Darboux theory of integrability for a class of nonautonomous vector fields

To decide when a differential system is integrable or not is one of the hardest problems of the theory of differential equations. The existence and the calculus of first integrals are in general a difficult problem. Many techniques have been applied in order to construct first integrals, such as Lie symmetries, Noether symmetries, the Painlevé analysis, the use of Lax pairs, the Darboux method, and the direct method. In 1878 Darboux in Ref. 9 presented a simple method to construct first integrals and integrating factors for planar polynomial vector fields using their invariant algebraic curves. This theory has been useful for studying different relevant problems of planar polynomial differential systems such as problems related to centers, limit cycles, and bifurcation problems, see, for instance, Refs. 13, 19, and 28. Also Darboux in Ref. 10 extended his method to polynomial vector fields in Cn where the existence of invariant algebraic surfaces is the key point to build up first integrals see also Ref. 18 and for some applications see, for instance, Refs. 20–23 . Nowadays Darboux’s method has been improved for polynomial vector fields basically taking into account the exponential factors and the multiplicity of the invariant algebraic hypersurfaces, see, for instance, Refs. 6, 7, and 22–26. There are works such as Ref. 11 which generalize the Darboux theory of integrability using the concept of generalized cofactors. In this paper we extend the Darboux theory of integrability from the polynomial vector fields to a class of nonautonomous vector fields. More precisely we deal with differential vector fields in the plane that are polynomials in the variables x and y and their coefficients are convenient C1 functions in the time, i.e., in the independent variable. As far as we know it is the first time in the literature that such generalization is considered. The main results of this paper are Theorems 1 and 2 where the ideas of Darboux to the mentioned class of nonautonomous vector fields are generalized. We prove that a sufficient number of invariant surfaces and exponential factors generate a linearly dependent set of cofactors over the field of the coefficients of the system. In particular, in the case where a property W related to a kind of Wronskian of the cofactors holds then a subset of the cofactors is linearly dependent over C. In this case we can construct one or even two invariants a first integral depending on time , see Theorem 2. These invariants are very special because they are generalized Darboux functions, see relation 6 and Theorem 1.