An Algebraic Framework Toward the Solution of the Abel Differential Equation

Source abstract

The Abel differential equation of the first kind has no general closed-form solution, and most recent results on its rational solutions give upper bounds on their number or describe qualitative properties. This paper describes a direct algebraic procedure for finding rational solutions when the coefficients are polynomials. Writing the unknown as $y=P/Q$ with coprime polynomials $P$ and $Q$, then applying the quotient rule and clearing denominators, one obtains the polynomial identity $(P'Q-PQ')Q=f_0Q^3+f_1PQ^2+f_2P^2Q+f_3P^3$. For prescribed degrees of $P$ and $Q$, comparing coefficients gives a finite system of polynomial equations in the unknown coefficients, and rational solutions of those degrees correspond exactly to solutions of this system; an inconsistent system rules them out. The procedure is illustrated on $y'=-2y^2+xy^3$, where it recovers $y=1/x$ and shows that this is the only non-zero solution of linear-fractional form. The same reduction is extended to equations $y'=\sum_{k=0}^{N}f_k(x)y^k$ with necessary degree conditions. The method is suited to computer algebra systems and complements the existing bound-based results.