Table of contents for Introduction to differential equations with dynamical systems / Stephen L. Campbell and Richard Haberman.


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CHAPTER I
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, 8 = 0)           128
2.5.3 Free Response with Friction (8 > 0)                  135
2.6  The Method of Undetermined Coefficients                    142
2.7  Mechanical Vibrations II: Forced Response                  159
2.7.1 Friction is Absent (3 = 0)                           159
2.7.2 Friction is Present (8 > 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



Library of Congress subject headings for this publication: Differentiable dynamical systems, Differential equations