Inverse problems in the theory of analytic planar vector

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 $m\le 2k$ 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.