# Solution of the parametric center problem for the Abel differential equation

### Fedor Pakovich

Ben Gurion University, Beer Sheva, Israel

## Abstract

The Abel differential equation $y_{′}=p(x)y_{2}+q(x)y_{3}$ with $p,q∈R[x]$ is said to have a center on a segment $[a,b]$ if all its solutions, with the initial value $y(a)$ small enough, satisfy the condition $y(b)=y(a)$. The problem of description of conditions implying that the Abel equation has a center may be interpreted as a simplified version of the classical Center-Focus problem of Poincaré. The Abel equation is said to have a “parametric center” if for each $ϵ∈R$ the equation $y_{′}=p(x)y_{2}+ϵq(x)y_{3}$ has a center. In this paper we show that the Abel equation has a parametric center if and only if the antiderivatives $P=∫p(x)dx,$ $Q=∫q(x)dx$ satisfy the equalities $P=P∘W$, $Q=Q ∘W$ for some polynomials $P,$ $Q ,$ and $W$ such that $W(a)=W(b)$. We also show that the last condition is necessary and sufficient for the “generalized moments” $∫_{a}P_{i}dQ$ and $∫_{a}Q_{i}dP$ to vanish for all $igeq0.$

## Cite this article

Fedor Pakovich, Solution of the parametric center problem for the Abel differential equation. J. Eur. Math. Soc. 19 (2017), no. 8, pp. 2343–2369

DOI 10.4171/JEMS/719