# Inverse problems in the theory of analytic planar vector

### Natalia Sadovskaia

Universidat Politècnica de Catalunya, Barcelona, Spain### Rafael O. Ramírez

Universitat Rovira i Virgili, Tarragona, Spain

## Abstract

In this communication we state and analyze the new inverse problems in the theory of differential equations related to the construction of an analytic planar vector field from a given, finite number of solutions, trajectories or partial integrals. Likewise we study the problem of determining a stationary complex analytic vector field from a given finite subset of terms in the formal power series

and from the subsidiary condition

where $G_{2k}$ is the Liapunov constant. The particular case when

and $(f_0, D \subset \mathbb C^2)$ is a canonic element in the neigbourhood of the origin of the complex analytic first integral $F$ is analyzed. The results are applied to the quadratic planar vector fields. In particular we constructed the all quadratic vector field tangent to the curve

where $q$ and $p$ are polynomials of degree $k$ and $m ≤ 2k$ respectively. We showed that the quadratic differential 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 field tangent to the given curve and it is Darboux's integrable.

## Cite this article

Natalia Sadovskaia, Rafael O. Ramírez, Inverse problems in the theory of analytic planar vector . Rev. Mat. Iberoam. 14 (1998), no. 3, pp. 481–517

DOI 10.4171/RMI/243