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
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 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
(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 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.