TY  - JOUR
AU  - Farias, Matheus dos Santos
PY  - 2026
DA  - 2026/10/25
TI  - An Algebraic Framework Toward the Solution of the Abel Differential Equation
JO  - Journal of Mathematics and Interdisciplinary Applications
T2  - Journal of Mathematics and Interdisciplinary Applications
JF  - Journal of Mathematics and Interdisciplinary Applications
VL  - 2
IS  - 4
SP  - 238
EP  - 243
DO  - 10.62762/JMIA.2026.356064
UR  - https://www.icck.org/article/abs/JMIA.2026.356064
KW  - Abel differential equation
KW  - rational solutions
KW  - polynomial identity
KW  - algebraic compatibility
KW  - nonlinear differential equations
AB  - 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.
SN  - 3070-393X
PB  - Institute of Central Computation and Knowledge
LA  - English
ER  - 

@article{Farias2026An,
  author = {Matheus dos Santos Farias},
  title = {An Algebraic Framework Toward the Solution of the Abel Differential Equation},
  journal = {Journal of Mathematics and Interdisciplinary Applications},
  year = {2026},
  volume = {2},
  number = {4},
  pages = {238-243},
  doi = {10.62762/JMIA.2026.356064},
  url = {https://www.icck.org/article/abs/JMIA.2026.356064},
  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.},
  keywords = {Abel differential equation, rational solutions, polynomial identity, algebraic compatibility, nonlinear differential equations},
  issn = {3070-393X},
  publisher = {Institute of Central Computation and Knowledge}
}