Darboux integrability and the inverse integrating factor

Polynomial and rational first integrals for planar homogeneous polynomial differential systems

“…To know more about the Darboux theory of integrability, see for instance [6,8,9,19] and the references therein. The proof of Theorem 3 is given in Section 2.…”

“…Using these results for the general case, one could give alternative proofs to Proposition 1 and Theorem 3 in the particular case of the homogeneous system (1). This could be done in the framework of remarkable values, see [6,12,13] and the references therein, and using the necessary conditions for the existence of invariant algebraic curves, see [7] and the references therein. In particular, in [12], there are several results about systems of the form (2) which have a Darboux first integral characterizing the existence of a polynomial inverse integrating factor.…”

Detection of Special Curves Via the Double Resultant

“…The remarkable values and remarkable curves of rational first integrals of planar differential systems were first introduced by Poincaré in [23], and afterwards studied by several authors, see [8,10,11]. It has been shown in the literature that the remarkable curves play an important role in the phase portrait as they are strongly related to its separatrices.…”

“…It has been shown in the literature that the remarkable curves play an important role in the phase portrait as they are strongly related to its separatrices. It is proved in [8] that there are a finite number of them. In [8,11] they are related with the inverse integrating factor.…”

“…It is proved in [8] that there are a finite number of them. In [8,11] they are related with the inverse integrating factor.…”