ELEMENTARY DIFFERENTIAL EQUATIONS

1 ELEMENTARY DIFFERENTIALEQUATIONSW illiam F. TrenchAndrew G. Cowles Distinguished Professor EmeritusDepartment of MathematicsTrinity UniversitySan Antonio, Texas, book has been judged to meet the evaluation criteria set by the Edi-torial Board of the American Instituteof Mathematics in connection withthe Institute sOpen Textbook Initiative. It may be copied, modified, re-distributed, translated, and built upon subject to the Creative CommonsAttribution-NonCommercial-ShareAl ike Unported DOWNLOAD: STUDENT SOLUTIONS MANUALFree Edition (December 2013)This book was published previously by Brooks/Cole Thomson Learning, 2001. This free edition is madeavailable in the hope that it will be useful as a textbook or reference. Reproduction is permitted forany valid noncommercial educational, mathematical, or scientific purpose. However, charges for profitbeyond reasonable printing costs are : This version of the textbook has been edited from its original for use by the University of CentralOklahoma.

2 Several sections (and one whole chapter) have been removed from the text. References toremoved material still appear in this text, as indicated by ?? .TO BEVERLYC ontentsChapter 1 Applications Leading to DIFFERENTIAL First Order Direction Fields for First Order Equations14 Chapter 2 First Order Linear First Order Separable Existence and Uniqueness of Solutions of Nonlinear Exact Integrating Factors63 Chapter 3 Numerical Euler s The Improved Euler Method and Related Methods85 Chapter 4 Applications of First Order Cooling and ELEMENTARY Autonomous Second Order Equations115 Chapter 5 Linear Second Order Homogeneous Linear Constant Coefficient Homogeneous Nonhomgeneous Linear The Method of Undetermined Coefficients The Method of Undetermined Coefficients Variation of Parameters178 Chapter 6 Applcations of Linear Second Order Spring Problems Spring Problems II197 Chapter 7 Series Solutions of Linear Second Order Review of Power Series Solutions Near an Ordinary

3 Point Series Solutions Near an Ordinary Point II231 Chapter 8 Laplace Introduction to the Laplace The Inverse Laplace Solution of Initial Value The Unit Step Constant Coefficient EQUATIONS with Piecewise Continuous Constant Cofficient EQUATIONS with A Brief Table of Laplace TransformsChapter 10 Linear Systems of DIFFERENTIAL Introduction to Systems of DIFFERENTIAL Linear Systems of DIFFERENTIAL Basic Theory of Homogeneous Linear Constant Coefficient Homogeneous Systems Constant Coefficient Homogeneous Systems Constant Coefficient Homogeneous Systems Variation of Parameters for Nonhomogeneous Linear Systems354 PrefaceElementary DIFFERENTIAL EQUATIONS with Boundary Value Problemsis written for students in science, en-gineering, and mathematics who have completed calculus through partial differentiation. If your syllabusincludes Chapter 10 (Linear Systems of DIFFERENTIAL EQUATIONS ), your students should have some prepa-ration in linear writing this book I have been guided by the these principles: An ELEMENTARY text should be written so the student can read it with comprehension without toomuch pain.

4 I have tried to put myself in the student s place, and have chosen to err on the side oftoo much detail rather than not enough. An ELEMENTARY text can t be better than its exercises. This text includes 1695 numbered exercises,many with several parts. They range in difficulty from routine to very challenging. An ELEMENTARY text should be written in an informal but mathematically accurate way, illustratedby appropriate graphics. I have tried to formulate mathematical concepts succinctly in languagethat students can understand. I have minimized the number of explicitly stated theorems and def-initions, preferring to deal with concepts in a more conversational way, copiously illustrated by250 completely worked out examples. Where appropriate, concepts and results are depicted in I believe that the computer is an immensely valuable tool for learning, doing, and writingmathematics, the selection and treatment of topics in this text reflects my pedagogical orientation alongtraditional lines.

5 However, I have incorporated what I believe to be the best use of modern technology,so you can select the level of technology that you want to include in your course. The text includes 336exercises identified by the symbolsCandC/G that call for graphics or computation and are also 73 laboratory exercises identified byL that require extensive use of technology. Inaddition, several sections include informal advice on the use of technology. If you prefer not to emphasizetechnology, simply ignore these exercises and the are two schools of thought on whether techniques and applications should be treated together orseparately. I have chosen to separate them; thus, Chapter 2 deals with techniques for solving first orderequations, and Chapter 4 deals with applications. Similarly, Chapter 5 deals with techniques for solvingsecond order EQUATIONS , and Chapter 6 deals with applications.

6 However, the exercise sets of the sectionsdealing with techniques include some applied oriented ELEMENTARY DIFFERENTIAL EQUATIONS texts are occasionally criticized as being col-lections of unrelated methods for solving miscellaneous problems. To some extent this is true; after all,no single method applies to all situations. Nevertheless, I believe that one idea can go a long way towardunifying some of the techniques for solving diverse problems: variation of parameters. I use variation ofparameters at the earliest opportunity in Section , to solve the nonhomogeneous linear equation, givena nontrivial solution of the complementary equation. You may find this annoying, since most of us learnedthat one should use integrating factors for this task, while perhaps mentioning the variation of parametersoption in an exercise. However, there s little difference between the two approaches, since an integratingfactor is nothing more than the reciprocal of a nontrivial solution of the complementary equation.

7 Theadvantage of using variation of parameters here is that it introduces the concept in its simplest form andfocuses the student s attention on the idea of seeking a solutionyof a DIFFERENTIAL equation by writing itasy=uy1,wherey1is a known solution of related equation anduis a function to be determined. I usethis idea in nonstandard ways, as follows: In Section to solve nonlinear first order EQUATIONS , such as Bernoulli EQUATIONS and nonlinearhomogeneous EQUATIONS . In Chapter 3 for numerical solution of semilinear first order In Section to avoid the necessity of introducing complex exponentials in solving a second or-der constant coefficient homogeneous equation with characteristic polynomials that have complexzeros. In Sections , , and for the method of undetermined coefficients. (If the method of an-nihilators is your preferred approach to this problem, compare the labor involved in solving, forexample,y +y +y=x4exby the method of annihilators and the method used in Section )Introducing variation of parameters as early as possible (Section ) prepares the student for the con-cept when it appears again in more complex forms in Section , where reduction of order is used notmerely to find a second solution of the complementary equation, but also to find the general solution of thenonhomogeneous equation, and in Sections , , and , that treat the usual variation of parametersproblem for second and higher order linear EQUATIONS and for linear may also find the following to be of interest: Section deals with integrating factors of the form =p(x)q(y), in addition to those of theform =p(x)and =q(y)discussed in most texts.

8 Section makes phase plane analysis of nonlinear second order autonomous EQUATIONS accessi-ble to students who have not taken linear algebra, since eigenvalues and eigenvectors do not enterinto the treatment. Phase plane analysis of constant coefficient linear systems is included in Sec-tions Section presents an extensive discussion of applications of DIFFERENTIAL EQUATIONS to curves. Section studies motion under a central force, which may be useful to students interested in themathematics of satellite orbits. Sections present the method of Frobenius in more detail than in most texts. The approachis to systematize the computations in a way that avoids the necessity of substituting the unknownFrobenius series into each equation. This leads to efficiency in the computation of the coefficientsof the Frobenius solution. It also clarifies the case where the roots of the indicial equation differ byan integer (Section ).

9 The free Student Solutions Manual contains solutions of most of the even-numbered exercises. The free Instructor s Solutions Manual is available by email subject toverification of the requestor s faculty following observations may be helpful as you choose your syllabus: Section is the only specific prerequisite for Chapter 3. To accomodate institutions that offer aseparate course in numerical analysis, Chapter 3 is not a prerequisite for any other section in thetext. The sections in Chapter 4 are independent of eachother, and are not prerequisites for any of thelater chapters. This is also true of the sections in Chapter 6, except that Section is a prerequisitefor Section Chapters 7, 8, and 9 can be covered in any order after the topics selected from Chapter 5. Forexample, you can proceed directly from Chapter 5 to Chapter 9. The second order Euler equation is discussed in Section , where it sets the stage for the methodof Frobenius.

10 As noted at the beginning of Section , if you want to include Euler EQUATIONS inyour syllabus while omitting the method of Frobenius, you can skip the introductory paragraphsin Section and begin with Definition You can then cover Section immediately afterSection F. TrenchCHAPTER 1 IntroductionIN THIS CHAPTER we begin our study of DIFFERENTIAL presents examples of applications that lead to DIFFERENTIAL introduces basic concepts and definitions concerning DIFFERENTIAL presents a geometric method for dealing with DIFFERENTIAL EQUATIONS that has been knownfor a very long time, but has become particularly useful and important with the proliferation of readilyavailable DIFFERENTIAL EQUATIONS Chapter LEADING TO DIFFERENTIAL EQUATIONSIn order to apply mathematical methods to a physical or real life problem, we must formulate the prob-lem in mathematical terms; that is, we must construct amathematical modelfor the problem.