Als die Gute Fee Hanschen fragte: "was wunschst Du dir?," antwortete er: "Keine Differentialgleichungen mehr in der Schule": Hans im Gluck! Jetzt konnen Sie auch auf eine Gute Fee warten, oder sich dieses Buch kaufen. Sie finden hier Hilfe sollten Sie mit linearen und nichtlinearen gewohnlichen Differentialgleichungen ihre liebe Muhe haben, seien sie nun erster, zweiter oder hoherer Ordnung. Sie lernen auch, was Sie zu Laplace Transformation, Potenzreihen und vielen anderen vertrackten Problemen wissen sollten. Sehen Sie der Realitat ins Auge, mit diesem Buch.
Author(s): Steven Holzner
Series: For Dummies
Edition: 1
Differential Equations For Dummies
About the Author
Dedication
Author’s Acknowledgments
Contents at a Glance
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
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
Seeing the Big Picture with Direction Fields
Classifying Differential 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
Solving Linear First Order Differential Equations with Integrating Factors
Determining Whether a Solution for a Linear First Order Equation Exists
Figuring Out Whether a Solution for a Nonlinear Differential Equation Exists
Chapter 3: Sorting Out Separable First Order Differential Equations
Beginning with the Basics of Separable Differential Equations
Trying Out Some Real World Separable Equations
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
Determining Whether a Differential Equation Is Exact
Conquering Nonexact Differential Equations with Integrating Factors
Getting Numerical with Euler’s Method
Delving into Difference Equations
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
Second Order Linear Homogeneous Equations with Constant Coefficients
Checking Out Characteristic Equations
Getting a Second Solution by Reduction of Order
Putting Everything Together with Some Handy Theorems
Chapter 6: Studying Second Order Linear Nonhomogeneous Differential Equations
The General Solution of Second Order Linear Nonhomogeneous Equations
Finding Particular Solutions with the Method of Undetermined Coefficients
Breaking Down Equations with the Variation of Parameters Method
Bouncing Around with Springs ’n’ Things
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
Tackling Different Types of Higher Order Linear Homogeneous Equations
Chapter 8: Taking On Higher Order Linear Nonhomogeneous Differential Equations
Mastering the Method of Undetermined Coefficients for Higher Order Equations
Solving Higher Order Equations with Variation of Parameters
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
Shifting the Series Index
Taking a Look at the Taylor Series
Solving Second Order Differential Equations with Power Series
Chapter 10: Powering through Singular Points
Pointing Out the Basics of Singular Points
Exploring Exciting Euler Equations
Figuring Series Solutions Near Regular Singular Points
Chapter 11: Working with Laplace Transforms
Breaking Down a Typical Laplace Transform
Deciding Whether a Laplace Transform Converges
Calculating Basic Laplace Transforms
Solving Differential Equations with Laplace Transforms
Factoring Laplace Transforms and Convolution Integrals
Surveying Step Functions
Chapter 12: Tackling Systems of First Order Linear Differential Equations
Introducing the Basics of Matrices
Mastering Matrix Operations
Having Fun with Eigenvectors ’n’ Things
Solving Systems of First-Order Linear Homogeneous Differential Equations
Solving Systems of First Order Linear Nonhomogeneous Equations
Chapter 13: Discovering Three Fail-Proof Numerical Methods
Number Crunching with Euler’s Method
Moving On Up with the Improved Euler’s Method
Adding Even More Precision with the Runge-Kutta Method
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
