The center problem and composition condition for Abel differential equations (Q281909)

For the polynomial Abel differential equation of the form NEWLINE\[NEWLINE{dy\over dx}= p(x) y^2+ q(x) y^3,\tag{1}NEWLINE\]NEWLINE where \(y\) is real, \(x\) is a real independent variable considered in a real interval \([a,b]\) and \(p(x)\) and \(q(x)\) are real polynomials in \(\mathbb{R}[x]\), there is presented a detailed discussion of the following conjecture:NEWLINENEWLINE For any polynomial Abel differential equation (1) the center variety and the composition center variety coincide.NEWLINENEWLINE In the present paper, the correctness of this conjecture is proved for the case \(\max(\delta p,\delta q)\leq 3\), where \(\delta p\), \(\delta q\) are the degrees of the polynomials \(p(x)\) and \(q(x)\). The correctness of this conjecture is also proved for the case NEWLINE\[NEWLINEp(x)= a_i x^i+ a_i x^j,\quad q(x)= a_m x^m+ a_n x^nNEWLINE\]NEWLINE with \(a_i, a_j, a_m, a_n\in\mathbb{R}\) and \(i,j,m,n\in \mathbb{N}\cup\{0\}\).