sequence, this section can be covered quickly. However, Ive found that a review is al-

AModernIntroduction toDifferential Equations Third Edition Instructor’sResourceManual Henry J. Ricardo 1 / 4

1

CHAPTER

Introduction to differential equations 1.1Basic terminology If students have seen an introduction to differential equations as part of the calculus sequence, this section can be covered quickly. However, I’ve found that a review is al- ways helpful. In particular, students who have recently taken two or more semesters of calculus sometimes have trouble distinguishing dependent variables from inde- pendent variables in differentiation problems and have trouble with dummy variables in integration. I want students to focus on theformof a differential equation, not on the particular variables used or on the derivative notation employed. Throughout the book I have deliberately mixed the Leibniz(d/dx), Newton (dot), and Lagrange (prime) notations for derivatives, although the dot notation becomes dominant in later chapters as the dynamical systems interpretation becomes more pronounced.Parametersin equations often cause difficulty, but the student should become more comfortable with this concept as the course progresses. Students will have trou- ble with the general form(s) of annth-order differential equation, especially if they are not familiar with functions of several variables. Concrete examples are necessary.Many students need time to understand the idea of alineardifferential equation. Later discussions of linearity (in particular, the Superposition Principle) should help.The idea of asystemof differential equations is introduced early because of its importance in later chapters, but sometimes I postpone any classroom mention of systems until Chapter 5 or 6. If the students are reading the book (Ha!, you say), they’ll see this on their own.The text is dedicated to the proposition that technology is a valuable tool that can aid a student’s understanding and that may be essential in solving certain problems.The November 1994 issue (Vol. 25, No. 5) of theCollege Mathematics Journalis devoted to the teaching of differential equations. However, I don’t want to spend a great deal of valuable class time teaching the intricacies of the syntax of any CAS or other software I may be using. For example, in usingMapleI’ve found that a few basic commands should be mastered and used over and over again, making min- imal changes to accommodate different problems. I’ve handed out summaries of these commands and have encouraged students to use the “Help” system. Getting comfortable with the various options (numerical, graphical, and analytic) may take some time, and I have learned to avoid embarrassment in class by preparing ahead of time and saving examples on a USB drive. I hand out hard copies of certainMaple worksheets for the students to use as templates. Students sometimes make their own

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• CHAPTER 1 Introduction to differential equations

electronic copies of worksheets that may be on a departmental server. There are many books dealing with ODEs and various computer algebra systems. However, it’s im- portant for students to realize that computers don’t have all the answers. I’ve found that showing problems that the CAS can’t solve or can only solve incompletely is a sound pedagogical technique.There is a wealth of ODE information on the Internet. See, for example, my ar- ticle “Internet Resources for Differential Equations” in the MAA newsletterFocus (May/June, 2002; August/September, p. 24). (Past issues ofFocuscan be found at www.maa.org/press/periodicals/maa-focus#Past.) Some of the links may no longer be viable, but there are still some good search tips in the article.For the history of differential equations I recommend the classic bookOrdinary

Differential Equationsby E.L. Ince (New York: Dover Publications, 1956);Analy-

sis by Its Historyby E. Hairer and G. Wanner (New York: Springer-Verlag, 1996);

andMathematics of the 19th Centuryby A.N. Kolmogorov and A.P. Yushkevich (Boston: Birkhäuser, 1998). Two Internet sources arewww.eoht.info/page/History+ of+differential+equationsand a brief history (with references) by John E. Sasser:

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=C56A1A79F28D5CD1E8

73E9AEA1610BAB?doi=10.1.1.112.3646&rep=rep1&type=pdf.“Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equa- tions” by Gian-Carlo Rota (https://web.williams.edu/Mathematics/lg5/Rota.pdf )isan amusing and insightful article by a master.Finally, “The Dynamical Systems Approach to Differential Equations” by Morris

• Hirsch (Bulletinof the American Mathematical Society,11, Number 1, July 1984),

although somewhat dated, is an authoritative survey of the early history of the subject, whereas “Dynamical Systems Theory: What in the World Is It?” by Mike Hochman

(http://math.huji.il/~mhochman/research-expo.html) is a more modernapologiafor

this field of study.A 1.(a) The independent variable isxand the dependent variable isy; (b) first-order; (c) linear 2.(a) The independent variable isxand the dependent variable isy; (b) first-order; (c) linear 3.(a) The independent variable is not indicated, but the dependent variable isx; (b) second-order; (c) nonlinear because of the term exp(−x)—the equation can- not be written in the form (1.1.1), whereyis replaced byxandxis replaced by the independent variable.

4.(a) The independent variable isxand the dependent variable isy; (b) first-order; (c) nonlinear because of the term(y π ) 2 =y π (x)·y π (x)—the equation cannot be written in the form (1.1.1).

5.(a) The independent variable isxand the dependent variable isy; (b) first-order; (c) nonlinear because you get the termsx 2 (y π ) 2 andxy π ywhen you remove the parentheses. 3 / 4

1.1Basic terminology3 6.(a) The independent variable istand the dependent variable isr; (b) second- order; (c) linear 7.(a) The independent variable isxand the dependent variable isy; (b) fourth- order; (c) linear 8.(a) The independent variable istand the dependent variable isy; (b) second- order; (c) nonlinear because of the term−y π (y 2 −1) 9.(a) The independent variable istand the dependent variable isx; (b) third-order; (c) linear 10.(a) The independent variable istand the dependent variable isx; (b) seventh- order; (c) linear 11.(a) The independent variable isxand the dependent variable isy; (b) first-order; (c) nonlinear because of the terme y π 12.(a) The independent variable istand the dependent variable isR; (b) third-order; (c) linear

• a.Nonlinear; the first equation is nonlinear because of the term 4xy=

4x(t)y(t).b.Linear c.Nonlinear; the first and second equations are nonlinear because each con- tains a product of dependent variables.d.Linear 14.Using the “B” definition of the FTC in Section A.4, we havey(x)= π x 1 sintdt, y π (x)=sinx,y ππ (x)=cosx,y

πππ

(x)=−sinx. Therefore,y

πππ

(x)+y π (x)= −sinx+sinx=0.B 15.The terms(a 2 −a)x dx dt andte (a−1)x make the equation nonlinear. Ifa 2 −a= 0—that is, ifa=0ora=1—then the first troublesome term disappears. How- ever, only the valuea=1 makes the second nonlinear term vanish as well. Thus, a=1 is the answer.

• a.

dx dt =ln(2 x )=xln 2=(ln 2)x, a linear equation b.x π = δ x 2 −1x−1 forxδ=1 2forx=1 = δ x+1forx=1 2forx=1 =x+1 for allx, which is linear c.x π = δ x 4 −1 x 2 −1 forxδ=1 2forx=1 = δ x 2 +1forxδ=1 2forx=1 =x 2 +1 for allx, which is nonlinear.C 17.Using the formula for arc length and the formula for area under a curve, our given conditions translate to π x a α 1+{f π (t)} 2 dt= π x a f(t)dt. Differentiating

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