Polynomial Solutions of Equivariant Polynomial Abel Differential Equations

1 Introduction and Statement of the Main Results

Abel differential equations of the first kind, namely,

with b3⁢(x)≠0, appear in many textbooks of ordinary differential equations as one of the first non-trivial examples of nonlinear differential equations, see, for instance, [11]. Here the dot denotes the derivative with respect to the independent variable x. If b3⁢(x)=b0⁢(x)=0 or b2⁢(x)=b0⁢(x)=0, the Abel differential equation reduces to a Bernoulli differential equation, while if b3⁢(x)=0, the Abel differential equation reduces to a Riccati differential equation.

Abel differential equations of the form (1) have been studied intensively, either by calculating their solutions (see, for instance, [8, 12, 14, 15]), or by classifying their centers (see [2, 3, 4]), and recently in [7, 9, 10, 13], the polynomial solutions of the differential equation y′=∑i=0nai⁢(x)⁢yi were studied as well.

The analysis of particular solutions (as polynomial or rational solutions) of differential equations is important for understanding the set of their solutions. In 1936 Rainville [16] characterized the Riccati differential equation y˙=b0⁢(x)+b1⁢(x)⁢y+y2, with b0⁢(x) and b1⁢(x) being polynomials in the variable x, having polynomial solutions.

Campbell and Golomb [5] in 1954 provided an algorithm for determining the polynomial solutions of the Riccati differential equation a⁢(x)⁢y′=b0⁢(x)+b1⁢(x)⁢y+b2⁢(x)⁢y2, where a,b0,b1,b2 are polynomials in the variable x. Behloul and Cheng [1] in 2006 gave a different algorithm for finding the rational solutions of the differential equation a⁢(x)⁢y′=∑i=0nbi⁢(x)⁢yi, where a,bi are polynomials in the variable x.

Here we consider the Abel differential equation (1), where a⁢(x)∈𝔽⁢[x]∖{0}, bi⁢(x)∈𝔽⁢[x], i=0,1,2,3, b3⁢(x)≠0, and 𝔽=ℝ,ℂ, with 𝔽⁢[x] being the ring of polynomials in the variable x with coefficients in 𝔽. We also assume that a⁢(x) is not constant. The case where a⁢(x) is constant has been studied in [10]. We say that the Abel differential equation (1) has degree η.

Equation (1) is reversible with respect to the change of variables (x,y)→(x,-y) if the following equation coincides with equation (1):

In particular, this implies b1⁢(x)=b3⁢(x)=0, and since b3⁢(x)=0, we do not consider these reversible differential equations.

The Abel differential equation (1) is equivariant with respect to the change of variables (x,y)→(x,-y) if the following equation coincides with equation (1):

This implies b0⁢(x)=b2⁢(x)=0. In this paper first we focus our study in these kind of equivariant polynomial Abel equations, i.e., in equations of the form

The proof of Theorem 1 is given in Section 2.

Our second objective in this paper is on the Abel differential equations of the second kind, i.e., on the equations of the form

where a⁢(x),bi⁢(x)∈𝔽⁢[x] for i=0,1,2, with a⁢(x) and b2⁢(x) being non-zero. We also consider the ones that are equivariant with respect to the change (x,y)→(x,-y). Then we have that b1⁢(x)=0, and so equation (3) becomes

We also assume that b2⁢(x)≠0 (otherwise the differential equation would be of separation of variables) and that a⁢(x) is not constant, since the case where a⁢(x) is constant has been studied in [7]. We say that equation (4) is the equivariant polynomial Abel differential equation of the second kind.

The proof of Theorem 2 is given in Section 3.

2 Proof of Theorem 1

First we recall that if y⁢(x)≠0 is a solution of equation (2), then -y⁢(x) is too, which is different from y⁢(x).

In view of Lemma 3, if y1⁢(x) and y2⁢(x) are polynomial solutions of equation (2) such that y1⁢(x)≢0, y2⁢(x)≢0 and y2⁢(x)≠-y1⁢(x), then any other polynomial solution different from them is of the form given in (5) for some appropriate constant c such that c∉{0,1}. In particular, c⁢y~12⁢(x)+(1-c)⁢y~22⁢(x) or y~12⁢(x)+(1-c)⁢y~22⁢(x)/c is a square of a polynomial P, and P divides g. In view of Theorem 7 (see Appendix A), there is at most one constant c∉{0,1} such that c⁢y~12+(1-c)⁢y~22 is a square of a polynomial, meaning that equation (2) has at most seven different polynomial solutions 0, ±y1, ±y2 and y0, as in (5).

3 Proof of Theorem 2

First we recall that if y⁢(x)≠0 is a solution of equation (4), then -y⁢(x) is too, which is different from y⁢(x).

In view of Lemma 3, if y1⁢(x) and y2⁢(x) are polynomial solutions of equation (4) such that y1⁢(x)≢0, y2⁢(x)≢0 and y2⁢(x)≠-y1⁢(x), then any other polynomial solution is of the form (6) for some appropriate constant c. In particular, c⁢y~12⁢(x)+(1-c)⁢y~22⁢(x) is a square of a polynomial P. Again, in view of Theorem 7, this c is unique, and we conclude that equation (4) has at most seven different polynomial solutions 0, ±y1, ±y2 and y0, as in (6).