Introduction to differential equations with dynamical systems

Cover......Page 1
Title......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 10
1.1 Introduction to Ordinary Differential Equations......Page 16
1.2 The Definite Integral and the Initial Value Problem......Page 19
1.2.1 The Initial Value Problem and the Indefinite Integral......Page 20
1.2.2 The Initial Value Problem and the Definite Integral......Page 21
1.2.3 Mechanics I: Elementary Motion of a Particle with Gravity Only......Page 23
1.3 First-Order Separable Differential Equations......Page 28
1.3.1 Using Definite Integrals for Separable Differential Equations......Page 31
1.4 Direction Fields......Page 34
1.4.1 Existence and Uniqueness......Page 40
1.5 Euler’s Numerical Method (optional)......Page 46
1.6.1 Form of the General Solution......Page 52
1.6.2 Solutions of Homogeneous First-Order Linear Differential Equations......Page 54
1.6.3 Integrating Factors for First-Order Linear Differential Equations......Page 57
1.7.1 Homogeneous Linear Differential Equations with Constant Coefficients......Page 63
1.7.2 Constant Coefficient Linear Differential Equations with Constant Input......Page 65
1.7.4 Constant Coefficient Differential Equations with Discontinuous Input......Page 67
1.8.1 A First Model of Population Growth......Page 74
1.8.2 Radioactive Decay......Page 80
1.8.3 Thermal Cooling......Page 83
1.9.1 Mixture Problems with a Fixed Volume......Page 89
1.9.2 Mixture Problems with Variable Volumes......Page 92
1.10 Electronic Circuits......Page 97
1.11 Mechanics II: Including Air Resistance......Page 103
1.12 Orthogonal Trajectories (optional)......Page 107
2.1 General Solution of Second-Order Linear Differential Equations......Page 111
2.2 Initial Value Problem (for Homogeneous Equations)......Page 115
2.3 Reduction of Order......Page 122
2.4 Homogeneous Linear Constant Coefficient Differential Equations (Second Order)......Page 127
2.4.1 Homogeneous Linear Constant Coefficient Differential Equations (nth-Order)......Page 137
2.5.1 Formulation of Equations......Page 139
2.5.2 Simple Harmonic Motion (No Damping, δ = 0)......Page 143
2.5.3 Free Response with Friction (δ > 0)......Page 150
2.6 The Method of Undetermined Coefficients......Page 157
2.7.1 Friction is Absent (δ = 0)......Page 174
2.7.2 Friction is Present (δ > 0) (Damped Forced Oscillations)......Page 183
2.8 Linear Electric Circuits......Page 189
2.9 Euler Equation......Page 194
2.10 Variation of Parameters (Second-Order)......Page 200
2.11 Variation of Parameters (nth-Order)......Page 208
3.1 Definition and Basic Properties......Page 212
3.1.1 The Shifting Theorem (Multiplying by an Exponential)......Page 220
3.1.2 Derivative Theorem (Multiplying by t)......Page 225
3.2 Inverse Laplace Transforms (Roots, Quadratics, and Partial Fractions)......Page 228
3.3 Initial Value Problems for Differential Equations......Page 240
3.4 Discontinuous Forcing Functions......Page 249
3.4.1 Solution of Differential Equations......Page 254
3.5 Periodic Functions......Page 263
3.6 Integrals and the Convolution Theorem......Page 268
3.6.1 Derivation of the Convolution Theorem (optional)......Page 272
3.7 Impulses and Distributions......Page 275
4.1 Introduction......Page 280
4.2 Introduction to Linear Systems of Differential Equations......Page 283
4.2.1 Solving Linear Systems Using Eigenvalues and Eigenvectors of the Matrix......Page 284
4.2.2 Solving Linear Systems if the Eigenvalues are Real and Unequal......Page 287
4.2.3 Finding General Solutions of Linear Systems in the Case of Complex Eigenvalues......Page 291
4.2.4 Special Systems with Complex Eigenvalues (optional)......Page 294
4.2.5 General Solution of a Linear System if the Two Real Eigenvalues are Equal (Repeated) Roots......Page 296
4.2.6 Eigenvalues and Trace and Determinant (optional)......Page 298
4.3.1 Introduction to the Phase Plane for Linear Systems of Differential Equations......Page 302
4.3.2 Phase Plane for Linear Systems of Differential Equations......Page 310
4.3.3 Real Eigenvalues......Page 311
4.3.4 Complex Eigenvalues......Page 319
4.3.5 General Theorems......Page 325
5.1 First-Order Differential Equations......Page 330
5.2.1 Equilibrium......Page 331
5.2.2 Stability......Page 332
5.2.4 Linear Stability Analysis......Page 333
5.3 One-Dimensional Phase Lines......Page 337
5.4 Application to Population Dynamics: The Logistic Equation......Page 342
6.1 Introduction......Page 347
6.2 Equilibria of Nonlinear Systems, Linear Stability Analysis of Equilibrium, and the Phase Plane......Page 350
6.2.1 Linear Stability Analysis and the Phase Plane......Page 351
6.2.2 Nonlinear Systems: Summary, Philosophy, Phase Plane, Direction Field, Nullclines......Page 357
6.3 Population Models......Page 364
6.3.1 Two Competing Species......Page 365
6.3.2 Predator-Prey Population Models......Page 371
6.4.1 Nonlinear Pendulum......Page 378
6.4.3 Conservative Systems and the Energy Integral......Page 379
6.4.4 The Phase Plane and the Potential......Page 382
Answers to Odd-Numbered Exercises......Page 394
L......Page 444
Y......Page 445