Many textbooks on differential equations are written to be interesting to the teacher rather than the student. Introduction to Differential Equations with Dynamical Systems is directed toward students. This concise and up-to-date textbook addresses the challenges that undergraduate mathematics, engineering, and science students experience during a first course on differential equations. And, while covering all the standard parts of the subject, the book emphasizes linear constant coefficient equations and applications, including the topics essential to engineering students. Stephen Campbell and Richard Haberman--using carefully worded derivations, elementary explanations, and examples, exercises, and figures rather than theorems and proofs--have written a book that makes learning and teaching differential equations easier and more relevant. The book also presents elementary dynamical systems in a unique and flexible way that is suitable for all courses, regardless of length.

Preface ix CHAPTER 1: First-Order Differential Equations and Their Applications 11.1 Introduction to Ordinary Differential Equations 11.2 The Definite Integral and the Initial Value Problem 41.2.1 The Initial Value Problem and the Indefinite Integral 51.2.2 The Initial Value Problem and the Definite Integral 61.2.3 Mechanics I: Elementary Motion of a Particle with Gravity Only 81.3 First-Order Separable Differential Equations 131.3.1 Using Definite Integrals for Separable Differential Equations 161.4 Direction Fields 191.4.1 Existence and Uniqueness 251.5 Euler's Numerical Method (optional) 311.6 First-Order Linear Differential Equations 371.6.1 Form of the General Solution 371.6.2 Solutions of Homogeneous First-Order Linear Differential Equations 391.6.3 Integrating Factors for First-Order Linear Differential Equations 421.7 Linear First-Order Differential Equations with Constant Coefficients and Constant Input 481.7.1 Homogeneous Linear Differential Equations with Constant Coefficients 481.7.2 Constant Coefficient Linear Differential Equations with Constant Input 501.7.3 Constant Coefficient Differential Equations with Exponential Input 521.7.4 Constant Coefficient Differential Equations with Discontinuous Input 521.8 Growth and Decay Problems 591.8.1 A First Model of Population Growth 591.8.2 Radioactive Decay 651.8.3 Thermal Cooling 681.9 Mixture Problems 741.9.1 Mixture Problems with a Fixed Volume 741.9.2 Mixture Problems with Variable Volumes 771.10 Electronic Circuits 821.11 Mechanics II: Including Air Resistance 881.12 Orthogonal Trajectories (optional) 92
CHAPTER 2: Linear Second- and Higher-Order Differential Equations 962.1 General Solution of Second-Order Linear Differential Equations 962.2 Initial Value Problem (for Homogeneous Equations) 1002.3 Reduction of Order 1072.4 Homogeneous Linear Constant Coefficient Differential Equations (Second Order) 1122.4.1 Homogeneous Linear Constant Coefficient Differential Equations (nth-Order) 1222.5 Mechanical Vibrations I: Formulation and Free Response 1242.5.1 Formulation of Equations 1242.5.2 Simple Harmonic Motion (No Damping, d =0) 1282.5.3 Free Response with Friction (d > 0) 1352.6 The Method of Undetermined Coefficients 1422.7 Mechanical Vibrations II: Forced Response 1592.7.1 Friction is Absent (d = 0) 1592.7.2 Friction is Present (d > 0) (Damped Forced Oscillations) 1682.8 Linear Electric Circuits 1742.9 Euler Equation 1792.10 Variation of Parameters (Second-Order) 1852.11 Variation of Parameters (nth-Order) 193
CHAPTER 3: The Laplace Transform 1973.1 Definition and Basic Properties 1973.1.1 The Shifting Theorem (Multiplying by an Exponential) 2053.1.2 Derivative Theorem (Multiplying by t ) 2103.2 Inverse Laplace Transforms (Roots, Quadratics, and Partial Fractions) 2133.3 Initial Value Problems for Differential Equations 2253.4 Discontinuous Forcing Functions 2343.4.1 Solution of Differential Equations 2393.5 Periodic Functions 2483.6 Integrals and the Convolution Theorem 2533.6.1 Derivation of the Convolution Theorem (optional) 2563.7 Impulses and Distributions 260
CHAPTER 4: An Introduction to Linear Systems of Differential Equations and Their Phase Plane 2654.1 Introduction 2654.2 Introduction to Linear Systems of Differential Equations 2684.2.1 Solving Linear Systems Using Eigenvalues and Eigenvectors of the Matrix 2694.2.2 Solving Linear Systems if the Eigenvalues are Real and Unequal 2724.2.3 Finding General Solutions of Linear Systems in the Case of Complex Eigenvalues 2764.2.4 Special Systems with Complex Eigenvalues (optional) 2794.2.5 General Solution of a Linear System if the Two Real Eigenvalues are Equal (Repeated) Roots 2814.2.6 Eigenvalues and Trace and Determinant (optional) 2834.3 The Phase Plane for Linear Systems of Differential Equations 2874.3.1 Introduction to the Phase Plane for Linear Systems of Differential Equations 2874.3.2 Phase Plane for Linear Systems of Differential Equations 2954.3.3 Real Eigenvalues 2964.3.4 Complex Eigenvalues 3044.3.5 General Theorems 310
CHAPTER 5: Mostly Nonlinear First-Order Differential Equations 3155.1 First-Order Differential Equations 3155.2 Equilibria and Stability 3165.2.1 Equilibrium 3165.2.2 Stability 3175.2.3 Review of Linearization 3185.2.4 Linear Stability Analysis 3185.3 One-Dimensional Phase Lines 3225.4 Application to Population Dynamics: The Logistic Equation 327
CHAPTER 6: Nonlinear Systems of Differential Equations in the Plane 3326.1 Introduction 3326.2 Equilibria of Nonlinear Systems, Linear Stability Analysis of Equilibrium, and the Phase Plane 3356.2.1 Linear Stability Analysis and the Phase Plane 3366.2.2 Nonlinear Systems: Summary, Philosophy, Phase Plane, Direction Field, Nullclines 3416.3 Population Models 3496.3.1 Two Competing Species 3506.3.2 Predator-Prey Population Models 3566.4 Mechanical Systems 3636.4.1 Nonlinear Pendulum 3636.4.2 Linearized Pendulum 3646.4.3 Conservative Systems and the Energy Integral 3646.4.4 The Phase Plane and the Potential 367Answers to Odd-Numbered Exercises 379
Index 429