Introduction to Differential Equations with Dynamical Systems

Preface ix CHAPTER 1: First-Order Differential Equations and Their Applications 1 1.1 Introduction to Ordinary Differential Equations 1 1.2 The Definite Integral and the Initial Value Problem 4 1.2.1 The Initial Value Problem and the Indefinite Integral 5 1.2.2 The Initial Value Problem and the Definite Integral 6 1.2.3 Mechanics I: Elementary Motion of a Particle with Gravity Only 8 1.3 First-Order Separable Differential Equations 13 1.3.1 Using Definite Integrals for Separable Differential Equations 16 1.4 Direction Fields 19 1.4.1 Existence and Uniqueness 25 1.5 Euler's Numerical Method (optional) 31 1.6 First-Order Linear Differential Equations 37 1.6.1 Form of the General Solution 37 1.6.2 Solutions of Homogeneous First-Order Linear Differential Equations 39 1.6.3 Integrating Factors for First-Order Linear Differential Equations 42 1.7 Linear First-Order Differential Equations with Constant Coefficients and Constant Input 48 1.7.1 Homogeneous Linear Differential Equations with Constant Coefficients 48 1.7.2 Constant Coefficient Linear Differential Equations with Constant Input 50 1.7.3 Constant Coefficient Differential Equations with Exponential Input 52 1.7.4 Constant Coefficient Differential Equations with Discontinuous Input 52 1.8 Growth and Decay Problems 59 1.8.1 A First Model of Population Growth 59 1.8.2 Radioactive Decay 65 1.8.3 Thermal Cooling 68 1.9 Mixture Problems 74 1.9.1 Mixture Problems with a Fixed Volume 74 1.9.2 Mixture Problems with Variable Volumes 77 1.10 Electronic Circuits 82 1.11 Mechanics II: Including Air Resistance 88 1.12 Orthogonal Trajectories (optional) 92 CHAPTER 2: Linear Second- and Higher-Order Differential Equations 96 2.1 General Solution of Second-Order Linear Differential Equations 96 2.2 Initial Value Problem (for Homogeneous Equations) 100 2.3 Reduction of Order 107 2.4 Homogeneous Linear Constant Coefficient Differential Equations (Second Order) 112 2.4.1 Homogeneous Linear Constant Coefficient Differential Equations (nth-Order) 122 2.5 Mechanical Vibrations I: Formulation and Free Response 124 2.5.1 Formulation of Equations 124 2.5.2 Simple Harmonic Motion (No Damping, delta =0) 128 2.5.3 Free Response with Friction (delta > 0) 135 2.6 The Method of Undetermined Coefficients 142 2.7 Mechanical Vibrations II: Forced Response 159 2.7.1 Friction is Absent (delta = 0) 159 2.7.2 Friction is Present (delta > 0) (Damped Forced Oscillations) 168 2.8 Linear Electric Circuits 174 2.9 Euler Equation 179 2.10 Variation of Parameters (Second-Order) 185 2.11 Variation of Parameters (nth-Order) 193 CHAPTER 3: The Laplace Transform 197 3.1 Definition and Basic Properties 197 3.1.1 The Shifting Theorem (Multiplying by an Exponential) 205 3.1.2 Derivative Theorem (Multiplying by t ) 210 3.2 Inverse Laplace Transforms (Roots, Quadratics, and Partial Fractions) 213 3.3 Initial Value Problems for Differential Equations 225 3.4 Discontinuous Forcing Functions 234 3.4.1 Solution of Differential Equations 239 3.5 Periodic Functions 248 3.6 Integrals and the Convolution Theorem 253 3.6.1 Derivation of the Convolution Theorem (optional) 256 3.7 Impulses and Distributions 260 CHAPTER 4: An Introduction to Linear Systems of Differential Equations and Their Phase Plane 265 4.1 Introduction 265 4.2 Introduction to Linear Systems of Differential Equations 268 4.2.1 Solving Linear Systems Using Eigenvalues and Eigenvectors of the Matrix 269 4.2.2 Solving Linear Systems if the Eigenvalues are Real and Unequal 272 4.2.3 Finding General Solutions of Linear Systems in the Case of Complex Eigenvalues 276 4.2.4 Special Systems with Complex Eigenvalues (optional) 279 4.2.5 General Solution of a Linear System if the Two Real Eigenvalues are Equal (Repeated) Roots 281 4.2.6 Eigenvalues and Trace and Determinant (optional) 283 4.3 The Phase Plane for Linear Systems of Differential Equations 287 4.3.1 Introduction to the Phase Plane for Linear Systems of Differential Equations 287 4.3.2 Phase Plane for Linear Systems of Differential Equations 295 4.3.3 Real Eigenvalues 296 4.3.4 Complex Eigenvalues 304 4.3.5 General Theorems 310 CHAPTER 5: Mostly Nonlinear First-Order Differential Equations 315 5.1 First-Order Differential Equations 315 5.2 Equilibria and Stability 316 5.2.1 Equilibrium 316 5.2.2 Stability 317 5.2.3 Review of Linearization 318 5.2.4 Linear Stability Analysis 318 5.3 One-Dimensional Phase Lines 322 5.4 Application to Population Dynamics: The Logistic Equation 327 CHAPTER 6: Nonlinear Systems of Differential Equations in the Plane 332 6.1 Introduction 332 6.2 Equilibria of Nonlinear Systems, Linear Stability Analysis of Equilibrium, and the Phase Plane 335 6.2.1 Linear Stability Analysis and the Phase Plane 336 6.2.2 Nonlinear Systems: Summary, Philosophy, Phase Plane, Direction Field, Nullclines 341 6.3 Population Models 349 6.3.1 Two Competing Species 350 6.3.2 Predator-Prey Population Models 356 6.4 Mechanical Systems 363 6.4.1 Nonlinear Pendulum 363 6.4.2 Linearized Pendulum 364 6.4.3 Conservative Systems and the Energy Integral 364 6.4.4 The Phase Plane and the Potential 367 Answers to Odd-Numbered Exercises 379 Index 429