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Tables of Contents for Differential Equations & Linear Algebra
Chapter/Section Title
Page #
Page Count
Computing Projects
vi
Preface
ix
First-Order Differential Equations
1
74
Differential Equations and Mathematical Models
1
9
Integrals as General and Particular Solutions
10
8
Direction Fields and Solution Curves
18
12
Separable Equations and Applications
30
14
Linear First-Order Equations
44
14
Substitution Methods and Exact Equations
58
17
Mathematical Models and Numerical Methods
75
65
Population Models
75
13
Equilibrium Solutions and Stability
88
6
Acceleration-Velocity Models
94
11
Numerical Approximation: Euler's Method
105
11
A Closer Look at the Euler Method
116
12
The Runge--Kutta Method
128
12
Linear Systems and Matrices
140
86
Introduction to Linear Systems
140
10
Matrices and Gaussian Elimination
150
12
Reduced Row-Echelon Matrices
162
9
Matrix Operations
171
13
Inverses of Matrices
184
15
Determinants
199
17
Linear Equations and Curve Fitting
216
10
Vector Spaces
226
46
The Vector Space R3
226
12
The Vector Space Rn and Subspaces
238
8
Linear Combinations and Independence of Vectors
246
8
Bases and Dimension for Vector Spaces
254
8
General Vector Spaces
262
10
Linear Equations of Higher Order
272
81
Introduction: Second-Order Linear Equations
272
15
General Solutions of Linear Equations
287
13
Homogeneous Equations with Constant Coefficients
300
12
Mechanical Vibrations
312
11
Undetermined Coefficients and Variation of Parameters
323
16
Forced Oscillations and Resonance
339
14
Eigenvalues and Eigenvectors
353
32
Introduction to Eigenvalues
353
10
Diagonalization of Matrices
363
9
Applications Involving Powers of Matrices
372
13
Linear Systems of Differential Equations
385
91
First-Order Systems and Applications
385
12
Matrices and Linear Systems
397
12
The Eigenvalue Method for Linear Systems
409
15
Second-Order Systems and Mechanical Applications
424
14
Multiple Eigenvalue Solutions
438
20
Numerical Methods for Systems
458
18
Matrix Exponential Methods
476
38
Matrix Exponentials and Linear Systems
476
15
Nonhomogeneous Linear Systems
491
8
Spectral Decomposition Methods
499
15
Nonlinear Systems and Phenomena
514
58
Stability and the Phase Plane
514
13
Linear and Almost Linear Systems
527
15
Ecological Models: Predators and Competitors
542
13
Nonlinear Mechanical Systems
555
17
Laplace Transform Methods
572
55
Laplace Transforms and Inverse Transforms
572
12
Transformation of Initial Value Problems
584
12
Translation and Partial Fractions
596
10
Derivatives, Integrals, and Products of Transforms
606
9
Periodic and Piecewise Continuous Forcing Functions
615
12
Power Series Methods
627
56
Introduction and Review of Power Series
627
14
Power Series Solutions
641
13
Frobenius Series Solutions
654
17
Bessel's Equation
671
12
References for Further Study
683
Appendix A: Existence and Uniqueness of Solutions
1
16
Appendix B: Theory of Determinants
17
10
Answers to Selected Problems
27
Index
1