## Differential Equations: Theory and ApplicationsThis book was written as a comprehensive introduction to the theory of ordinary di?erential equations with a focus on mechanics and dynamical systems as time-honored and important applications of this theory. H- torically, these were the applications that spurred the development of the mathematical theory and in hindsight they are still the best applications for illustrating the concepts, ideas, and impact of the theory. While the book is intended for traditional graduate students in mat- matics, thematerial is organized sothatthebookcan alsobeusedin awider setting within today’s modern university and society (see “Ways to Use the Book” below). In particular, it is hoped that interdisciplinary programs with courses that combine students in mathematics, physics, engineering, and other sciences can bene?t from using this text. Working professionals in any of these ?elds should be able to pro?t too by study of this text. An important, but optional component of the book (based on the - structor’s or reader’s preferences) is its computer material. The book is one of the few graduate di?erential equations texts that use the computer to enhance the concepts and theory normally taught to ?rst- and second-year graduate students in mathematics. I have made every attempt to blend - gether the traditional theoretical material on di?erential equations and the new, exciting techniques a?orded by computer algebra systems (CAS), like Maple, Mathematica, or Matlab. The electronic material for mastering and enjoying the computer component of this book is on Springer’s website. |

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### Contents

1 | |

Chapter 2 | 36 |

Chapter 3 | 79 |

Chapter 4 | 119 |

Chapter 5 | 220 |

Chapter 6 | 267 |

Chapter 7 | 333 |

Chapter 8 | 371 |

Chapter 9 | 475 |

Appendix A | 535 |

Appendix B | 553 |

Appendix C | 579 |

Appendix D | 613 |

615 | |

621 | |

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### Common terms and phrases

algebra angular asymptotically body canonical center of mass Chapter complete conservation laws constant converges corresponding defined definition denote derivative determine diffeomorphism differential equations dynamical systems easy eigenspace eigenvalues eigenvectors equivalent Euler’s example exercise Existence and Uniqueness fixed points flow formula function F fundamental matrix geometric given gives graph Hamiltonian system inequality inertia initial conditions integrable systems integral curve intersection interval inverse Jacobian matrix Jordan form level curves Liapunov function linear system Linearization Theorem linearly independent matrix exponential neighborhood Newton’s norm notation Note open set orbit origin orthogonal particle phase portrait Picard iterates plane plot Poincaré map polar coordinates positive proof Proposition rigid-body satisfies sequence shown in Figure sketch solution stability subspace Suppose symplectic system x transformation vector field velocity worksheet zero