The Darbouxian theory of integrability allows to determine when a polynomial differential system in C2 has a first integral of the kind f λ1 1 ···f λp p exp(g/h) where fi , g and h are polynomials in C[x, y], and λi ∈ C for i = 1, . . . ,p. The functions of this form are called Darbouxian functions.
Here, we solve the inverse problem, i.e. we characterize the polynomial vector fields in C2 having a given Darbouxian function as a first integral.
On the other hand, using information about the degree of the invariant algebraic curves of a polynomial vector field, we improve the conditions for the existence of an integrating factor in the Darbouxian theory of integrability.
CitationLlibre, Jaume; Pantazi, Chara. “Polynomial differential systems having a given Darbouxian first integral”. Bulletin des sciences mathématiques, 2004, vol. 128, núm. 9, p. 775-788.
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