  Abel equations of the form $x'(t)=f(t)x^3(t)+g(t)x^2(t)$, $t \in [-a,a]$,
where $a>0$ is a constant, $f$ and $g$ are continuous functions, are of
interest because of their close relation to planar vector fields. If $f$ and
$g$ are odd functions, we prove, in this paper, that the Abel equation has a
center at the origin. We also consider a class of polynomial differential
equations $\dot{x} = -y+P_n(x,y)$ and $\dot{y} = x+Q_n(x,y)$, where $P_n$ and
$Q_n$ are homogeneous polynomials of degree $n$. Using the results obtained for
Abel's equation, we obtain a new subclass of systems having a center at the
origin.
