1
1.2 Solutions of Some Differential Equations 10
1.3 Classification of Differential Equations 19
1.4 Historical Remarks 26
Chapter 2 First Order Differential Equations 31
2.1 Linear Equations; Method of Integrating Factors 31
2.2 Separable Equations 42
2.3 Modeling with First Order Equations 50
2.4 Differences Between Linear and Nonlinear Equations 68
2.5 Autonomous Equations and Population Dynamics 78
2.6 Exact Equations and Integrating Factors 94
2.7 Numerical Approximations: Euler’s Method 101
2.8 The Existence and Uniqueness Theorem 111
2.9 First Order Difference Equations 121
Chapter 3 Second Order Linear Equations 137
3.1 Homogeneous Equations with Constant Coefficients 137
3.2 Solutions of Linear Homogeneous Equations; the Wronskian 145
3.3 Complex Roots of the Characteristic Equation 157
3.4 Repeated Roots; Reduction of Order 166
3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients 174
3.6 Variation of Parameters 185
3.7 Mechanical and Electrical Vibrations 191
3.8 Forced Vibrations 206
Chapter 4 Higher Order Linear Equations 219
4.1 General Theory of nth Order Linear Equations 219
4.2 Homogeneous Equations with Constant Coefficients 226
4.3 The Method of Undetermined Coefficients 234
4.4 The Method of Variation of Parameters 239
Chapter 5 Series Solutions of Second Order Linear Equations 243
5.1 Review of Power Series 243
5.2 Series Solutions Near an Ordinary Point, Part I 250
5.3 Series Solutions Near an Ordinary Point, Part II 261
5.4 Euler Equations; Regular Singular Points 268
5.5 Series Solutions Near a Regular Singular Point, Part I 278
5.6 Series Solutions Near a Regular Singular Point, Part II 284
5.7 Bessel’s Equation 292
Chapter 6 The Laplace Transform 305
6.1 Definition of the Laplace Transform 305
6.2 Solution of Initial Value Problems 312
6.3 Step Functions 323
6.4 Differential Equations with Discontinuous Forcing Functions 331
6.5 Impulse Functions 339
6.6 The Convolution Integral 345
Chapter 7 Systems of First Order Linear Equations 355
7.1 Introduction 355
7.2 Review of Matrices 364
7.3 Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 373
7.4 Basic Theory of Systems of First Order Linear Equations 385
7.5 Homogeneous Linear Systems with Constant Coefficients 390
7.6 Complex Eigenvalues 401
7.7 Fundamental Matrices 413
7.8 Repeated Eigenvalues 422
7.9 Nonhomogeneous Linear Systems 432
Chapter 8 Numerical Methods 443
8.1 The Euler or Tangent Line Method 443
8.2 Improvements on the Euler Method 454
8.3 The Runge–Kutta Method 459
8.4 Multistep Methods 464
8.5 More on Errors; Stability 470
8.6 Systems of First Order Equations 480
Chapter 9 Nonlinear Differential Equations and Stability 485
9.1 The Phase Plane: Linear Systems 485
9.2 Autonomous Systems and Stability 497
9.3 Locally Linear Systems 508
9.4 Competing Species 520
9.5 Predator–Prey Equations 533
9.6 Liapunov’s Second Method 543
9.7 Periodic Solutions and Limit Cycles 554
9.8 Chaos and Strange Attractors: The Lorenz Equations 566
Chapter 10 Partial Differential Equations and Fourier Series 577
10.1 Two-Point Boundary Value Problems 577
10.2 Fourier Series 584
10.3 The Fourier Convergence Theorem 595
10.4 Even and Odd Functions 602
10.5 Separation of Variables; Heat Conduction in a Rod 611
10.6 Other Heat Conduction Problems 620
10.7 TheWave Equation: Vibrations of an Elastic String 631
10.8 Laplace’s Equation 646
AppendixA Derivation of the Heat Conduction Equation 657
Appendix B Derivation of theWave Equation 661
Chapter 11 Boundary Value Problems 665
11.1 The Occurrence of Two-Point Boundary Value Problems 665
11.2 Sturm–Liouville Boundary Value Problems 673
11.3 Nonhomogeneous Boundary Value Problems 687
11.4 Singular Sturm–Liouville Problems 702
11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion 709
11.6 Series of Orthogonal Functions: Mean Convergence 716
Answers to Problems 727
Index 786
