# Elementary Differential Equations with Boundary Value Problems (6th Edition)

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## Book Preface

The evolution of the present text in successive editions is based on experience teaching the introductory differential equations course with an emphasis on conceptual ideas and the use of applications and projects to involve students in active problem-solving experiences. At various points our approach reflects the widespread use of technical computing environments like Maple, Mathematica, and MATLAB for the graphical, numerical, or symbolic solution of differential equations. Nevertheless, we continue to believe that the traditional elementary analytical methods of solution are important for students to learn and use. One reason is that effective and reliable use of computer methods often requires preliminary analysis using standard symbolic techniques; the construction of a realistic computational model often is based on the study of a simpler analytical model.

C O N T E N T S

CHAPTER 1
1 First-Order Differential Equations 1
1.1 Differential Equations and Mathematical Models 1
1.2 Integrals as General and Particular Solutions 10
1.3 Slope Fields and Solution Curves 19
1.4 Separable Equations and Applications 32
1.5 Linear First-Order Equations 46
1.6 Substitution Methods and Exact Equations 59
1.7 Population Models 74
1.8 Acceleration-Velocity Models 85
CHAPTER 2
2 Linear Equations of Higher Order 100
2.1 Introduction: Second-Order Linear Equations 100
2.2 General Solutions of Linear Equations 113
2.3 Homogeneous Equations with Constant Coefficients 124
2.4 Mechanical Vibrations 135
2.5 Nonhomogeneous Equations and Undetermined Coefficients 148
2.6 Forced Oscillations and Resonance 162
2.7 Electrical Circuits 173
2.8 Endpoint Problems and Eigenvalues 180
CHAPTER 3
3 Power Series Methods 194
3.1 Introduction and Review of Power Series 194
3.2 Series Solutions Near Ordinary Points 207
3.3 Regular Singular Points 218
3.4 Method of Frobenius: The Exceptional Cases 233
3.5 Bessel’s Equation 248
3.6 Applications of Bessel Functions 257
CHAPTER 4
4 Laplace Transform Methods 266
4.1 Laplace Transforms and Inverse Transforms 266
4.2 Transformation of Initial Value Problems 277
4.3 Translation and Partial Fractions 289
4.4 Derivatives, Integrals, and Products of Transforms 297
4.5 Periodic and Piecewise Continuous Input Functions 304
4.6 Impulses and Delta Functions 316
CHAPTER 5
5 Linear Systems of Differential Equations 326
5.1 First-Order Systems and Applications 326
5.2 The Method of Elimination 338
5.3 Matrices and Linear Systems 347
5.4 The Eigenvalue Method for Homogeneous Systems 366
5.5 Second-Order Systems and Mechanical Applications 381
5.6 Multiple Eigenvalue Solutions 393
5.7 Matrix Exponentials and Linear Systems 407
5.8 Nonhomogeneous Linear Systems 420
CHAPTER 6
6 Numerical Methods 430
6.1 Numerical Approximation: Euler’s Method 430
6.2 A Closer Look at the Euler Method 442
6.3 The Runge-Kutta Method 453
6.4 Numerical Methods for Systems 464
CHAPTER 7
7 Nonlinear Systems and Phenomena 480
7.1 Equilibrium Solutions and Stability 480
7.2 Stability and the Phase Plane 488
7.3 Linear and Almost Linear Systems 500
7.4 Ecological Models: Predators and Competitors 513
7.5 Nonlinear Mechanical Systems 526
7.6 Chaos in Dynamical Systems 542
CHAPTER 8
8 Fourier Series Methods 554
8.1 Periodic Functions and Trigonometric Series 554
8.2 General Fourier Series and Convergence 563
8.3 Fourier Sine and Cosine Series 570
8.4 Applications of Fourier Series 581
8.5 Heat Conduction and Separation of Variables 586
8.6 Vibrating Strings and the One-Dimensional Wave Equation 599
8.7 Steady-State Temperature and Laplace’s Equation 611
CHAPTER 9
9 Eigenvalues and Boundary Value Problems 622
9.1 Sturm-Liouville Problems and Eigenfunction Expansions 622
9.2 Applications of Eigenfunction Series 633
9.3 Steady Periodic Solutions and Natural Frequencies 642
9.4 Cylindrical Coordinate Problems 650
9.5 Higher-Dimensional Phenomena 664