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**Differential equations: their solution using symmetries. Edited by Malcolm MacCallum.**
*(English)*
Zbl 0704.34001

Cambridge etc.: Cambridge University Press. xii, 260 p. £37.50/hbk; $ 44.50/hbk; £12.95/pbk; $ 19.95/pbk (1989).

This book is devoted to the study of symmetries of differential equations with emphasis on how to use symmetries to find solutions. The basic ideas in this book are due to Lie. The first part of this book is concerned with ordinary differential equations. Paragraphs 2 and 3 contain introductory notions about point transformation and Lie point symmetries of ordinary differential equations. Paragraphs 4 and 5 show us how to find the Lie point symmetries of an ordinary differential equation and how to use these symmetries. Paragraph 6 gives some basic properties and examples of Lie algebras. The next four paragraphs are devoted to the study of second order differential equations and systems using the Lie point symmetries. Paragraph 11 contains generalizations of Lie point symmetries and applications of these notions. The next two paragraphs give the basic definitions and properties concerning dynamical symmetries and how to find and use these symmetries for systems possessing a Lagrangean. The first part of the book ends with the study of systems of first order differential equations with a fundamental system of solutions - a subject not concerned with symmetries. The second part of this book is devoted to the applications of symmetries in the study of partial differential equations. Paragraphs 15, 16, 17, 18 and 19 give the basic ideas about Lie point transformations and symmetries and how to find and use these symmetries of partial differential equations. The connections between symmetries and separability of partial differential equations are developed in the 20th paragraph. Paragraph 21 contains basic ideas about contact transformations and contact symmetries. The last paragraphs are concerned with Lie-Bäcklund symmetries and their applications. This book also includes more than 100 exercises.

Reviewer: S.Anita

### MSC:

34-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations |

34A99 | General theory for ordinary differential equations |

35C99 | Representations of solutions to partial differential equations |

35R99 | Miscellaneous topics in partial differential equations |