by Steven Holzner,PhD - Softouch

1 By Steven Holzner, PhDDifferentialEquationsFORDUMmIES 4/28/08 11:27 PM Page 4/28/08 11:27 PM Page iiDifferentialEquationsFORDUMmIES 4/28/08 11:27 PM Page 4/28/08 11:27 PM Page iiby Steven Holzner, PhDDifferentialEquationsFORDUMmIES 4/28/08 11:27 PM Page iiiDifferential Equations For Dummies Published byWiley Publishing, River , NJ 2008 by Wiley Publishing, Inc., Indianapolis, IndianaPublished simultaneously in CanadaNo part of this publication may be reproduced, stored in a retrieval system, or transmitted in any formor by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as per-mitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior writtenpermission of the Publisher, or authorization through payment of the appropriate per-copy fee to theCopyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax to the Publisher for permission should be addressed to the Legal Department, Wiley Publishing,Inc.

2 , 10475 Crosspoint Blvd., Indianapolis, IN 46256, 317-572-3447, fax 317-572-4355, or online : Wiley, the Wiley Publishing logo, For Dummies, the Dummies Man logo, A Reference for theRest of Us!, The Dummies Way, Dummies Daily, The Fun and Easy Way, and related tradedress are trademarks or registered trademarks of John Wiley & Sons, Inc. and/or its affiliates in the UnitedStates and other countries, and may not be used without written permission. All other trademarks are theproperty of their respective owners. Wiley Publishing, Inc., is not associated with any product or vendormentioned in this OF LIABILITY/DISCLAIMER OF WARRANTY: THE PUBLISHER AND THE AUTHOR MAKE NO REP-RESENTATIONS OR WARRANTIES WITH RESPECT TO THE ACCURACY OR COMPLETENESS OF THE CON-TENTS OF THIS WORK AND SPECIFICALLY DISCLAIM ALL WARRANTIES, INCLUDING WITHOUTLIMITATION WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE.

3 NO WARRANTY MAY BE CRE-ATED OR EXTENDED BY SALES OR PROMOTIONAL MATERIALS. THE ADVICE AND STRATEGIES CON-TAINED HEREIN MAY NOT BE SUITABLE FOR EVERY SITUATION. THIS WORK IS SOLD WITH THEUNDERSTANDING THAT THE PUBLISHER IS NOT ENGAGED IN RENDERING LEGAL, ACCOUNTING, OROTHER PROFESSIONAL SERVICES. IF PROFESSIONAL ASSISTANCE IS REQUIRED, THE SERVICES OF ACOMPETENT PROFESSIONAL PERSON SHOULD BE SOUGHT. NEITHER THE PUBLISHER NOR THEAUTHOR SHALL BE LIABLE FOR DAMAGES ARISING HEREFROM. THE FACT THAT AN ORGANIZATIONOR WEBSITE IS REFERRED TO IN THIS WORK AS A CITATION AND/OR A POTENTIAL SOURCE OF FUR-THER INFORMATION DOES NOT MEAN THAT THE AUTHOR OR THE PUBLISHER ENDORSES THE INFOR-MATION THE ORGANIZATION OR WEBSITE MAY PROVIDE OR RECOMMENDATIONS IT MAY , READERS SHOULD BE AWARE THAT INTERNET WEBSITES LISTED IN THIS WORK MAY HAVECHANGED OR DISAPPEARED BETWEEN WHEN THIS WORK WAS WRITTEN AND WHEN IT IS general information on our other products and services, please contact our Customer CareDepartment within the at 800-762-2974, outside the at 317-572-3993, or fax technical support, please visit also publishes its books in a variety of electronic formats.

4 Some content that appears in print maynot be available in electronic of Congress Control Number: 2008925781 ISBN: 978-0-470-17814-0 Manufactured in the United States of America10 9 8 7 6 5 4 3 2 4/28/08 11:27 PM Page ivAbout the AuthorSteven Holzneris an award-winning author of science, math, and technicalbooks. He got his training in differential equations at MIT and at CornellUniversity, where he got his PhD. He has been on the faculty at both MIT andCornell University, and has written such bestsellers as Physics For Dummiesand Physics Workbook For 4/28/08 11:27 PM Page 4/28/08 11:27 PM Page viDedicationTo Nancy, always and s AcknowledgmentsThe book you hold in your hands is the work of many people. I d especiallylike to thank Tracy Boggier, Georgette Beatty, Jessica Smith, technicalreviewer Jamie Song, PhD, and the folks in Composition Services who put thebook together so 4/28/08 11:27 PM Page viiPublisher s AcknowledgmentsWe re proud of this book; please send us your comments through our Dummies online registrationform located at of the people who helped bring this book to market include the following:Acquisitions, Editorial, and Media DevelopmentProject Editor: Georgette BeattyAcquisitions Editor: Tracy BoggierCopy Editor: Jessica SmithEditorial Program Coordinator:Erin Calligan MooneyTechnical Editor: Jamie Song, PhDEditorial Manager: Michelle HackerEditorial Assistants:Joe Niesen, Leeann HarneyCartoons: Rich Tennant( )Composition ServicesProject Coordinator:Erin SmithLayout and Graphics:Carrie A.

5 Cesavice,Stephanie D. JumperProofreaders:Caitie Kelly, Linda D. MorrisIndexer:Broccoli Information ManagementPublishing and Editorial for Consumer DummiesDiane Graves Steele, Vice President and Publisher, Consumer DummiesJoyce Pepple, Acquisitions Director, Consumer DummiesKristin A. Cocks, Product Development Director, Consumer DummiesMichael Spring, Vice President and Publisher, TravelKelly Regan,Editorial Director, TravelPublishing for Technology DummiesAndy Cummings,Vice President and Publisher, Dummies Technology/General UserComposition ServicesGerry Fahey,Vice President of Production ServicesDebbie Stailey,Director of Composition 4/28/08 11:27 PM Page viiiContents at a GlanceIntroduction ..1 Part I: Focusing on First Order Differential 1: Welcome to the World of Differential Equations ..7 Chapter 2: Looking at Linear First Order Differential 3: Sorting Out Separable First Order Differential 4: Exploring Exact First Order Differential Equations and Euler s II: Surveying Second and Higher OrderDifferential Equations.

6 89 Chapter 5: Examining Second Order Linear Homogeneous Differential Equations ..91 Chapter 6: Studying Second Order Linear Nonhomogeneous Differential Equations ..123 Chapter 7: Handling Higher Order Linear Homogeneous Differential Equations ..151 Chapter 8: Taking On Higher Order Linear Nonhomogeneous Differential Equations ..173 Part III: The Power Stuff: Advanced Techniques ..189 Chapter 9: Getting Serious with Power Series and Ordinary 10: Powering through Singular Points ..213 Chapter 11: Working with Laplace Transforms ..239 Chapter 12: Tackling Systems of First Order Linear Differential Equations ..265 Chapter 13: Discovering Three Fail-Proof Numerical Methods ..293 Part IV: The Part of Tens ..315 Chapter 14: Ten Super-Helpful Online Differential Equation 15: Ten Really Cool Online Differential Equation Solving Tools ..321 Index .. 4/28/08 11:28 PM Page 4/28/08 11:28 PM Page xTable of ContentsIntroduction.

7 1 About This Used in This Book ..1 What You re Not to Read ..2 Foolish Assumptions ..2 How This Book Is I: Focusing on First Order Differential II: Surveying Second and Higher Order Differential III: The Power Stuff: Advanced Techniques ..3 Part IV: The Part of Used in This Book ..4 Where to Go from I: Focusing on First Order Differential Equations ..5 Chapter 1: Welcome to the World of Differential Equations ..7 The Essence of Differential : The Foundation of Differential Equations ..11 Derivatives that are that are powers ..12 Derivatives involving trigonometry ..12 Derivatives involving multiple functions ..12 Seeing the Big Picture with Direction a direction field ..13 Connecting slopes into an integral curve ..14 Recognizing the equilibrium Differential Equations ..17 Classifying equations by order ..17 Classifying ordinary versus partial linear versus nonlinear First Order Differential Equations.

8 19 Tackling Second Order and Higher Order Differential Equations ..20 Having Fun with Advanced Techniques .. 4/28/08 11:28 PM Page xiChapter 2: Looking at Linear First Order Differential Equations ..23 First Things First: The Basics of Solving Linear First OrderDifferential Equations ..24 Applying initial conditions from the start ..24 Stepping up to solving differential equations involving a couple of constants to the Linear First Order Differential Equations with Integrating Factors ..26 Solving for an integrating factor ..27 Using an integrating factor to solve a differential equation ..28 Moving on up: Using integrating factors in differentialequations with functions ..29 Trying a special shortcut ..30 Solving an advanced Whether a Solution for a Linear First Order Equation Exists ..35 Spelling out the existence and uniqueness theorem for linear differential equations ..35 Finding the general solution.

9 36 Checking out some existence and uniqueness examples ..37 Figuring Out Whether a Solution for a Nonlinear Differential Equation Exists ..38 The existence and uniqueness theorem for nonlinear differential couple of nonlinear existence and uniqueness examples ..39 Chapter 3: Sorting Out Separable First Order Differential Equations ..41 Beginning with the Basics of Separable Differential Equations ..42 Starting easy: Linear separable equations ..43 Introducing implicit solutions ..43 Finding explicit solutions from implicit solutions ..45 Tough to crack: When you can t find an explicit solution ..48A neat trick: Turning nonlinear separable equations into linear separable equations ..49 Trying Out Some Real World Separable in control with a sample flow problem ..52 Striking it rich with a sample monetary problem ..55 Break It Up! Using Partial Fractions in Separable Equations ..59 Differential Equations For Dummies 4/28/08 11:28 PM Page xiiChapter 4: Exploring Exact First Order Differential Equations and Euler s Method.

10 63 Exploring the Basics of Exact Differential Equations ..63 Defining exact differential equations ..64 Working out a typical exact differential equation ..65 Determining Whether a Differential Equation Is out a useful theorem ..66 Applying the theorem ..67 Conquering Nonexact Differential Equations with Integrating Factors ..70 Finding an integrating an integrating factor to get an exact finishing touch: Solving the exact equation ..74 Getting Numerical with Euler s Method ..75 Understanding the method ..76 Checking the method s accuracy on a into Difference handy terminology ..84 Iterative solutions ..84 Equilibrium solutions ..85 Part II: Surveying Second and Higher OrderDifferential 5: Examining Second Order Linear Homogeneous Differential Equations ..91 The Basics of Second Order Differential equations ..92 Homogeneous equations ..93 Second Order Linear Homogeneous Equations with Constant Coefficients.