An Introduction to Ordinary Differential Equations

1. Front Matter

   Pages 1-12

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2. Introduction

   Pages 1-6

3. Historical Notes

   Pages 7-12

4. Exact Equations

   Pages 13-20

5. Elementary First-Order Equations

   Pages 21-27

6. First-Order Linear Equations

   Pages 28-34

7. Second-Order Linera Equations

   Pages 35-44

8. Preliminaries to Existence and Uniqueness of Solutions

   Pages 45-52

9. Picard's Method of Successive Approximations

   Pages 53-60

10. Existence Theorems

    Pages 61-67

11. Uniqueness Theorems

    Pages 68-76

12. Differential Inequalities

    Pages 77-83

13. Continuous Dependence on Initial Conditions

    Pages 84-90

14. Preliminary Results from Algebra and Analysis

    Pages 91-96

15. Preliminary Results from Algebra and Analysis (Contd.)

    Pages 97-102

16. Existence and Uniqueness of Solutions of Systems

    Pages 103-108

17. Existence and Uniqueness of Solutions of Systems (Contd.)

    Pages 109-115

18. General Properties of Linear Systems

    Pages 116-123

19. Fundamental Matrix Solution

    Pages 124-132

20. Systems with Constant Coefficients

    Pages 133-143

Ordinary di?erential equations serve as mathematical models for many exciting “real-world” problems, not only in science and technology, but also in such diverse ?elds as economics, psychology, defense, and demography. Rapid growth in the theory of di?erential equations and in its applications to almost every branch of knowledge has resulted in a continued interest in its study by students in many disciplines. This has given ordinary di?er- tial equations a distinct place in mathematics curricula all over the world and it is now being taught at various levels in almost every institution of higher learning. Hundredsofbooksonordinarydi?erentialequationsareavailable. H- ever, the majority of these are elementary texts which provide a battery of techniquesfor?ndingexplicitsolutions. Thesizeofsomeofthesebookshas grown dramatically—to the extent that students are often lost in deciding wheretostart. Thisisallduetotheadditionofrepetitiveexamplesand- ercises, and colorful pictures. The advanced books are either on specialized topics or are encyclopedic in character. In fact, there are hardly any rig- ousandperspicuousintroductorytextsavailablewhichcanbeuseddirectly in class for students of applied sciences. Thus, in an e?ort to bring the s- ject to a wide audience we provide a compact, but thorough, introduction to the subject in An Introduction to Ordinary Di?erential Equations. This book is intended for readers who have had a course in calculus, and hence it canbeusedforaseniorundergraduatecourse. Itshouldalsobesuitablefor a beginning graduate course, because in undergraduate courses, students do not have any exposure to various intricate concepts, perhaps due to an inadequate level of mathematical sophistication.

• Algebra
• Boundary value problem
• calculus
• differential equation
• ordinary differential equation
• stability
• ordinary differential equations
• partial differential equations

From the reviews:

"Presents a thorough treatment of the classical material traditionally covered in an advanced book on ordinary differential equations, including a number of interesting historical notes. … The authors also discuss Lyapunov functions, Green’s functions comparison and separation theorems, maximum principle, Sturm-Liouville problems, Fredholm alternative, and Floquet theory. In addition, the book addresses results by Perron, Kamke, Osgood, Nagumo, Krasnoselski-Krein, and Van Kampen which are not found in some similar works. … Summing Up: Recommended. Upper-division undergraduates, graduate students, researchers, and faculty." (J. D. Fehribach, Choice, Vol. 46 (8), April, 2009)

"The textbook is devoted to a systematic and rigorous introduction to the theory of ordinary differential equations. … the practical part include numerous exercises with answers or hints. Written by two prolific leaders in the field of ordinary differential equations and nonlinear analysis, the textbook provides a very clear, well-organized and lucid introduction to ordinary differential equations, with an implicit orientation towards the most recent research topics and methods in the field and related areas." (Radu Precup, Zentralblatt MATH, Vol. 1158, 2009)

“This text book provides an excellent introduction to the subject accessible to second-year undergraduate or graduate-level students. Its structure in the form of a succession of 42 class-tested lectures makes it not only an inspiring source for self-study but gives also a good framework for the organization of course material. … a highly recommendable book for students in mathematics, sciences, or engineering as well as for teachers on college and university level.” (G. Hörmann, Monatshefte für Mathematik, Vol. 159 (4), March, 2010)

• Dept. Mathematical Sciences, Florida Institute of Technology, Melbourne, U.S.A.

  Ravi P. Agarwal

• University College Galway, National University of Ireland, Galway, Ireland

  Donal O'Regan

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