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IMA Journal of Applied Mathematics Advance Access published online on February 14, 2007
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxm001
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© The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

C{infty}-symmetries and nonlocal symmetries of exponential type

C Muriel{dagger} and JL Romero

Department of Mathematics, University of Cadiz, 11510 Puerto Real, Spain

{dagger} Email: concepcion.muriel{at}uca.es

Received on 6 July 2006; Accepted on 27 November 2006


  Abstract

Nonlocal symmetries generated by type I hidden symmetries are identified as specific C{infty}-symmetries of an nth-order ordinary differential equation. The general method of reduction associated to these C{infty}-symmetries allows us to give explicit transformations to reduce the order if n > 1. As a consequence, we give a complete classification of the equations of arbitrary order that admit this kind of nonlocal symmetries. We illustrate these results with several equations that have no Lie point symmetries. For n = 1, the method provides the linearization of first-order equations. This is applied to some examples of Riccati equations and Abel equations of the second kind.

Keywords: ordinary differential equations; hidden symmetries; C{infty}-symmetries.


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