# American Institute of Mathematical Sciences

March  2011, 3(1): 1-22. doi: 10.3934/jgm.2011.3.1

## Superposition rules and second-order Riccati equations

 1 Departamento de Física Teórica and IUMA, Facultad de Ciencias, Universidad de Zaragoza, Pedro Cerbuna 12, 50.009, Zaragoza, Spain 2 Institute of Mathematics, Polish Academy of Sciences, Śniadeckish 8, P.O. Box 21, 00-956, Warszawa, Poland

Received  June 2010 Revised  April 2011 Published  April 2011

A superposition rule is a particular type of map that enables one to express the general solution of certain systems of first-order ordinary differential equations, the so-called Lie systems, out of generic families of particular solutions and a set of constants. The first aim of this work is to propose various generalisations of this notion to second-order differential equations. Next, several results on the existence of such generalisations are given and relations with the theories of Lie systems and quasi-Lie schemes are found. Finally, our methods are used to study second-order Riccati equations and other second-order differential equations of mathematical and physical interest.
Citation: José F. Cariñena, Javier de Lucas Araujo. Superposition rules and second-order Riccati equations. Journal of Geometric Mechanics, 2011, 3 (1) : 1-22. doi: 10.3934/jgm.2011.3.1
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