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• Title Page
• Copyright Page
• Table of Contents
• Introduction
• About This Book
• Conventions Used in This Book
• What You’re Not to Read
• Foolish Assumptions
• How This Book Is Organized
• Part I: Focusing on First Order Differential Equations
• Part II: Surveying Second and Higher Order Differential Equations
• Part III: The Power Stuff: Advanced Techniques
• Part IV: The Part of Tens
• Icons Used in This Book
• Where to Go from Here
• Part I: Focusing on First Order Differential Equations
• Chapter 1: Welcome to the World of Differential Equations
• The Essence of Differential Equations
• Derivatives: The Foundation of Differential Equations
• Derivatives that are constants
• Derivatives that are powers
• Derivatives involving trigonometry
• Derivatives involving multiple functions
• Seeing the Big Picture with Direction Fields
• Plotting a direction field
• Connecting slopes into an integral curve
• Recognizing the equilibrium value
• Classifying Differential Equations
• Classifying equations by order
• Classifying ordinary versus partial equations
• Classifying linear versus nonlinear equations
• Solving First Order Differential Equations
• Tackling Second Order and Higher Order Differential Equations
• Having Fun with Advanced Techniques
• Chapter 2: Looking at Linear First Order Differential Equations
• First Things First: The Basics of Solving Linear First Order Differential Equations
• Applying initial conditions from the start
• Stepping up to solving differential equations involving functions
• Adding a couple of constants to the mix
• Solving Linear First Order Differential Equations with Integrating Factors
• Solving for an integrating factor
• Using an integrating factor to solve a differential equation
• Moving on up: Using integrating factors in differential equations with functions
• Trying a special shortcut
• Solving an advanced example
• Determining Whether a Solution for a Linear First Order Equation Exists
• Spelling out the existence and uniqueness theorem for linear differential equations
• Finding the general solution
• Checking out some existence and uniqueness examples
• Figuring Out Whether a Solution for a Nonlinear Differential Equation Exists
• The existence and uniqueness theorem for nonlinear differential equations
• A couple of nonlinear existence and uniqueness examples
• Chapter 3: Sorting Out Separable First Order Differential Equations
• Beginning with the Basics of Separable Differential Equations
• Starting easy: Linear separable equations
• Introducing implicit solutions
• Finding explicit solutions from implicit solutions
• Tough to crack: When you can’t find an explicit solution
• A neat trick: Turning nonlinear separable equations into linear separable equations
• Trying Out Some Real World Separable Equations
• Getting in control with a sample flow problem
• Striking it rich with a sample monetary problem
• Break It Up! Using Partial Fractions in Separable Equations
• Chapter 4: Exploring Exact First Order Differential Equations and Euler’s Method
• Exploring the Basics of Exact Differential Equations
• Defining exact differential equations
• Working out a typical exact differential equation
• Determining Whether a Differential Equation Is Exact
• Checking out a useful theorem
• Applying the theorem
• Conquering Nonexact Differential Equations with Integrating Factors
• Finding an integrating factor
• Using an integrating factor to get an exact equation
• The finishing touch: Solving the exact equation
• Getting Numerical with Euler’s Method
• Understanding the method
• Checking the method’s accuracy on a computer
• Delving into Difference Equations
• Some handy terminology
• Iterative solutions
• Equilibrium solutions
• Part II: Surveying Second and Higher Order Differential Equations
• Chapter 5: Examining Second Order Linear Homogeneous Differential Equations
• The Basics of Second Order Differential Equations
• Linear equations
• Homogeneous equations
• Second Order Linear Homogeneous Equations with Constant Coefficients
• Elementary solutions
• Initial conditions
• Checking Out Characteristic Equations
• Real and distinct roots
• Complex roots
• Identical real roots
• Getting a Second Solution by Reduction of Order
• Seeing how reduction of order works
• Trying out an example
• Putting Everything Together with Some Handy Theorems
• Superposition
• Linear independence
• The Wronskian
• Chapter 6: Studying Second Order Linear Nonhomogeneous Differential Equations
• The General Solution of Second Order Linear Nonhomogeneous Equations
• Understanding an important theorem
• Putting the theorem to work
• Finding Particular Solutions with the Method of Undetermined Coefficients
• When g(x) is in the form of erx
• When g(x) is a polynomial of order n
• When g(x) is a combination of sines and cosines
• When g(x) is a product of two different forms
• Breaking Down Equations with the Variation of Parameters Method
• Nailing down the basics of the method
• Solving a typical example
• Applying the method to any linear equation
• What a pair! The variation of parameters method meets the Wronskian
• Bouncing Around with Springs ’n’ Things
• A mass without friction
• A mass with drag force
• Chapter 7: Handling Higher Order Linear Homogeneous Differential Equations
• The Write Stuff: The Notation of Higher Order Differential Equations
• Introducing the Basics of Higher Order Linear Homogeneous Equations
• The format, solutions, and initial conditions
• A couple of cool theorems
• Tackling Different Types of Higher Order Linear Homogeneous Equations
• Real and distinct roots
• Real and imaginary roots
• Complex roots
• Duplicate roots
• Chapter 8: Taking On Higher Order Linear Nonhomogeneous Differential Equations
• Mastering the Method of Undetermined Coefficients for Higher Order Equations
• When g(x) is in the form erx
• When g(x) is a polynomial of order n
• When g(x) is a combination of sines and cosines
• Solving Higher Order Equations with Variation of Parameters
• The basics of the method
• Working through an example
• Part III: The Power Stuff: Advanced Techniques
• Chapter 9: Getting Serious with Power Series and Ordinary Points
• Perusing the Basics of Power Series
• Determining Whether a Power Series Converges with the Ratio Test
• The fundamentals of the ratio test
• Plugging in some numbers
• Shifting the Series Index
• Taking a Look at the Taylor Series
• Solving Second Order Differential Equations with Power Series
• When you already know the solution
• When you don’t know the solution beforehand
• A famous problem: Airy’s equation
• Chapter 10: Powering through Singular Points
• Pointing Out the Basics of Singular Points
• Finding singular points
• The behavior of singular points
• Regular versus irregular singular points
• Exploring Exciting Euler Equations
• Real and distinct roots
• Real and equal roots
• Complex roots
• Putting it all together with a theorem
• Figuring Series Solutions Near Regular Singular Points
• Identifying the general solution
• The basics of solving equations near singular points
• A numerical example of solving an equation near singular points
• Taking a closer look at indicial equations
• Chapter 11: Working with Laplace Transforms
• Breaking Down a Typical Laplace Transform
• Deciding Whether a Laplace Transform Converges
• Calculating Basic Laplace Transforms
• The transform of 1
• The transform of eat
• The transform of sin at
• Consulting a handy table for some relief
• Solving Differential Equations with Laplace Transforms
• A few theorems to send you on your way
• Solving a second order homogeneous equation
• Solving a second order nonhomogeneous equation
• Solving a higher order equation
• Factoring Laplace Transforms and Convolution Integrals
• Factoring a Laplace transform into fractions
• Checking out convolution integrals
• Surveying Step Functions
• Defining the step function
• Figuring the Laplace transform of the step function
• Chapter 12: Tackling Systems of First Order Linear Differential Equations
• Introducing the Basics of Matrices
• Setting up a matrix
• Working through the algebra
• Examining matrices
• Mastering Matrix Operations
• Equality
• Addition
• Subtraction
• Multiplication of a matrix and a number
• Multiplication of two matrices
• Multiplication of a matrix and a vector
• Identity
• The inverse of a matrix
• Having Fun with Eigenvectors ’n’ Things
• Linear independence
• Eigenvalues and eigenvectors
• Solving Systems of First-Order Linear Homogeneous Differential Equations
• Understanding the basics
• Making your way through an example
• Solving Systems of First Order Linear Nonhomogeneous Equations
• Assuming the correct form of the particular solution
• Crunching the numbers
• Winding up your work
• Chapter 13: Discovering Three Fail-Proof Numerical Methods
• Number Crunching with Euler’s Method
• The fundamentals of the method
• Using code to see the method in action
• Moving On Up with the Improved Euler’s Method
• Understanding the improvements
• Coming up with new code
• Plugging a steep slope into the new code
• Adding Even More Precision with the Runge-Kutta Method
• The method’s recurrence relation
• Working with the method in code
• Part IV: The Part of Tens
• Chapter 14: Ten Super-Helpful Online Differential Equation Tutorials
• AnalyzeMath.com’s Introduction to Differential Equations
• Harvey Mudd College Mathematics Online Tutorial
• John Appleby’s Introduction to Differential Equations
• Kardi Teknomo’s Page
• Martin J. Osborne’s Differential Equation Tutorial
• Midnight Tutor’s Video Tutorial
• The Ohio State University Physics Department’s Introduction to Differential Equations
• Paul’s Online Math Notes
• S.O.S. Math
• University of Surrey Tutorial
• Chapter 15: Ten Really Cool Online Differential Equation Solving Tools
• AnalyzeMath.com’s Runge-Kutta Method Applet
• Coolmath.com’s Graphing Calculator
• Direction Field Plotter
• An Equation Solver from QuickMath Automatic Math Solutions
• First Order Differential Equation Solver
• GCalc Online Graphing Calculator
• JavaView Ode Solver
• Math @ CowPi’s System Solver
• A Matrix Inverter from QuickMath Automatic Math Solutions
• Visual Differential Equation Solving Applet
• Index
• EULA