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Tables of Contents for Introduction to Differential Equations and Dynamical Systems
Chapter/Section Title
Page #
Page Count
Preface
xi
Introductory Survey
1
13
First-Order Equations
14
76
Solutions and Initial Values
14
9
Elementary Solution Formulas
15
2
Initial Values and Solution Families
17
6
Direction Fields
23
14
Definition and Examples
24
2
Isoclines
26
5
Continuity, Existence, and Uniqueness
31
6
Integration
37
23
y' = f(x)
37
7
Variables Separable
44
9
Transformations (Optional)
53
7
Exact Equations (Optional)
60
11
Solutions by Integration
60
3
Exactness Test
63
3
Integrating Factors
66
5
Linear Equations
71
19
Exponential Integrating Factor
72
3
Linear Models
75
8
Linearization (Optional)
83
4
Chapter Review and Supplement
87
3
Applied Dynamics and Equilibrium
90
53
Population Dynamics
90
2
Radioactive Decay
92
5
Velocity in a Resisting Medium
97
7
Variable Mass and Rocket Propulsion
104
13
Equilibrium Solutions (Optional)
117
7
Numerical Methods
124
19
Euler Method
125
2
Improved Euler Method
127
7
Newton's Method
134
9
Equations of Order Two or More
143
60
Introduction
143
11
Exponential Solutions
144
2
Factoring Operators
146
8
Complex Solutions
154
19
Complex Exponentials
154
2
Complex Characteristic Roots
156
6
Order More Than Two
162
6
Sinusoidal Oscillations (Optional)
168
5
Nonhomogeneous Linear Equations
173
13
General Solution
173
1
Undetermined-Coefficient Method
174
6
Green's Functions
180
6
Formal Integration Methods (Optional)
186
17
Linearization
187
3
Independence of Position: y = f(t,y)
190
2
Independence of Time: y = f(y,y)
192
8
Chapter Review and Supplement
200
3
Applied Dynamics and Phase Space
203
91
Gravitational Forces
204
7
Pendulum
211
8
Mechanical Oscillators
219
19
Electric Circuits
238
10
Phase Space and Periodicity
248
16
Phase Space
248
6
Existence of Periodic Solutions
254
10
Energy and Stability
264
13
Potential Energy
264
4
Stability
268
9
Numerical Methods
277
17
Euler Methods
278
3
Stiffness
281
6
Time Sections
287
7
Introduction to Systems
294
35
Vector Equations
294
25
Geometric Setting
294
5
Vector Fields
299
2
Order Reduction and Normal Form
301
8
Equilibrium Solutions
309
3
Existence, Uniqueness, and Flows (Optional)
312
7
Linear Systems
319
10
Definitions
319
1
Elimination Method
320
2
General Form of Solutions
322
5
Chapter Review and Supplement
327
2
Applied Dynamics and Stability
329
105
Multicompartment Mixing
329
7
Interacting Populations
336
6
Electric Networks
342
5
Mechanical Oscillations
347
10
Spring-Linked Masses in Linear Motion
347
4
Nonlinear Spring Analysis
351
6
Inverse-Square Law
357
13
Energy
370
12
Numerical Methods
382
10
Euler's Method
382
2
Improved Euler Method
384
8
Stability for Autonomous Systems
392
30
Linear Systems
392
9
Nonlinear Systems and Linearization
401
12
Liapunov's Method
413
9
Calculus of Variations
422
12
Euler-Lagrange Equations
423
6
Vector Equations
429
5
Matrix Methods
434
44
Eigenvalues and Eigenvectors
434
11
Exponential Solutions
435
5
Eigenvector Matrices and Initial Conditions
440
5
Matrix Exponentials
445
11
Solving Systems
448
2
Relationship to Eigenvectors
450
3
Independent Solutions
453
3
Computing etA in General
456
7
Nonhomogeneous Systems
463
7
Solution Formula
463
1
Linearity
464
1
Variation of Parameters
465
2
Summary of Methods
467
3
Stability for Autonomous Systems
470
8
Linear Systems
471
1
Nonlinear Systems
472
6
Laplace Transforms
478
22
Basic Properties
478
10
Convolution
488
5
Generalized Functions
493
7
The Delta Function
493
3
Solution of Equations
496
4
Variable-Coefficient Methods
500
58
Independent Solutions
501
7
Nonhomogeneous Equations
508
8
Variation of Parameters
508
3
Green's Functions
511
1
Summary
512
4
Boundary Problems
516
9
Two-Point Boundaries
516
3
Nonhomogeneous Equations
519
2
Nonhomogeneous Boundary Conditions
521
4
Shooting
525
5
Power Series Solutions
530
3
Undetermined Coefficients
533
9
Singular Points
542
16
Euler's Equation
544
3
Bessel's Equation
547
11
Partial Differential Equations
558
48
Introduction
558
6
Equations and Solutions
558
3
Exponential Solutions
561
3
Fourier Series
564
9
Introduction
564
1
Computing Coefficients
565
3
Convergence
568
5
Adapted Fourier Expansions
573
7
General Intervals
573
2
Sine and Cosine Expansions
575
5
Heat and Wave Equations
580
17
One-Dimensional Heat Equation
580
5
Equilibrium Solutions
585
3
One-Dimensional Wave Equation
588
9
Laplace Equation
597
9
Chapter Review and Supplement
603
3
Sturm-Liouville Expansions
606
26
Orthogonal Functions
606
7
Eigenfunctions and Symmetry
613
10
Fourier Inner Product
614
4
Weighted Inner Products
618
5
Vibrational Modes
623
9
Rectangular Membrane
623
3
Circular Membrane
626
6
APPENDIX A COMPLEX NUMBERS
632
7
APPENDIX B MATRIX ALGEBRA
639
23
Sum and Scalar Multiple
640
3
Matrix Products
643
3
Identity Matrices
646
1
Invertibility
647
1
Powers of a Square Matrix
648
3
Determinants
651
11
APPENDIX C EXISTENCE OF SOLUTIONS
662
7
Picard Method
662
3
Existence Theorems
665
4
APPENDIX D INDEFINITE INTEGRALS
669
6
Identity Substitutions
669
1
Substitution for the Integration Variable
670
1
Substitution for the Part of the Integrand
671
1
Integration by Parts
671
1
Integral Table
672
3
Answers to Selected Exercises
675
12
Index
687