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All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections or of the United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Rosewood Drive, Danvers, MA , , fax The cover was printed by Hamilton Printing.

This book is printed on acid free paper. II General comments on the purpose of each section and its classroom use, with mathematical and didactic information on teaching practice and pedagogical aspects. Some of the comments refer to whole chapters and are indicated accordingly. Changes in Problem Sets The major changes in this edition of the text are listed and explained in the Preface of the book.

They include global improvements produced by updating and streamlining chapters as well as many local improvements aimed at simplification of the whole text. Speedy orientation 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.

Resulting effects of these changes on the problem sets are as follows. The problems have been changed. Thus, the danger of overemphasizing minor techniques and ideas is avoided as much as possible. These changes in the problem sets will help students in solving problems as well as in gaining a better understanding of practical aspects in the text. It will also enable instructors to explain ideas and methods in terms of examples supplementing and illustrating theoretical discussions—or even replacing some of them if so desired.

Of course, it is intended to prevent the student from simply producing answers by a CAS instead of trying to understand the underlying mathematics. Orientation on Computers Comments on computer use are included in the Preface of the book.

Software systems are listed in the book at the beginning of Chap. Part A. Some additions on population dynamics appear in Sec.

Electric circuits are shifted to Chap. This avoids repetitions that are unnecessary and practically irrelevant. Basic Concepts. Modeling, page 2 Purpose. To give the students a first impression what an ODE is and what we mean by solving it. Background Material. For the whole chapter we need integration formulas and techniques, which the student should review.

General Comments This section should be covered relatively rapidly to get quickly to the actual solution methods in the next sections. Problem Set 1.

Some books use the term to mean a solution that includes all solutions, that is, both the particular and the singular ones. We do not adopt this definition for two reasons. First, it is frequently quite difficult to prove that a formula includes all solutions; hence, this definition of a general solution is rather useless in practice.

Second, linear differential equations satisfying rather general conditions on the coefficients have no singular solutions as mentioned in the text , so that for these equations a general solution as defined does include all solutions.

Second order. First order. In Prob. Times of fall are 4. We see that the difference between daily compounding and continuous compounding is very small. Direction Fields, page 9 Purpose.

To give the student a feel for the nature of ODEs and the general behavior of fields of solutions. This amounts to a conceptual clarification before entering into formal manipulations of solution methods, the latter being restricted to relatively small—albeit important—classes of ODEs.

This approach is becoming increasingly important, especially because of the graphical power of computer software. It is the analog of conceptual studies of the derivative and integral in calculus as opposed to formal techniques of differentiation and integration.

Comment on Isoclines These could be omitted because students sometimes confuse them with solutions. In the computer approach to direction fields they no longer play a role. Comment on Order of Sections This section could equally well be presented later in Chap. Logistic equation Verhulst equation; Sec. Linear ODE.

The solution involves the error function. CAS Project. The direction fields should turn out in perfect shape. Much will depend on the choice of x- and y-intervals, a method of trial and error. Isoclines may be preferable if the explicit form of the ODE contains roots on the right. Separable ODEs. Modeling, page 12 Purpose. The section includes standard applications that lead to separable ODEs, namely, 1. In reducing to separability we consider 6. This also illustrates that it is natural to consider solutions of ODEs on open rather than on closed intervals.

Scott, American Math. Monthly 92 , —]. Simple cases are easy to decide, but this may save time in cases of more complicated ODEs, some of which may perhaps be of practical interest. You may perhaps ask your students to derive such a criterion.

Comments on Application Each of those examples can be modified in various ways, for example, by changing the application or by taking another form of the tank, so that each example characterizes a whole class of applications. Comment on Footnote 3 Newton conceived his method of fluxions calculus in —, at the age of Philosophiae Naturalis Principia Mathematica was his most influential work.

Leibniz invented calculus independently in and introduced notations that were essential to the rapid development in this field. His first publication on differential calculus appeared in The variables are now separated. Subtract 2u2 on both sides and separate the variables. Problem

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II General comments on the purpose of each section and its classroom use, with mathematical and didactic information on teaching practice and pedagogical aspects. Some of the comments refer to whole chapters and are indicated accordingly. Changes in Problem Sets The major changes in this edition of the text are listed and explained in the Preface of the book. They include global improvements produced by updating and streamlining chapters as well as many local improvements aimed at simplification of the whole text. Speedy orientation 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. Resulting effects of these changes on the problem sets are as follows. The problems have been changed.

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