Integration of polynomial ordinary differential equations in the real plane

Research paper by A. E. Zernov, B. A. Scárdua

Indexed on: 01 Feb '01Published on: 01 Feb '01Published in: Aequationes mathematicae


Which differential equations can be integrated using functions that appear in the Differential Calculus? This is an ancient problem that has been considered by I. Newton, H. Poincaré, H. Dulac and P. Painlevé ([7],[8]). We address this question for differential equations of the form (1) \( {dy_1\over dx_1} = {P_1(x_1,y_1)\over Q_1(x_1,y_1)} \) where \( P_1,Q_1\in {\Bbb R}[x_1,y_1] \) are polynomials in two real variables. The classification of such an integrable equation is strongly related to the study of its complexification, mainly its complex singularities. For instance, we prove that if the complexification has only generic singularities then a polynomial differential equation (1) as above, which is integrable in the sense of Liouville (see §2 for the definition), must come from a linear differential equation after introduction of complex coefficients. Relaxing slightly the hypothesis on the singularities we obtain also Bernoulli differential equations (see §3 for these main results). These conclusions enforce original ideas of P. Painlevé (see Remark 3 in §3). Based on this classification, an interesting application to certain polynomial vector fields whose orbits have an algebraic curve as limit set in \( {\Bbb R}^2 \) is given in §4. It is proved that under appropriate hypothesis these equations also come from linear or Bernoulli equations.