On the number of polynomial solutions of Bernoulli and Abel polynomial differential equations

Abstract

Abstract. In this paper we determine the maximum number of polynomial solutions of Bernoulli differential equations and of some integrable polynomial Abel differential equations. As far as we know, the tools used to prove our results have not been utilized before for studying this type of questions. We show that the addressed problems can be reduced to know the number of polynomial solutions of a related polynomial equation of arbitrary degree. Then we approach to these equations either applying several tools developed to study extended Fermat problems for polynomial equations, or reducing the question to the computation of the genus of some associated planar algebraic curves.

Transformation Method for Generating Periodic Solutions of Abel’s Differential Equation

“…Proof. (1) Firstly, we prove the existence of a positive −periodic continuous solution of Equation (6). Suppose = { ( ) ∈ ( , ) | ( + ) = ( )} .…”

“…Similar to the proofs of Theorems 5 and 6, we can get the following. (6); ( ), ( ) are −periodic continuous functions; suppose that the following conditions hold:…”

“…plays an important role in many physical and technical applications [1,2]. The mathematical properties of Equation (1) have been intensively investigated in the mathematical and physical literature [3][4][5][6][7][8][9][10][11]. S. S. Misthry, S. D. Maharaj, and P. G. L. Leach [12] introduced a new transformation at the boundary that leads to an Abel's equation and showed explicitly that a variety of exact solutions can be generated from the Abel equation.…”

“…(66) Consider Equation (6); ( ), ( ) are −periodic continuous functions; suppose that the following conditions hold:( 1 ) ( ) > 0,( 2 ) ( ) < 0. (67)Then Equation(6) has a unique positive −periodic continuous solution ( ), and Advances in Mathematical Physics…”

The Fixed Point Theory and the Existence of the Periodic Solution on a Nonlinear Differential Equation

“…Recently, A. Cima, A. Gasull, and F. Manosas [8] gave the maximum number of polynomial solutions of some integrable polynomial Abel differential equations; Jaume Giné Claudia and Valls [9] studied the center problem for Abel polynomial differential equations of second kind; Jianfeng Huang and Haihua Liang [10] were devoted to the investigation of Abel equation by means of Lagrange interpolation formula; they gave a criterion to estimate the number of limit cycles of the Abel's equations; Berna Bülbül and Mehmet Sezer [11] introduced a numerical power series algorithm which is based on the improved Taylor matrix method for the approximate solution of Abel-type differential equations; Ni et al [12] discussed the existence and stability of the periodic solutions of (1) and obtained the sufficient conditions which guaranteed the existence and stability of the periodic solutions for (1) from a particular one.…”