## Elementary Lie group analysis and ordinary differential equationsLie group analysis, based on symmetry and invariance principles, is the only systematic method for solving nonlinear differential equations analytically. One of Lie's striking achievements was the discovery that the majority of classical devices for integration of special types of ordinary differential equations could be explained and deduced by his theory. Moreover, this theory provides a universal tool for tackling considerable numbers of differential equations when other means of integration fail. * This is the first modern text on ordinary differential equations where the basic integration methods are derived from Lie group theory * Includes a concise and self contained introduction to differential equations * Easy to follow and comprehensive introduction to Lie group analysis * The methods described in this book have many applications The author provides students and their teachers with a flexible text for undergraduate and postgraduate courses, spanning a variety of topics from the basic theory through to its many applications. The philosophy of Lie groups has become an essential part of the mathematical culture for anyone investigating mathematical models of physical, engineering and natural problems. |

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### Contents

Introduction to differential equations | 1 |

Elementary methods of integration | 25 |

General properties of solutions | 45 |

Copyright | |

15 other sections not shown

### Common terms and phrases

algebra L3 analytic arbitrary constants arbitrary function calculations canonical form canonical variables change of variables Chap characteristic classical coefficients commutator complete integral Consider const contact transformations continuous group coordinates defined Definition denote derivatives determining equation differential functions differential invariants dx dy equations admitting equivalence transformations Example exponential map first-order equation formula geometry given group analysis group G Hence independent variables infinitesimal symmetries infinitesimal transformation integral curves invariant solutions inverse invoking Lemma Lie algebra Lie equations Lie group Lie-Backlund Lie's linear equation linearizable mathematical method non-homogeneous notation obtained one-parameter group operators ordinary differential equations parameter partial differential equations particular solution plane point transformations problem proof provides reduces respect Riccati equation rotation second-order equations Section singular solving Sophus Lie spanned subalgebra substitution symmetry group Theorem theory tions transformation groups vector velocity whence written yields