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Handbook of exact solutions for ordinary differential equations. Transl. from the Russian. (English) Zbl 0855.34001
Boca Raton, FL: CRC Press. 720 p. (1995).
This is an extremely useful book for everybody interested in exact solutions of ordinary differential equations. It presents the actual state of the art in this field, and it shows that since the famous handbook of E. Kamke [Differentialgleichungen. Lösungsmethoden und Lösungen. 1. Gewöhnliche Differentialgleichungen. Leipzig (1942; Zbl 0026.31801)] a lot of things have happened. As the authors write in the foreword, this book contains nearly 5000 ordinary differential equations and their solutions, among them the following numbers of specific second- and higher-order differential equations: 1227 second order, 587 third order, 75 fourth order, 160 higher order (the corresponding numbers from the 1976 Russian edition of the Kamke handbook to which the authors refer are 249, 13, 3, 3). When selecting the material, the authors gave preference to (i) equations that traditionally attracted the attention of many researchers (Abel equations, Emden-Fowler equations, Painlevé equations, etc.), and (ii) equations that encountered in various applications (heat and mass transfer, nonlinear mechanics, hydrodynamics, nonlinear oscillations, combustion, chemical engineering, etc.). Special attention is paid to equations containing arbitrary functions or at least one or more arbitrary parameters. So, actually, this book deals with whole families of ordinary differential equations. In the first moment, the reader may be puzzled by the notation used by the authors. In spite of the fact that the book exclusively deals with ordinary differential equations for a scalar function \(y(x)\), the fourth derivative \(d^{(4)} y/dx^4\) is denoted by \(y''''_{xxxx}\), etc. The formulation “the handbooks by G. Beitmen and A. Erdelyi (1953-1955)” in the foreword (which, of course, refers to [A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions. (H. Batemen Manuscript Project.) Vol. I (1953; Zbl 0051.30303); Vol. II (1953; Zbl 0052.29502); Vol. III (1955; Zbl 0064.06302)]) indicates that this is a translation from some Russian source.
Reviewer: W.Müller (Berlin)

34-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to ordinary differential equations
34A05 Explicit solutions, first integrals of ordinary differential equations