Preface

Preliminaries

Vectors and Matrices

MATLAB

Special Kinds of Matrices

The Geometry of Vector Operations

Solving Linear Equations

The Geometry of Low-Dimensional Solutions

Linear Equations with Special Coefficients

Uniqueness of Reduced Echelon Form

Matrices and Linearity

Matrix Mappings

The Principle of Superposition

Determinants of 2 × 2 Matrices

Solving Ordinary Differential Equations

A Single Differential Equation

Graphing Solutions to Differential Equations

Phase Space Pictures and Equilibria

Separation of Variables

Uncoupled Linear Systems of Two Equations

Coupled Linear Systems

The Initial Value Problem and Eigenvectors

Eigenvalues of 2 × 2 Matrices

Initial Value Problems Revisited

Vector Spaces

Vector Spaces and Subspaces

Construction of Subspaces

Dimension and Bases

Closed Form Solutions for Planar ODEs

The Initial Value Problem

Closed Form Solutions by the Direct Method

Solutions Using Matrix Exponentials

Linear Normal Form Planar Systems

Similar Matrices

Formulas for Matrix Exponentials

Second Order Equations

Qualitative Theory of Planar ODEs

Sinks, Saddles, and Sources

Phase Portraits of Sinks

Phase Portraits of Nonhyperbolic Systems

Determinants and Eigenvalues

Determinants

Eigenvalues

Existence of Determinants

Linear Maps and Changes of Coordinates

Linear Mappings and Bases

Orthogonality

Least Squares Fitting of Data

Symmetric Matrices

Autonomous Planar Nonlinear Systems

Bifurcation Theory

The Continuous Flow Stirred Tank Reactor

Saddle-Node Bifurcations Revisited

Matrix Normal Forms

Markov Matrix Theory

Proof of Jordan Normal Form

Higher Dimensional Systems

MATLAB ODE

Linear Differential Equations

Solving Systems in Original Coordinates

Higher Order Equations

Linear Differential Operators

Undetermined Coefficients

Periodic Forcing and Resonance

Laplace Transforms

The Method of Laplace Transforms

Laplace Transforms and Their Computation

Partial Fractions

RLC Circuits

Additional Techniques for Solving ODEs

Nonconstant Coefficient Linear Equations

Variation of Parameters for Systems

The Wronskian

Higher Order Equations

Simplification by Substitution

Exact Differential Equations

Numerical Solutions of ODEs

A Description of Numerical Methods

Error Bounds for Euler’s Method

Local and Global Error Bounds

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