Abstract
In this paper we identify systems of an arbitrary number \(N\) of first-order Ordinary Differential Equations with nonlinear homogeneous right-hand sides of an arbitrary (integer, positive or nonpositive) degree \(M\), which feature very simple explicit solutions; as well as variants of these systems—with right-hand sides no more homogeneous—some of which feature periodic solutions. A novelty of these findings is to consider systems characterized by constraints involving their parameters and/or their initial data.
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Acknowledgments
It is a pleasure to thank our colleagues Robert Conte, François Leyvraz, and Andrea Giansanti for very useful discussions. Finally, we also like to thank Fernanda Lupinacci who, in these difficult times—with extreme efficiency and kindness—facilitated all the arrangements necessary for the presence of FP with her family in Rome.
Funding
We like to acknowledge with thanks two grants, facilitating our collaboration—mainly developed via e-mail exchanges—by making it possible for FP to visit twice the Department of Physics of the University of Rome “La Sapienza”: one granted by that University, and one granted jointly by the Istituto Nazionale di Alta Matematica (INdAM) of that University and by the International Institute of Theoretical Physics (ICTP) in Trieste in the framework of the ICTP–INdAM “Research in Pairs” Programme.
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Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 213, pp. 5–19 https://doi.org/10.4213/tmf10222.
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Calogero, F., Payandeh, F. Explicitly solvable systems of first-order ordinary differential equations with homogeneous right-hand sides, and their periodic variants. Theor Math Phys 213, 1317–1330 (2022). https://doi.org/10.1134/S0040577922100026
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DOI: https://doi.org/10.1134/S0040577922100026