Solution of the System of Two Coupled First-Order ODEs with Second-Degree Polynomial Right-Hand Sides

The explicit solution x n t , $x_{n}\left (t\right ) ,$ = 1,2, of the initial-values problem is exhibited a subclass autonomous system 2 coupled first-order ODEs with second-degree polynomial right-hand sides, hence featuring 12 priori arbitrary (time-independent) coefficients: ̇ c 1 + 3 4 5 6 . $$ \dot{x}_{n}=c_{n1}\left( x_{1}\right)^{2}+c_{n2}x_{1}x_{2}+c_{n3}\left( x_{2}\right)^{2}+c_{n4}x_{1}+c_{n5}x_{2}+c_{n6}~,~~~n=1,2~. explicitly provided if coefficients cnj (n 1,2; j 1,2,3,4,5,6) are expressed by formulas in terms 10 parameters; inverse to express these parameters also solved, but it found imply—as were, posteriori—that must then satisfy algebraic constraints, which exhibited. Special subcases identified general solutions completely periodic period independent initial data (“isochrony”), or characterized additional restrictions on identify particularly interesting models.