The transvectants and the integrals for Darboux systems of differential equations (original) (raw)

We apply the algebraic theory of invariants of differential equations to integrate the polynomial differential systems dx/dt = P1(x, y) + x C(x, y), dy/dt = Q1(x, y) + y C(x, y), where real homogeneous polynomials P1 and Q1 have the first degree and C(x, y) is a real homogeneous polynomial of degree r ≥ 1. In generic cases the invariant algebraic curves and the first integrals for these systems are constructed. The constructed invariant algebraic curves are expressed by comitants and invariants of investigated systems.