Introduction to Ordinary Differential Equations (eBook)
XII, 326 Seiten
Springer New York (Verlag)
978-0-387-71276-5 (ISBN)
Ordinary di?erential equations serve as mathematical models for many exciting "e;real-world"e; 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.
Preface 6
Contents 9
Introduction 11
Historical Notes 17
Exact Equations 23
Elementary First-Order Equations 31
First-Order Linear Equations 38
Second-Order Linear Equations 45
Preliminaries to Existence and Uniqueness of Solutions 55
Picard’s Method of Successive Approximations 63
Existence Theorems 71
Uniqueness Theorems 78
Differential Inequalities 87
Continuous Dependence on Initial Conditions 94
Preliminary Results from Algebra and Analysis 101
Preliminary Results from Algebra and Analysis ( Contd.) 107
Existence and Uniqueness of Solutions of Systems 113
Existence and Uniqueness of Solutions of Systems ( Contd.) 119
General Properties of Linear Systems 126
Fundamental Matrix Solution 134
Systems with Constant Coefficients 143
Periodic Linear Systems 154
Asymptotic Behavior of Solutions of Linear Systems 162
Asymptotic Behavior of Solutions of Linear Systems ( Contd.) 169
Preliminaries to Stability of Solutions 178
Stability of Quasi- Linear Systems 185
Two-Dimensional Autonomous Systems 191
Two-Dimensional Autonomous Systems ( Contd.) 197
Limit Cycles and Periodic Solutions 206
Lyapunov’s Direct Method for Autonomous Systems 214
Lyapunov’s Direct Method for Nonautonomous Systems 221
Higher-Order Exact and Adjoint Equations 227
Oscillatory Equations 235
Linear Boundary Value Problems 243
Green’s Functions 250
Degenerate Linear Boundary Value Problems 260
Maximum Principles 268
Sturm–Liouville Problems 275
Sturm–Liouville Problems ( Contd.) 281
Eigenfunction Expansions 289
Eigenfunction Expansions ( Contd.) 296
Nonlinear Boundary Value Problems 305
Nonlinear Boundary Value Problems ( Contd.) 310
Topics for Further Studies 318
References 324
Index 327
Erscheint lt. Verlag | 10.12.2008 |
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Reihe/Serie | Universitext |
Sprache | englisch |
Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
Technik | |
Schlagworte | Algebra • Boundary value problem • Calculus • differential equation • ordinary differential equation • stability |
ISBN-10 | 0-387-71276-3 / 0387712763 |
ISBN-13 | 978-0-387-71276-5 / 9780387712765 |
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