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I frequently encounter in the literature the statement that Abel's equation $$\frac{dy}{dx}=x+y^3$$ is not integrable. This is always stated without reference. My questions are

a) What is the precise statement? (I suppose it should be that there is no non-constant meromorphic function (first integral) $F(x,y)$ on $C^2$ which is constant on each solution $y(x)$).

b) Where is this proved?

When I search in the reference books or internet, they list many known cases of integrability of Abel's equations, but no one addresses proofs of non-integrability.

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  • $\begingroup$ encyclopediaofmath.org/wiki/Abel_differential_equation gives as a reference the book: E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden", 1. Gewöhnliche Differentialgleichungen, Chelsea, reprint (1971) $\endgroup$ Jan 31, 2022 at 23:06
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    $\begingroup$ I looked into Kamke before asking. It only lists known cases of integrability. $\endgroup$ Feb 1, 2022 at 0:24
  • $\begingroup$ Abel differential equations admitting a certain first integral, J. Math. Anal. Appl. 370 (2010) 187–199 (sciencedirect.com/science/article/pii/S0022247X10003483/pdf) has references to old work on the topic, IMHO. $\endgroup$ Feb 1, 2022 at 12:53
  • $\begingroup$ @Dima Pesechnik: Which work in the reference list contains a proof of NON-integrability? A brief inspection only reveals "classification of Abel equations having a first integral of particular type". $\endgroup$ Feb 1, 2022 at 13:07
  • $\begingroup$ actually, looking at cited there arxiv.org/abs/math-ph/0001037 tempts me to say that no proof of non-integrability is known. By the way, the latter refers to the original work by Abel on solvable cases of his equation, perhaps he said something on your case too? $\endgroup$ Feb 1, 2022 at 13:28

2 Answers 2

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Maple's help on Abel equations as well as the text describing this part of Maple ODE code suggests that indeed no proofs of unsolvability of this (and other(?)) Abel equation are known. To quote:

The most general method available at the moment to solve Abel ODEs
 seems to be the method of "Abel's invariant", described in E. Kamke, p. 26, as sub-method (g) due to M. Chini. The invariant of an Abel equation with f2=0 is the following quantity:
>   
Abel_invariant := -1/27/f3(x)^4*(-diff(f0(x),x)*f3(x)+f0(x)*diff(f3(x),x)+
3*f0(x)*f3(x)*f1(x))^3/f0(x)^5

We have $f_3=1$, $f_0=x$, so the latter quantity depends on $x$, the case for which this "most general" method does not work.

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  • $\begingroup$ I hoped that some MO participants may know more on the subject than Kamke or Maple do. $\endgroup$ Feb 1, 2022 at 17:39
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The paper by Primitivo B. Acosta-Humánez, J. Tomás Lázaro, Juan J. Morales-Ruiz and Chara Pantazi Differential Galois theory and non-integrability of plane polynomial vector fields, J. Differential Equations 264 (2018) 7183- 7212 https://doi.org/10.1016/j.jde.2018.02.016 contains both:

a) a definition of integrability (which appears to match your conjectured meaning of the term); and b) a general theorem for non-integrability for planar polynomial vector fields.

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