For Abel's differential equation \(dr/dt=a(t)r^2+b(t)r^3,\) \(t\in[t_0,t_1],\) it is said that the solution \(r=0\) is a center if for \(c\) small enough, the solution passing for \(r=c\) when \(t=t_0,\) \(r(t,c),\) also satisfies \(r(t_1,c)=c.\) The aim of this paper is to study conditions for Abel's differential equation to have a center at \(r=0.\) The main result is a theorem which allows to compute in a recurrent way the solution \(r(t,c)=c+\sum_{i\geq 2}r_i(t)c^i,\) as \(r_{n+1}(t)=A(t)r_n(t)+C_n(t),\) being \(A(t)=\int_{t_0}^ta(s)ds,\) \(r_1(t)\equiv 1\) and \(C_m(t)=\sum_{i+j=m}r_i(t)\int_{t_0}^t b(s)r_j(s)ds.\) It is proved by writing the Abel differential equation as the equivalent integral equation \(r(t,c)=c(1+A(t)r(t,c)+r(t,c)\int_{t_0}^tb(s)r(s,c) ds).\) As a consequence of the recurrent expression given in the theorem, the authors prove that when \(a(t)=2t,\) \(b(t)=\varepsilon \tilde b(t),\) being \(\tilde b(t)\) a polynomial, and \([t_0,t_1]=[-1,1]\) then, for \(\varepsilon\) small enough, the Abel differential equation has a center at \(r=0\) if and only if \(\tilde b(t)\) is an odd polynomial.