Abstract
New relationships between generalized symmetries, commuting symmetries, and generalized \(\mathscr {C}^\infty \)–symmetries for second-order ordinary differential equations are established. The sets of solutions of the respective determining equations are interrelated, which provides new strategies for solving them. Particular solutions of these determining equations can be appropriately combined in order to provide first integrals and Jacobi last multipliers for the equation.
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Acknowledgements
The authors acknowledge the financial support from the University of Cádiz by means of the project PR2017-090 and from the Junta de Andalucía research group FQM-377.
A. Ruiz acknowledges the financial support from the Ministerio de Educación, Cultura y Deporte of Spain by means of a FPU grant (FPU15/02872).
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Muriel, C., Romero, J.L., Ruiz, A. (2018). The Calculation and Use of Generalized Symmetries for Second-Order Ordinary Differential Equations. In: Kac, V., Olver, P., Winternitz, P., Özer, T. (eds) Symmetries, Differential Equations and Applications. Springer Proceedings in Mathematics & Statistics, vol 266. Springer, Cham. https://doi.org/10.1007/978-3-030-01376-9_8
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