An Introduction to Ordinary Differential Equations

This textbook provides a rigorous and lucid introduction to the theory of ordinary differential equations (ODEs), which serve as mathematical models for many exciting real-world problems in science, engineering, and other disciplines.

Key Features of this textbook:

Effectively organizes the subject into easily manageable sections in the form of 42 class-tested lectures

Provides a theoretical treatment by organizing the material around theorems and proofs

Uses detailed examples to drive the presentation

Includes numerous exercise sets that encourage pursuing extensions of the material, each with an "answers or hints" section

Covers an array of advanced topics which allow for flexibility in developing the subject beyond the basics

Provides excellent grounding and inspiration for future research contributions to the field of ODEs and related areas

This book is ideal for a senior undergraduate or a graduate-level course on ordinary differential equations. Prerequisites include a course in calculus.

Series: Universitext

Ravi P. Agarwal received his Ph.D. in mathematics from the Indian Institute of Technology, Madras, India. He is a professor of mathematics at the Florida Institute of Technology. His research interests include numerical analysis, inequalities, fixed point theorems, and differential and difference equations. He is the author/co-author of over 800 journal articles and more than 20 books, and actively contributes to over 40 journals and book series in various capacities.

Donal O'Regan received his Ph.D. in mathematics from Oregon State University, Oregon, U.S.A. He is a professor of mathematics at the National University of Ireland, Galway. He is the author/co-author of 14 books and has published over 650 papers on fixed point theory, operator, integral, differential and difference equations. He serves on the editorialboard of many mathematical journals.

Previously, the authors have co-authored/co-edited the following books with Springer: Infinite Interval Problems for Differential, Difference and Integral Equations; Singular Differential and Integral Equations with Applications; Nonlinear Analysis and Applications: To V. Lakshmikanthan on his 80th Birthday. In addition, they have collaborated with others on the following titles: Positive Solutions of Differential, Difference and Integral Equations; Oscillation Theory for Difference and Functional Differential Equations; Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations.

Résumé
Ordinary di?erential equations serve as mathematical models for many exciting &quote;real-world&quote; 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.

Contenu

Historical Notes.- Exact Equations.- Elementary First-Order Equations.- First-Order Linear Equations.- Second-Order Linera Equations.- Preliminaries to Existence and Uniqueness of Solutions.- Picard#x0027;s Method of Successive Approximations.- Existence Theorems.- Uniqueness Theorems.- Differential Inequalities.- Continuous Dependence on Initial Conditions.- Preliminary Results from Algebra and Analysis.- Preliminary Results from Algebra and Analysis (Contd.).- Existence and Uniqueness of Solutions of Systems.- Existence and Uniqueness of Solutions of Systems (Contd.).- General Properties of Linear Systems.- Fundamental Matrix Solution.- Systems with Constant Coefficients.- Periodic Linear Systems.- Asymptotic Behavior of Solutions of Linear Systems.- Asymptotic Behavior of Solutions of Linear Systems (Contd.).- Preliminaries to Stability of Solutions.- Stability of Quasi-Linear Systems.- Two-Dimensional Autonomous Systems.- Two-Dimensional Autonomous Systems (Contd.).- Limit Cycles and Periodic Solutions.- Lyapunov#x0027;s Direct Method for Autonomous Systems.- Lyapunov#x0027;s Direct Method for Nonautonomous Systems.- Higher-Order Exact and Adjoint Equations.- Oscillatory Equations.- Linear Boundary Value Problems.- Green#x0027;s Functions.- Degenerate Linear Boundary Value Problems.- Maximum Principles.- Sturm#x2014;Liouville Problems.- Sturm#x2013;Liouville Problems (Contd.).- Eigenfunction Expansions.- Eigenfunction Expansions (Contd.).- Nonlinear Boundary Value Problems.- Nonlinear Boundary Value Problems (Contd.).- Topics for Further Studies.