# Revista Matemática Iberoamericana

Volume 14, Issue 3, 1998, pp. 481–517
DOI: 10.4171/RMI/243

Published online: 1998-12-31

Inverse problems in the theory of analytic planar vector

Natalia Sadovskaia[1] and Rafael O. Ramírez[2]

(1) Universidat Politècnica de Catalunya, Barcelona, Spain
(2) Universitat Rovira i Virgili, Tarragona, Spain

In this communication we state and analyze the new inverse problems in the theory of differential equations related to the construction of an analytic planar vector field from a given, finite number of solutions, trajectories or partial integrals. Likewise we study the problem of determining a stationary complex analytic vector field  from a given finite subset of terms in the formal power series $$V(z, w) = \lambda (z^2 + w^2) + \sum^\infty_{k=3} H_k (z, w), H-k (az, aw) = a^kH_k(z,w),$$ and from the subsidiary condition $$\Gamma (V) = \sum^\infty_{k=1} G_{2k}(z^2 + w^2)^{k+1},$$ where $G_{2k}$ is the Liapunov constant. The particular case when $$V (z,w) = f_0(z,w) – f_0(0,0)$$ and $(f_0, D \subset \mathbb C^2)$ is a canonic element in the neigbourhood of the origin of the complex analytic first integral $F$ is analyzed. The results are applied to the quadratic planar vector fields. In particular we constructed the all quadratic vector field tangent to the curve $$(y – q (x))^2 – p(x) = 0.$$ where $q$ and $p$ are polynomials of degree $k$ and $m ≤ 2k$ respectively. We showed that the quadratic differential systems admits a limit cycle of this type only when the algebraic curve is of the fourth degree. For the case when $k > 5$ it proved that there exist an unique quadratic vector field tangent to the given curve and it is Darboux's integrable.