Table of Contents

Chapter 1 Basic Concepts
     1.1 Simple Differential Equations and Explicit Solutions
     1.2 Graphical Solutions Using Calculus
     1.3 Slope Fields and Isoclines
     1.4 Functions and Power Series Expansions

Chapter 2 Autonomous Differential Equations
     2.1 Autonomous Equations
     2.2 Simple Models
     2.3 The Logistic Equation
     2.4 Existence and Uniqueness of Solutions, and Words of Caution
     2.5 Qualitative Behavior of Solutions Using Phase Lines
     2.6 Bifurcation Diagrams

Chapter 3 First Order Differential Equations—Qualitative and Quantitative Aspects
     3.1 Graphical Solutions Using Calculus
     3.2 Symmetry of Slope Fields
     3.3 Numerical Solutions and Chaos
     3.4 Comparing Solutions of Differential Equations
     3.5 Finding Power Series Solutions

Chapter 4 Models and Applications Leading to New Techniques
     4.1 Solving Separable Differential Equations
     4.2 Solving Differential Equations with Homogeneous Coefficients
     4.3 Models - Deriving Differential Equations From Data
     4.4 Models - Objects in Motion
     4.5 Application - Orthogonal Trajectories
     4.6 Piecing Together Differential Equations

Chapter 5 First Order Linear Differential Equations and Models
     5.1 Solving Linear Differential Equations
     5.2 Models That Use Linear Equations
     5.3 Models That Use Bernoulli's Equation

Chapter 6 Interplay Between First Order Systems and Second Order Equations
     6.1 Simple Models
     6.2 How First Order Systems and Second Order Equations Are Related
     6.3 Second Order Linear Differential Equations with Constant Coefficients
     6.4 Modeling Physical Situations
     6.5 Interpreting the Phase Plane
     6.6 How Explicit Solutions Are Related to Orbits
     6.7 The Motion of a Nonlinear Pendulum

Chapter 7 Second Order Linear Differential Equations with Forcing Functions
     7.1 The General Solution
     7.2 Finding Solutions by the Method of Undetermined Coefficients
     7.3 Applications and Models

Chapter 8 Second Order Linear Differential Equations—Qualitative and Quantitative Aspects
     8.1 Qualitative Behavior of Solutions
     8.2 Finding Solutions by Reduction of Order
     8.3 Finding Solutions by Variation of Parameters
     8.4 The Importance of Linear Independence and Dependence
     8.5 Solving Cauchy-Euler Equations
     8.6 Boundary Value Problems and the Shooting Method
     8.7 Solving Higher Order Homogeneous Differential Equations
     8.8 Solving Higher Order Nonhomogeneous Differential Equations

Chapter 9 Linear Autonomous Systems
     9.1 Solving Linear Autonomous Systems
     9.2 Classification of Solutions via Stability
     9.3 When Do Straight-Line Orbits Exist?
     9.4 Qualitative Behavior Using Nullclines
     9.5 Matrix Formulation of Solutions
     9.6 Compartmental Models

Chapter 10 Nonlinear Autonomous Systems
     10.1 Introduction to Nonlinear Autonomous Systems
     10.2 Qualitative Behavior Using Nullclines Analysis
     10.3 Qualitative Behavior Using Linearization
     10.4 Models Involving Nonlinear Autonomous Equations
     10.5 Bungee Jumping
     10.6 Linear Versus Nonlinear Differential Equations
     10.7 Autonomous Versus Nonautonomous Differential Equations

Chapter 11 Using Laplace Transforms
     11.1 Motivation
     11.2 Constructing New Laplace Transforms from Old
     11.3 The Inverse Laplace Transform and the Convolution Theorem
     11.4 Functions That Jump
     11.5 Models Involving First Order Linear Differential Equations
     11.6 Models Involving Higher Order Linear Differential Equations
     11.7 Applications to Systems of Linear Differential Equations
     11.8 When Do Laplace Transforms Exist?

Chapter 12 Using Power Series
     12.1 Solutions Using Taylor Series
     12.2 Solutions Using Power Series
     12.3 What To Do When Power Series Fail
     12.4 Solutions Using The Method of Frobenius

     A.1 Background Material
     A.2 Partial Fractions
     A.3 Infinite Series, Power Series, and Taylor Series
     A.4 Complex Numbers
     A.5 Elementary Matrix Operations
     A.6 Least Squares Approximation
     A.7 Proofs of the Oscillation Theorems

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