## Abstract

The *explicit* solution \(x_{n}\left (t\right ) ,\) *n* = 1,2, of the *initial-values* problem is exhibited of a *subclass* of the *autonomous* system of 2 coupled *first-order* ODEs with *second-degree* polynomial right-hand sides, hence featuring 12 *a priori arbitrary* (time-independent) coefficients:

The solution is *explicitly* provided if the 12 coefficients *c*_{nj} (*n* = 1,2; *j* = 1,2,3,4,5,6) are expressed by *explicitly* provided formulas in terms of 10 *a priori arbitrary* parameters; the *inverse* problem to express these 10 parameters in terms of the 12 coefficients *c*_{nj} is also *explicitly* solved, but it is found to imply—as it were, *a posteriori*—that the 12 coefficients *c*_{nj} must then satisfy 4 *algebraic constraints*, which are *explicitly* exhibited. Special subcases are also identified the *general* solutions of which are *completely periodic* with a period independent of the initial data (“isochrony”), or are characterized by additional restrictions on the coefficients *c*_{nj} which identify particularly interesting models.

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## References

- 1.
Garnier, R.: Sur des systèmes différentielles du second ordre dont l’intégrale générale est uniforme. C. R. Acad. Sci. Paris

**249**, 1982–1986 (1959) and Ann. École Norm.**77**(2), 123–144 (1960) - 2.
Dumortier, F., Llibre, J., Artés, J.C.: Qualitative Theory of Planar Differential Systems. Springer, Berlin (2006)

- 3.
Artés, J.C., Llibre, J., Schlomiuk, D., Vulpe, N.: Geometric Configurations of Singularities of Planar Polynomial Differential Systems: A Global Classification in the Quadratic Case. Springer, Berlin (2021)

- 4.
Calogero, F., Conte, R., Leyvraz, F.: New algebraically solvable systems of two autonomous first-order ordinary differential equations with purely quadratic right-hand sides. J. Math. Phys.

**61**, 102704 (2020). https://doi.org/10.1063/5.0011257, arXiv:2009.11200 - 5.
Calogero, F., Payandeh, F.: Polynomials with multiple zeros and solvable dynamical systems including models in the plane with polynomial interactions. J. Math. Phys.

**60**(23 pages), 082701 (2019). https://doi.org/101063.1.5082249, arXiv:1904.00496v1 [math-ph] 31 Mar 2019 - 6.
Calogero, F.: Isochronous Systems. Oxford University Press, Oxford (2012). U. K., 2008 (hardback), (paperback)

- 7.
Gómez-Ullate, D., Sommacal, M.: Periods of the goldfish many-body problem. J. Nonlinear Math. Phys. Suppl.

**1**, 351–362 (2005) - 8.
Calogero, F., Gómez-Ullate, D.: Asymptotically isochronous systems. J. Nonlinear Math. Phys.

**15**, 410–426 (2008) - 9.
Calogero, F., Payandeh, F.: Solvable system of two coupled first-order ODEs with homogeneous cubic polynomial right-hand sides. J. Math. Phys.

**62**, 012701 (2021). https://doi.org/10.1063/5.0031963

## Acknowledgements

The results reported in this paper have been mainly obtained by a collaboration at a distance among its two authors (essentially via e-mails). We would like to acknowledge with thanks 2 grants, facilitating our future collaboration by allowing FP to visit (hopefully more than once) in 2021 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. Finally, we gratefully acknowledge a special contribution by François Leyvraz, who pointed out a serious flaw in a preliminary version of this paper, the elimination of which also entailed a substantial simplification of its presentation.

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## Appendices

### Appendix : A

In this Appendix A we tersely demonstrate the following elementary fact, which clearly implies the result (6): that the solution of the *initial-values* problem for the ODE

is provided by the formula

with *y*_{±} defined as follows:

Indeed the ODE (74a) can clearly be reformulated as follows:

with *y*_{±} defined by (74c); and then (again, via (74c)) this ODE can be rewritten as follows:

The integration of this ODE for the dependent variable \(y\left (t^{\prime }\right ) \) over the independent variable \(t^{\prime }\)—from \(t^{\prime }=0\) to \(t^{\prime }=t\)—clearly yields

which coincides—after exponentiation—with (74b). Q. E. D.

Finally let us emphasize that the results reported above are valid for generic values of the parameters; the interested reader shall have no difficulty to figure out the results in special cases, for instance those with *a*_{2} = 0 or *β* = 0.

### Appendix : B

In this Appendix ?? we tersely outline the derivation of the expressions (??) and (??) of the 12 parameters *c*_{nj} in terms of the 10 parameters *A*_{nm} and *a*_{nℓ}.

The first step is to invert the relations (??), getting

where the quantity *D* is defined as above, see (??).

The second step is to note that the relations (??) imply

hence, via the ODEs (??),

The third and last step is to insert the expressions (??) of *y*_{1} and *y*_{2} in terms of *x*_{1} and *x*_{2} in the right-hand sides of these ODEs. Then, via a bit of trivial if tedious algebra, there obtains the system (??) with the definitions (??) and (??) of the 12 coefficients *c*_{nj}. Q. E. D.

### Appendix : C

In this Appendix ?? we show that the solutions \(x_{n}\left (t\right ) \) of the dynamical system (??)—as treated above, see Sections ?? and ??—do *not* depend on the *free* parameters *λ*_{n} introduced via the positions (??).

To this end we insert the expressions (??) of the parameters *A*_{nm} in terms of the free parameters *λ*_{n} in the formulas (??) and (??) expressing the 6 parameters *a*_{nk}; in order to display the very simple dependence of these 8 parameters from the 2 free parameters *λ*_{n}. We thus easily find the following formulas:

where the notation *C* indicates—above and hereafter—the set of the 12 parameters *c*_{nj}, and the 6 functions \(\alpha _{n\ell }\left (C\right ) \) are *explicitly* displayed in Section ??, see (??); of course to this end we also used the definitions (??) of the 2 auxiliary parameters *z*_{n} in terms of the 4 coefficients *c*_{nm} (*n* = 1, 2; *m* = 1, 2).

The next step is to insert the formulas (79a) in the expressions (??), getting thereby

again with the functions \(w_{n\pm }\left (C\right ) \) and \(\beta _{n}\left (C\right ) \) *explicitly* displayed in Section ??, see (??) and (??).

The insertion of these formulas in the expressions (??) of the solutions \(y_{n}\left (t\right ) \) of the auxiliary dynamical system (??) evidences the following, very simple, dependence of these functions from the *free* parameters *λ*_{n}:

where again the 2 functions \(w_{n}\left (C,t\right ) \) are *explicitly* displayed in Section ??, see (??).

And via the insertion in the expressions (??) of \(x_{n}\left (t\right ) \) of these formulas (80), together with the expressions (??) of *A*_{nm}, we conclude that the solutions \(x_{n}\left (t\right ) \) are independent of the *free* parameters *λ*_{n}; as indeed displayed in Section 4, see the set of eqs. from (39) to (44). Q. E. D.

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Calogero, F., Payandeh, F. Solution of the System of Two Coupled First-Order ODEs with Second-Degree Polynomial Right-Hand Sides.
*Math Phys Anal Geom* **24, **29 (2021). https://doi.org/10.1007/s11040-021-09400-7

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### Keywords

- Solvable dynamical systems
- Solvable systems of nonlinearly-coupled ordinary differential equations

### Mathematics Subject Classification (2010)

- 34A34