Abstract
The construction of first integrals for SL(2,R)-invariant nth-order ordinary differential equations is a non-trivial problem due to the nonsolvability of the underlying symmetry algebra sl(2,R). Firstly, we provide for n=2 an explicit expression for two non-constant first integrals through algebraic operations involving the symmetry generators of sl(2,R), and without any kind of integration. Moreover, although there are cases when the two first integrals are functionally independent, it is proved that a second functionally independent first integral arises by a single quadrature. This result is extended for n>2, provided that a solvable structure for an integrable distribution generated by the differential operator associated to the equation and one of the prolonged symmetry generators of sl(2,R) is known. Several examples illustrate the procedures.
Highlights
The study of nth-order ordinary differential equations admitting the unimodular Lie groupSL(2, R) is a non-trivial problem due to the nonsolvability of the underlying symmetry algebra sl(2, R).This problem has been tackled by different approaches during the last decades, we review only some of the most relevant for the purposes of this work.Most results in the literature refer to third-order SL(2, C)-invariant ODEs, which can be solved via a pair of quadratures and the solution to a Riccati equation
In this paper we present new relevant results about first integrals of differential operators associated to SL(2, R)-invariant ODEs, showing how some of these first integrals can be constructed without any kind of integration at all
New methods to construct first integrals of differential operators associated to SL(2, R)-invariant
Summary
The study of nth-order ordinary differential equations admitting the unimodular Lie group. Olver [1], who connected the three inequivalent actions of SL(2, C) in the complex plane via the standard prolongation process It was demonstrated in Reference [2] that the fourth action that appears in the real case can be obtained from the same source, and this study was extended to 2D and 3D Lie algebras of symmetries, providing interesting results on the linearization of second-order ODEs via contact transformations [2,3]. By using techniques based on solvable structures [6,7,8,9,10,11,12], the general solution in parametric form for each one of the four canonical third-order SL(2, R)-invariant ODEs were obtained in References [13,14] Such solution is given in terms of a fundamental set of solutions to a second-order linear ODE, which is explicitly given for each one of the four different actions of SL(2, R). We present illustrative examples in order to show how these new results can be applied in practice
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