Efnisyfirlit

• 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

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