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Ordinary Differential Equations Ordinary Differential Equations for Engineers and Scientists Gregg Waterman Oregon Institute of Technology c 2017 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International license. The essence of the license is that You are free to: Share copy and redistribute the material in any medium or format • Adapt remix, transform, and build upon the material for any purpose, even commercially. • The licensor cannot revoke these freedoms as long as you follow the license terms. Under the following terms: Attribution You must give appropriate credit, provide a link to the license, and indicate if changes • were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Notices: You do not have to comply with the license for elements of the material in the public domain or where your use is permitted by an applicable exception or limitation. No warranties are given. The license may not give you all of the permissions necessary for your intended use. For example, other rights such as publicity, privacy, or moral rights may limit how you use the material. For any reuse or distribution, you must make clear to others the license terms of this work. The best way to do this is with a link to the web page below. To view a full copy of this license, visit https://creativecommons.org/licenses/by/4.0/legalcode. Contents 0 Introduction to This Book 1 0.1 GoalsandEssentialQuestions . .... 1 0.2 AnIllustrativeExample . .. .. .. 3 1 Functions and Derivatives, Variables and Parameters 7 1.1 FunctionsandVariables . .. .. .. 9 1.2 Derivatives and Differential Equations . ........ 18 1.3 ParametersandVariables . .. 23 1.4 Initial Conditions and Boundary Conditions . ......... 26 1.5 Differential Equations and Their Solutions . ........ 30 1.6 Classification of Differential Equations . ......... 36 1.7 Initial Value Problems and Boundary Value Problems . .......... 40 1.8 Chapter1Summary ................................ 45 1.9 Chapter1Exercises ................................ 47 2 First Order Equations 49 2.1 SolvingBySeparationofVariables . ..... 51 2.2 SolutionCurvesandDirectionFields . ...... 57 2.3 SolvingWithIntegratingFactors . ..... 62 2.4 PhasePortraitsandStability. ..... 66 2.5 ApplicationsofFirstOrderODEs . .... 71 2.6 Chapter2Summary ................................ 81 2.7 Chapter2Exercises ................................ 82 3 Second Order Linear ODEs 87 3.1 Homogeneous Second-Order Equations . ..... 90 3.2 Free,UndampedVibration. .. 96 3.3 Free,DampedVibration ............................. 100 3.4 ParticularSolutions,PartOne . ..... 104 3.5 DifferentialOperators ............................. 108 3.6 Initial Value Problems and Forced, Damped Vibration . .......... 113 3.7 Chapter3Summary ................................ 116 3.8 Chapter3Exercises ................................ 118 4 More on Second Order Differential Equations 119 4.1 LinearIndependenceofSolutions . ..... 121 4.2 ReductionofOrder................................. 124 4.3 ParticularSolutions,PartTwo. ..... 126 4.4 Forced,UndampedVibration . 130 4.5 Chapter4Summary ................................ 132 5 Boundary Value Problems 137 5.1 DeflectionofHorizontalBeams . 139 5.2 Second-Order Boundary Value Problems, Eigenfunctions and Eigenvalues . 144 5.3 Eigenvalue Problems, Deflection of Vertical Columns . ........... 151 5.4 TheHeatEquationinOneDimension. 159 5.5 Chapter5Summary ................................ 163 i A Formula Sheet 165 B Review of Calculus and Algebra 167 B.1 ReviewofDifferentiation. 167 B.2 ReviewofIntegration .............................. 172 B.3 SolvingSystemsofEquations . 176 B.4 PartialFractionDecomposition . ..... 181 B.5 SeriesandEuler’sFormula. 183 C Numerical Solutions to ODEs 185 D Solutions to Exercises 193 D.1 Chapter1Solutions ................................ 193 D.2 Chapter2Solutions ................................ 198 D.3 Chapter3Solutions ................................ 203 D.4 Chapter4Solutions ................................ 206 D.5 Chapter5Solutions ................................ 208 D.6 SolutionsforAppendices. 210 ii 0 Introduction to This Book 0.1 Goals and Essential Questions Differential equations are perhaps the most central mathematical topic of science and engineering. Our quest in those areas is to understand and predict the behavior of some sort of “system” consisting of a collection of “parts” that could be things like electrical or mechanical components, living organisms, or some part of the natural world. We often wish to construct a mathematical model that describes the behavior of the system reasonably well; such a model usually consists of one of three things: An equation or a set of equations (an analytical model). • A general but imprecise description or a graph (a qualitative model). • “Snapshots” of the state of the system at discrete points in time and/or space (a numerical • model). The problem is that we generally can’t construct directly the equation or equations making up an analytical model of a system. What we will usually have at our disposal are pieces of information about how a system is changing and/or how forces are acting on and within the system. Those pieces of information are combined to form an equation containing the changes or forces, in the form of derivatives. Such an equation containing derivatives is called a differential equation. Once a differential equation is obtained, we hope we can use some mathematical technique to extract a model without derivatives that describes the behavior of the system. This book is a fairly straightforward introduction to differential equations, with an applied emphasis. The student should be aware that this is a huge subject, with lifetimes of study possible. Our hope is that this collection of explanations, examples and exercises will create a solid foundation for understanding differential equations when they are encountered in subject-specific courses, and for further study of differential equations themselves. In the past an introduction to differential equations has usually consisted of learning specific tech- niques for solving a variety differential equations. It should be no surprise that those techniques are easily forgotten in short order! We will look at techniques for obtaining solutions - that is an essential part of the subject. However, we will also attend to the “bigger picture,” in the hopes of giving the student an overall understanding of the subject that will be more lasting than just a bunch of ‘recipes” for obtaining solutions. Our study of the subject of differential equations will be guided by some overarching goals, and essential questions related to those goals. Goals Upon completion of his/her study, the student should understand what differential equations, initial value problems, and boundary value problems are, and what their solutions consist of. For ordinary differential equations (ODEs) and associated initial value or boundary value problems, the student should understand where such problems come from, • what their solutions consist of, • how solutions are obtained, • how parameters of a system and initial or boundary conditions influence the nature of solutions. • Our pursuit of these goals will take place through the consideration of some related essential questions. 1 Essential Questions: What are differential equations and why do we need them? • What is a solution to a differential equation? What do we mean by a family of solutions to a • differential equation? What are initial value problems, and what are boundary value problems? How are the two alike • and how are they different? What is meant by an analytical solution? A qualitative solution? A numerical solution? • How do we go about finding solutions to differential equations? • How do parameters differ from variables? What is the role of parameters in differential equations? • What is a mathematical model? How do differential equations and their solutions model systems • and their responses? It has been demonstrated experimentally that retaining the things we learn can be enhanced if those things are learned through spaced repetition. To that end, I have attempted to write this book like a novel in which most of the characters are introduced early on, and are then developed and fleshed out as the plot unfolds. A large number of the important concepts of differential equations (including at least a little bit about partial differential equations) are first seen in Chapter 1, then taken up again at later points in the book, where they are reinforced and expanded upon. Those having prior knowledge of ordinary differential equations (in most cases, the instructor) will notice that the focus of this book is more on the important concepts related to differential equations (both ordinary and partial) rather than techniques for solving a broad range of types of equations. This is based on my conviction that most students will quickly forget the specific procedures for solving differential equations unless those techniques are used in other courses taken shortly after this one. However, a person with a good working understanding of differential equations, initial value problems and boundary value problems should be able to go to any of the many resources available and quickly remind themselves of techniques previously learned, or even techniques not seen in this course! 2 0.2
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