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 &gt; 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.

First Integrals of Differential Operators from {SL}(2,R) Symmetries / P. Morando, C. Muriel, A. Ruiz. - In: MATHEMATICS. - ISSN 2227-7390. - 8:12(2020), pp. 2167.1-2167.18. [10.3390/math8122167]

### First Integrals of Differential Operators from {SL}(2,R) Symmetries

#### 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.
##### Scheda breve Scheda completa Scheda completa (DC) differential operator; first integral; solvable structure; integrable distribution
Settore MAT/07 - Fisica Matematica
Settore MAT/05 - Analisi Matematica
MATHEMATICS
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/2434/799109`
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