**Traditional First-Order Differential Equations **

Introduction to First-Order Equations

Separable Differential Equations

Linear Equations

Some Physical Models Arising as Separable Equations

Exact Equations

Special Integrating Factors and Substitution Methods

Bernoulli Equation

Homogeneous Equations of the Form g(y=x)

**Geometrical and Numerical Methods for First-Order Equations **

Direction Fields|the Geometry of Differential Equations

Existence and Uniqueness for First-Order Equations

First-Order Autonomous Equations|Geometrical Insight

Graphing Factored Polynomials

Bifurcations of Equilibria

Modeling in Population Biology

Nondimensionalization

Numerical Approximation: Euler and Runge-Kutta Methods

An Introduction to Autonomous Second-Order Equations

**Elements of Higher-Order Linear Equations **

Introduction to Higher-Order Equations

Operator Notation

Linear Independence and the Wronskian

Reduction of Order|the Case n = 2

Numerical Considerations for nth-Order Equations

Essential Topics from Complex Variables

Homogeneous Equations with Constant Coe□cients

Mechanical and Electrical Vibrations

**Techniques of Nonhomogeneous Higher-Order Linear Equations **

Nonhomogeneous Equations

Method of Undetermined Coe□cients via Superposition

Method of Undetermined Coe□cients via Annihilation

Exponential Response and Complex Replacement

Variation of Parameters

Cauchy-Euler (Equidimensional) Equation

Forced Vibrations

**Fundamentals of Systems of Differential Equations **

Useful Terminology

Gaussian Elimination

Vector Spaces and Subspaces

The Nullspace and Column Space

Eigenvalues and Eigenvectors

A General Method, Part I: Solving Systems with Real and Distinct or Complex

Eigenvalues

A General Method, Part II: Solving Systems with Repeated Real Eigenvalues

Matrix Exponentials

Solving Linear Nonhomogeneous Systems of Equations

**Geometric Approaches and Applications of Systems of Differential Equations**

An Introduction to the Phase Plane

Nonlinear Equations and Phase Plane Analysis

Systems of More Than Two Equations

Bifurcations

Epidemiological Models

Models in Ecology

**Laplace Transforms **

Introduction

Fundamentals of the Laplace Transform

The Inverse Laplace Transform

Laplace Transform Solution of Linear Differential Equations

Translated Functions, Delta Function, and Periodic Functions

The s-Domain and Poles

Solving Linear Systems Using Laplace Transforms

The Convolution

**Series Methods **

Power Series Representations of Functions

The Power Series Method

Ordinary and Singular Points

The Method of Frobenius

Bessel Functions

**Boundary-Value Problems and Fourier Series **

Two-Point Boundary-Value Problems

Orthogonal Functions and Fourier Series

Even, Odd, and Discontinuous Functions

Simple Eigenvalue-Eigenfunction Problems

Sturm-Liouville Theory

Generalized Fourier Series

**Partial Differential Equations **

Separable Linear Partial Differential Equations

Heat Equation

Wave Equation

Laplace Equation

Non-Homogeneous Boundary Conditions

Non-Cartesian Coordinate Systems

**A An Introduction to MATLAB, Maple, and Mathematica **

MATLAB

Some Helpful MATLAB Commands

Programming with a script and a function in MATLAB

Maple

Some Helpful Maple Commands

Programming in Maple

Mathematica

Some Helpful Mathematica Commands

Programming in Mathematica

**B Selected Topics from Linear Algebra **

A Primer on Matrix Algebra

Matrix Inverses, Cramer's Rule

Calculating the Inverse of a Matrix

Cramer's Rule

Linear Transformations

Coordinates and Change of Basis

Similarity Transformations

Computer Labs: MATLAB, Maple, Mathematica