First Integrals of Differential Operators from SL (2, ? ) Symmetries

Paola Morando, Concepción Muriel and Adrián Ruiz
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Paola Morando: Dipartimento di Scienze Agrarie e Ambientali-Produzione, Territorio, Agroenergia, Università Degli Studi di Milano, 20133 Milano, Italy
Concepción Muriel: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cádiz, 11510 Puerto Real, Spain
Adrián Ruiz: Departamento de Matemáticas, Escuela de Ingenierías Marina, Náutica y Radioelectrónica, Universidad de Cádiz, 11510 Puerto Real, Spain

Mathematics, 2020, vol. 8, issue 12, 1-18

Abstract: The construction of first integrals for S L ( 2 , R ) -invariant n th-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.

Keywords: differential operator; first integral; solvable structure; integrable distribution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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