Integration of polynomial ordinary differential equations in the real plane (Q5931972)

Which differential equations can be integrated using functions that appear in the differential calculus? This ancient problem has motivated the introduction of Lie groups. Here, the authors consider this problem for polynomial differential equations. The classification of such an integrable equation is related to the study of its complexification, mainly its complex singularities. They prove that if the complexification has only generic singularities then the polynomial differential equation, which is integrable in the sense of Liouville, must come from a linear differential equation after introduction of complex coefficients. Relaxing slightly the hypothesis on the singularities they obtain Bernoulli differential equations. Based on this work, an interesting application to certain polynomial vector fields whose orbits have an algebraic curve as limit set \(\mathbb{R}^2\) is presented. It is proved that under appropriate hypothesis these equations come from linear or Bernoulli equations.