On Liouvillian integrability of the first–order polynomial ordinary differential equations

Abstract

a b s t r a c tRecently, the authors provided an example of an integrable Liouvillian planar polynomial differential system that has no finite invariant algebraic curves; see Giné and Llibre (2012) [8]. In this note, we prove that, if a complex differential equation of the form y ′ = a 0 (x) + a 1 (x)y + · · · + a n (x)y n , with a i (x) polynomials for i = 0, 1, . . . , n, a n (x) ̸ = 0, and n ≥ 2, has a Liouvillian first integral, then it has a finite invariant algebraic curve. So, this result applies to Riccati and Abel polynomial differential equations. We shall prove that in general this result is not true when n = 1, i.e., for linear polynomial differential equations.

A note on Liouvillian first integrals and invariant algebraic curves

“…In [11] it is proved that if a complex differential equation of the form dy/dx = a 0 (x) + a 1 (x)y + • • • + a n (x)y n with a i (x) polynomials for i = 0, 1, . .…”

“…where, without loss of generality, we have privileged in (6) the variable y with respect to variable x writing the polynomials P (x, y) and Q(x, y) of ( 1) as polynomials in y with coefficients polynomials in x. The particular case studied in [11] is system (6) with P (x, y) = 1.…”

“…The system (10) has ( , n) = (2s − 1, 2s − 1) which is an example of n = both odd integers. Finally, system (11) has ( , n) = (2s, 2s) which is an example of n = both even integers. System (7) with k = 0 has been studied in [10] where it was proved that the system has a Liouvillian first integral and no finite invariant algebraic curve.…”

“…Now we take system (7) with k = 0 and we do the transformation (u, v) → (x, y) with u = x and v = (x + y) s to get system (10). And finally we take again k = 0 and the transformation (u, v) → (x, y) with u = y and v = (x + y) s which gives system (11).…”

“…We remark that if one of the systems ( 8), ( 9), (10) or (11) has a finite invariant algebraic curve f (x, y) = 0 by the rational change of variables, then equation ( 14) has a particular algebraic solution. In particular, for system (8), we would get the particular solution f (u 1/s − v, v) = 0.…”