Preface

1Preliminaries

1.1Vectors and Matrices

1.2MATLAB

1.3Special Kinds of Matrices

1.4The Geometry of Vector Operations

2Solving Linear Equations

2.1

2.2The Geometry of Low-Dimensional Solutions

2.3

2.4

2.5Linear Equations with Special Coefficients

2.6Uniqueness of Reduced Echelon Form

3Matrices and Linearity

3.1

3.2Matrix Mappings

3.3

3.4The Principle of Superposition

3.5

3.6

3.7

3.8Determinants of 2 × 2 Matrices

4Solving Ordinary Differential Equations

4.1A Single Differential Equation

4.2Graphing Solutions to Differential Equations

4.3Phase Space Pictures and Equilibria

4.4Separation of Variables

4.5Uncoupled Linear Systems of Two Equations

4.6Coupled Linear Systems

4.7The Initial Value Problem and Eigenvectors

4.8Eigenvalues of 2 × 2 Matrices

4.9Initial Value Problems Revisited

4.10

5Vector Spaces

5.1Vector Spaces and Subspaces

5.2Construction of Subspaces

5.3

5.4

5.5Dimension and Bases

5.6

6Closed Form Solutions for Planar ODEs

6.1The Initial Value Problem

6.2Closed Form Solutions by the Direct Method

6.3Solutions Using Matrix Exponentials

6.4Linear Normal Form Planar Systems

6.5Similar Matrices

6.6Formulas for Matrix Exponentials

6.7Second Order Equations

7Qualitative Theory of Planar ODEs

7.1Sinks, Saddles, and Sources

7.2Phase Portraits of Sinks

7.3Phase Portraits of Nonhyperbolic Systems

8Determinants and Eigenvalues

8.1Determinants

8.2Eigenvalues

8.3Existence of Determinants

9Linear Maps and Changes of Coordinates

9.1Linear Mappings and Bases

9.2

9.3

9.4

10Orthogonality

10.1

10.2

10.3Least Squares Fitting of Data

10.4Symmetric Matrices

10.5

11Autonomous Planar Nonlinear Systems

11.1

11.2

11.3

11.4

12Bifurcation Theory

12.1

12.2

12.3The Continuous Flow Stirred Tank Reactor

12.4

12.5Saddle-Node Bifurcations Revisited

12.6

13Matrix Normal Forms

13.1

13.2

13.3

13.4

13.5Markov Matrix Theory

13.6Proof of Jordan Normal Form

14Higher Dimensional Systems

14.1

14.2

14.3MATLAB ODE

14.4

14.5

14.6

15Linear Differential Equations

15.1Solving Systems in Original Coordinates

15.2Higher Order Equations

15.3Linear Differential Operators

15.4Undetermined Coefficients

15.5Periodic Forcing and Resonance

16Laplace Transforms

16.1The Method of Laplace Transforms

16.2Laplace Transforms and Their Computation

16.3Partial Fractions

16.4

16.5RLC Circuits

17Additional Techniques for Solving ODEs

17.1Nonconstant Coefficient Linear Equations

17.2Variation of Parameters for Systems

17.3The Wronskian

17.4Higher Order Equations

17.5Simplification by Substitution

17.6Exact Differential Equations

17.7

18Numerical Solutions of ODEs

18.1A Description of Numerical Methods

18.2Error Bounds for Euler’s Method

18.3Local and Global Error Bounds

18.4

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