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

Abstract

The explicit solution x n t , $x_{n}\left (t\right ) ,$ 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: x ̇ n = c n 1 x 1 2 + c n 2 x 1 x 2 + c n 3 x 2 2 + c n 4 x 1 + c n 5 x 2 + c n 6 , n = 1 , 2 . $$ \dot{x}_{n}=c_{n1}\left( x_{1}\right)^{2}+c_{n2}x_{1}x_{2}+c_{n3}\left( x_{2}\right)^{2}+c_{n4}x_{1}+c_{n5}x_{2}+c_{n6}~,~~~n=1,2~. $$ The solution is explicitly provided if the 12 coefficients cnj (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 cnj is also explicitly solved, but it is found to imply—as it were, a posteriori—that the 12 coefficients cnj 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 cnj which identify particularly interesting models.