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|a Preface, This second edition, like the first, is an introduction to the basic methods, theory, and applications of differential equations. It presupposes a knowledge of elementary calculus. The detailed style of presentation which characterized the first edition has been retained. Indeed, many sections have been taken verbatim from the previous edition, while others have been rewritten or rearranged somewhat with the sole intention ot making them clearer and smoother. As in the first edition, the text contains many thoroughly worked-out examples. Further, over one hundred new exercises have been added. The book is divided into two main parts. The first part (Chapters I through 9) deals with the material often found in a one-semester introductory course in ordinary differential equations. The second part (Chapters 8 through 14) introduces the reader to certain specialized and more advanced methods and provides a systematic introduction to the fundamental theory. An examination of the table of contents will reveal just what topics are presented. Part 1 is also available separately as Introduction to Ordinary Differential Equations (Xerox College Publishing, Lexington, Massachusetts, 1974). The strictly new text material deals with additional theory and methods of Systems of linear differential equations (in Chapters 7 and 9) and with nonlinear autonomous systems (Chapter 13). I believe that Chapter 7 now provides considerable flexibility for the study of linear systems. Several possible presentations of this chapter are listed here: Sections 7.1 and 7.2 (methods and applications) Sections 7.1A, 7.3, and 7.4 (theory and methods for the casen = 2, but no 2 proofs) 3. Sections 7.1A, 7.3, 7.4, 7.5, 7.6, and 7.7 (same as presentation 2, plus the following: theory and methods for the general case, with almost all proofs) 4. Sections 7.1A, 7.5, 7.6, and 7.7 judiciously combined with 7.4 (an alternative 4. approach to presentation 3) It should be pointed out that Section 7.5 is a very elementary introduction to the surprisingly few concepts of vectors and matrices which are needed in Sections 7.6 and 7.7 and that these concepts are not used again in the rest ol the book. Section 7 may therefore be omitted or very rapidly reviewed if the class has already studiod elementary matrix algebra. The book can be used as a text in several diflerent types of courses. The more-or less traditional one-semester introductory course could be based on Chapter 1 throuo Section 7.4 of Chapter 7 if elementary applications are to be included. An alternative one-semester version omitting applications but including numerical methods and Laplace transforms could be based on Chapters I, 2, 4, 6, 7 (through Section 7.4). 8 and 9. Also, an introductory course designed to lead to the methods of partial differential equations as rapidly as pOSSi ble could be based on Chapters 1, 2 (in part). 4, 6, 12, and 14 (in part). The book can also be used as a text in various intermediate courses for juniors and seniors who have already had a one-semester introduction to the subject. An intermediate course emphasizing further methods could be based on Chapters 8, 9, 12, 13 and 14. An intermediate course designed as an introduction to fundamental theory could be based on the last three sections of Chapter 7 and all or part of Chapters 10 through 14. Further filexibility is possible by taking up certain of these later chapters various orders. In particular, Chapters 13 and 14 could be interchanged and the last three sections of Chapter 7 could be judiciously combined with Chapter 11. I am very pleased to record my great appreciation and sincere thanks to Professor Elmer Haskins of the State University of New York, Potsdam, N.Y., and Major Francis W. Farrell of the United States Military Academy, West Point, N.Y. Both Professor Haskins and Major Farrell carefully read the whole of Part 1 in manuseript form and made many valuable comments and suggestions which resulted in a variety of improvements-major and minor. I also express my appreciation to the mathematics department staff of West Point for their careful work in detecting and correcting errors in the first edition. Special thanks are also given to Professor Stanley M. Lukawecki of Clemson University, Clemson, S.C., who carefully read Chapter 7 and made a number of constructive suggestions which helped me considerably in reaching a final decision on the material and arrangement of this chapter. I am also indebted to him for his worthwhile comments and suggestions concerning the chapters of Part 2. Further thanks are given to Professor F. A. Ficken, New York University, New York, N.Y., and Professor Arnold Seiken, Union College, Schenectady, N.Y., for their thoughtful advice, comments, and suggestions. I am very grateful to Solange Abbott for her excellent work in typing the revised portions of the manuscript. I am pleased to record my appreciation to Arthur Evans, Marret McCorkle, and others of the staff of XerOx College Publishing for their constant helpfulness and cooperation. Special thanks go to my wife for her great encouragement, understanding, and patience, as well as for her considerable assistance in the many different tasks required in the writing and revising of a text. ¡Thanks again, Gin!
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