## An Introduction to Ordinary Differential EquationsOrdinary 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

1 | |

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 |

314 | |

319 | |

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An Introduction to Ordinary Differential Equations Ravi P. Agarwal,Donal O'Regan No preview available - 2008 |

### Common terms and phrases

Algebra Answers or Hints asymptotically stable boundary conditions boundary value problem Business Media Consider continuous function converges Corollary cosx critical point 0,0 deﬁned Deﬁnition differential inequality diﬀerential system 17.3 eigenfunctions eigenvalues Example ﬁnd ﬁrst first-order following result Fourier series function f(x,y function y(x fundamental matrix Further given Green's function hence homogeneous implies inequality inﬁnite infinite number initial value problem integrating factor Introduction to Ordinary least one solution Lecture Let f(x,y linearly independent linearly independent solutions Lipschitz condition Lipschitz condition 7.3 nonhomogeneous nonlinear nonnegative nontrivial solution O’Regan obtain Ordinary Differential Equations orthogonal particular solution periodic of period periodic solution polynomial positive definite Proof R.P. Agarwal satisﬁed satisfies second-order Show the following sinx solutions of 17.3 solved Springer Science Sturm–Liouville problem suﬃcient conditions tend to zero trajectories trivial solution unstable value problem 7.1 Verify directly Wronskian xo,oo xo,xo yi(x