NEW SOLVABLE SYSTEM OF 2 FIRST-ORDER NONLINEARLY-COUPLED ORDINARY DIFFERENTIAL EQUATIONS

The paper introduces the system ẋ1 = (x1 + α1 x2)/Q, ẋ2 = (−x2 + α2 x1)/Q with Q = β1 x1^2 + (α1β1 + α2β2) x1 x2 + β2 x2^2, and states a Proposition giving the explicit IVP solution x_n(t) = γ_{n1} sqrt(1 + t/t1) + γ_{n2} sqrt(1 + t/t2) together with closed-form expressions for γ’s, t1, t2, but provides no derivation and explicitly invites the reader to verify the formulas (2)–(3). The candidate solution supplies a detailed and correct verification via a linear-algebraic factorization (J,B) showing that U and V are B-null directions interchanged by J, that Q(x(t)) factorizes as 2(γ1^T B γ2) y1 y2, and that the ODE is matched when 1/t1 and 1/t2 are chosen as in the paper; it also discusses the natural domain/uniqueness restrictions away from Q=0, which the paper does not address beyond a brief remark about square-root branches and possible singularities.