An Introduction to Ordinary Differential Equations

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Springer Science & Business Media, Dec 10, 2008 - Mathematics - 322 pages
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.
 

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Contents

Introduction
1
Historical Notes
7
Exact Equations
13
Elementary FirstOrder Equations
21
FirstOrder Linear Equations
28
SecondOrder Linear Equations
35
Preliminaries to Existence and Uniqueness of Solutions
45
Picards Method of Successive Approximations
53
Stability of QuasiLinear Systems
175
TwoDimensional Autonomous Systems
181
TwoDimensional Autonomous Systems Contd
187
Limit Cycles and Periodic Solutions
196
Lyapunovs Direct Method for Autonomous Systems
204
Lyapunovs Direct Method for Nonautonomous Systems
211
HigherOrder Exact and Adjoint Equations
217
Oscillatory Equations
225

Existence Theorems
61
Uniqueness Theorems
68
Differential Inequalities
77
Continuous Dependence on Initial Conditions
84
Preliminary Results from Algebra and Analysis
91
Preliminary Results from Algebra and Analysis Contd
97
Existence and Uniqueness of Solutions of Systems
103
Existence and Uniqueness of Solutions of Systems Contd
109
General Properties of Linear Systems
116
Fundamental Matrix Solution
124
Systems with Constant Coefficients
133
Periodic Linear Systems
144
Asymptotic Behavior of Solutions of Linear Systems
152
Asymptotic Behavior of Solutions of Linear Systems Contd
159
Preliminaries to Stability of Solutions
168
Linear Boundary Value Problems
233
Greens Functions
240
Degenerate Linear Boundary Value Problems
250
Maximum Principles
258
SturmLiouville Problems
265
SturmLiouville Problems Contd
271
Eigenfunction Expansions
279
Eigenfunction Expansions Contd
286
Nonlinear Boundary Value Problems
295
Nonlinear Boundary Value Problems Contd
300
Topics for Further Studies
308
References
314
Index
319
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Agarwal is an environmentalist and photographer.

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