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    ERWIN KREYSZIGProfessor of Mathematics Ohio State University Columbus, Ohio


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  • Copyright 2006 by John Wiley & Sons, Inc. All rights reserved.

    No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by anymeans, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted underSections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of thePublisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center,222 Rosewood Drive, Danvers, MA 01923, (508) 750-8400, fax (508) 750-4470. Requests to the Publisher forpermission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street,Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, E-Mail: PERMREQ@WILEY.COM.

    ISBN-13: 978-0-471-72647-0ISBN-10: 0471-72647-8

    Printed in the United States of America

    10 9 8 7 6 5 4 3 2 1

    Vice President and Publisher: Laurie RosatoneEditorial Assistant: Daniel GraceAssociate Production Director: Lucille BuonocoreSenior Production Editor: Ken SantorMedia Editor: Stefanie LiebmanCover Designer: Madelyn LesureCover Photo: John Sohm/Chromosohm/Photo Researchers

    This book was set in Times Roman by GGS Information Services and printed and bound by Hamilton Printing. The cover was printed by Hamilton Printing.

    This book is printed on acid free paper.

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  • PREFACEGeneral Character and Purpose of the Instructors Manual

    This Manual contains:(I) Detailed solutions of the even-numbered problems.(II) General comments on the purpose of each section and its classroom use, with

    mathematical and didactic information on teaching practice and pedagogical aspects. Someof the comments refer to whole chapters (and are indicated accordingly).

    Changes in Problem SetsThe major changes in this edition of the text are listed and explained in the Preface of thebook. They include global improvements produced by updating and streamlining chaptersas well as many local improvements aimed at simplification of the whole text. Speedyorientation is helped by chapter summaries at the end of each chapter, as in the last edition,and by the subdivision of sections into subsections with unnumbered headings. Resultingeffects of these changes on the problem sets are as follows.

    The problems have been changed. The large total number of more than 4000 problemshas been retained, increasing their overall usefulness by the following:

    Placing more emphasis on modeling and conceptual thinking and less emphasis ontechnicalities, to parallel recent and ongoing developments in calculus.

    Balancing by extending problem sets that seemed too short and contracting othersthat were too long, adjusting the length to the relative importance of the materialin a section, so that important issues are reflected sufficiently well not only in thetext but also in the problems. Thus, the danger of overemphasizing minor techniquesand ideas is avoided as much as possible.

    Simplification by omitting a small number of very difficult problems that appearedin the previous edition, retaining the wide spectrum ranging from simple routineproblems to more sophisticated engineering applications, and taking into account thealgorithmic thinking that is developing along with computers.

    Amalgamation of text, examples, and problems by including the large number ofmore than 600 worked-out examples in the text and by providing problems closelyrelated to those examples.

    Addition of TEAM PROJECTS, CAS PROJECTS, and WRITING PROJECTS,whose role is explained in the Preface of the book.

    Addition of CAS EXPERIMENTS, that is, the use of the computer in experimentalmathematics for experimentation, discovery, and research, which often producesunexpected results for open-ended problems, deeper insights, and relations amongpractical problems.

    These changes in the problem sets will help students in solving problems as well as ingaining a better understanding of practical aspects in the text. It will also enable instructorsto explain ideas and methods in terms of examples supplementing and illustratingtheoretical discussionsor even replacing some of them if so desired.

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  • Show the details of your work.This request repeatedly stated in the book applies to all the problem sets. Of course, it isintended to prevent the student from simply producing answers by a CAS instead of tryingto understand the underlying mathematics.

    Orientation on ComputersComments on computer use are included in the Preface of the book. Software systems arelisted in the book at the beginning of Chap. 19 on numeric analysis and at the beginningof Chap. 24 on probability theory.


    vi Instructors Manual

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    CHAPTER 1 First-Order ODEs

    Major Changes

    There is more material on modeling in the text as well as in the problem set.Some additions on population dynamics appear in Sec. 1.5.Electric circuits are shifted to Chap. 2, where second-order ODEs will be available.

    This avoids repetitions that are unnecessary and practically irrelevant.Team Projects, CAS Projects, and CAS Experiments are included in most problem sets.

    SECTION 1.1. Basic Concepts. Modeling, page 2

    Purpose. To give the students a first impression what an ODE is and what we mean bysolving it.Background Material. For the whole chapter we need integration formulas andtechniques, which the student should review.

    General CommentsThis section should be covered relatively rapidly to get quickly to the actual solutionmethods in the next sections.

    Equations (1)(3) are just examples, not for solution, but the student will see thatsolutions of (1) and (2) can be found by calculus, and a solution y ex of (3) by inspection.

    Problem Set 1.1 will help the student with the tasks of

    Solving y (x) by calculus

    Finding particular solutions from given general solutions

    Setting up an ODE for a given function as solution

    Gaining a first experience in modeling, by doing one or two problems

    Gaining a first impression of the importance of ODEs

    without wasting time on matters that can be done much faster, once systematic methodsare available.

    Comment on General Solution and Singular SolutionUsage of the term general solution is not uniform in the literature. Some books use theterm to mean a solution that includes all solutions, that is, both the particular and thesingular ones. We do not adopt this definition for two reasons. First, it is frequently quitedifficult to prove that a formula includes all solutions; hence, this definition of a generalsolution is rather useless in practice. Second, linear differential equations (satisfying rathergeneral conditions on the coefficients) have no singular solutions (as mentioned in thetext), so that for these equations a general solution as defined does include all solutions.For the latter reason, some books use the term general solution for linear equations only;but this seems very unfortunate.


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    2. y e3x/3 c 4. y (sinh 4x) /4 c6. Second order. 8. First order.

    10. y ce0.5x, y(2) ce 2, c 2/e, y (2/e)e0.5x 0.736e0.5x

    12. y cex x 1, y(0) c 1 3, c 2, y 2ex x 114. y c sec x, y(0) c/cos 0 c 1_2, y

    1_2 sec x

    16. Substitution of y cx c2 into the ODE gives

    y2 xy y c2 xc (cx c2) 0.Similarly,

    y 1_4x2, y 1_2x, thus


    2 x(1_2x) 1_4x

    2 0.

    18. In Prob. 17 the constants of integration were set to zero. Here, by two integrations,

    y g, v y gt c1, y 1_2gt2 c1t c2, y(0) c2 y0,

    and, furthermore,

    v(0) c1 v0, hence y 1_2gt2 v0 t y0,

    as claimed. Times of fall are 4.5 and 6.4 sec, from t 100/4.9 and 200/4.9.20. y ky. Solution y y0e

    kx, where y0 is the pressure at sea level x 0. Nowy(18000) y0e

    k18000 1_2y0 (given). From this,

    ek18000 1_2, y(36000) y0ek218000 y0(e

    k18000)2 y0(1_2)

    2 1_4y0.

    22. For 1 year and annual, daily, and continuous compounding we obtain the values

    ya(1) 1060.00, yd(1) 1000(1 0.06/365)365 1061.83,

    yc(1) 1000e0.06 1061.84,

    respectively. Similarly for 5 years,

    ya(5) 1000 1.065 1338.23, yd(5) 1000(1 0.06/365)

    3655 1349.83,

    yc(5) 1000e0.065 1349.86.

    We see that the difference between daily compounding and continuous compoundingis very small.

    The ODE for continuous compounding is yc ryc.

    SECTION 1.2. Geometric Meaning of y (x, y). Direction Fields, page 9

    Purpose. To give the student a feel for the nature of ODEs and the general behavior offields of solutions. This amounts to a conceptual clarification before entering into formalmanipulations of solution methods, the latter being restricted to relatively smallalbeitimportantclasses of ODEs. This approach is becoming increasingly important, especiallybecause of the graphical power of computer software. It is the analog of conceptualstudies of the derivative and integral in calculus as opposed to formal techniques ofdifferentiation and integration.

    Comment on IsoclinesThese could be


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