Solution of the system of two coupled first-order ODEs with second-degree polynomial right-hand sides

The paper states and solves the IVP for the two-component quadratic ODE system under four algebraic constraints, giving an explicit linear change of variables x1 = z1 w1 + z2 w2, x2 = w1 + w2 that decouples the dynamics into two constant-coefficient Riccati equations with coefficients αnℓ, and provides closed-form solutions for wn and thus xn (eqs. (37)–(45), (39)–(44) in the paper). The candidate solution reproduces these formulas and independently verifies decoupling by showing mixed terms cancel using the same constraints and the quadratic relation defining z1, z2 (S(z) = 0, eq. (24a)-(24b)). Coefficients and constraints match the paper exactly, and the Riccati solution formula is the same as the paper’s (eq. (41), (43)). Hence both are correct; the model’s proof is a direct algebraic cancellation argument, while the paper’s proof proceeds via an inverse-parameter route and then specialization. Key points and formulas appear in the paper’s Section 4 summary and earlier derivations (system and constraints (37), (38); z1, z2 via (24b)/(40); explicit solution (39)–(44)) . The constraints themselves are derived in Section 3 (26), (32) and the nondegeneracy condition (27) ensures z1 ≠ z2 as used by the model .