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Final edit: The problem I had in mind is properly asked in THIS MO QUESTION, so I'll vote to close the present post e recommend anyone interested in the topic to visit that link.

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Below is the original question

Disclaimer: This question was originally posted in math.stackexchange.com. I also followed the instructions on this topic.

I'm now trying to understand how can a ordinary differential equation be tested to decide if it's integrable or not. Recently I become aware of the Painlevé property and start to read the following paper by Robert Conte:

The Painlevé Approach to Nonlinear Ordinary Differential Equations

In section 2.1, he states:

"A very deep result of L. Fuchs, Poincaré and Painlevé is that the class of first order ODEs... ...defines one and only one function... ...the elliptic function introduced earlier by Weierstrass..."

My initial question is: Does this means that the solutions of any integrable first order ODE can by expressed by elementary functions and the Weierstrass elliptic function?

My initial question is: Does this means that the solutions of any integrable first order ODE can by expressed by elementary functions, as in the above lines, and the Weierstrass elliptic function by composition and algebraic operations? If no, what that sentence exactly means?

I know that many functions that are solutions to second order ODE (Exponentials, Bessel functions, hypergeometric functions, Airy functions...) can be expressed by generalized hypergeometric series.

My main question is: Are there some MINIMAL SET of functions such that all solutions to linear second order ODE, with polynomial coefficients, can be expressed with? (may be: rational functions, exponentials and generalized hypergeometric series) If yes, where can I find a comprehensive list?

To be more clear: By that I mean a "MINIMAL LIST" of "families of functions" such that, by knowing this families, we can express all solutions to linear second order ODE with polynomial coefficients, only by taking composition and algebraic operations with the members of this list. Alexandre Eremenko said that the answer to this question is NO. To be sure that I'm not getting it wrong: This means that no matter how much we enlarge our initial set of "elementary functions", there always be linear second order ODE with polynomial coefficients such that we cannot express it's solutions only by composition and algebraic operations with our enlarged set?

Additional information: Since I posted the question I tried to find some information. Following the clues of the Conte's paper, I had encountered:

  1. Robert Conte -The Painleve property, one century later I found that the original paper is from this book. I tried to read and the chapter 1 gone fine. In chapter 2 the reading rapidly becomes too hard. Also the felling that this was not the way to the answer.

  2. J. Gray - Fuchs and the theory of differential equations I found this paper searching more about the Fuchs's work that Conte refers. Although I enjoyed too much the reading, it aso doesn't help at all. Same as the chapter 1 from Conte's book, my general impression is that the subject is more about analycity and singularity structure of solutions and nothing about HOW EXPRESS such solutions.

  3. Connemara Doran - Poincare’s Path to Uniformization Another very enjoyable reading, but in the end give no answer.

  4. Two answers of Loïc Teyssier here and here Although the questions are related, (and unless I'm getting all wrong) my question is fundamentally different: I'm not asking when solutions "be written in a closed form using integrals, coefficients of the equation, initial conditions and the standard functions". I'm asking for a FINITE AND MINIMAL set of functions that, with then, we can express solutions to linear second order ODE's, at least if we restrict ourselves to the ones with polynomial coefficients. Also if the statement from Conte is equivalent to this minimal set exists for first order equations and is given by elementary functions plus Weierstrass function.

Edits:

Edit 1: As Alexandre Eremenko pointed out, I was not clear about what I mean by "integrable". So, to be more clear, I'm thinking in Goriely's definition that integrable is: "sufficiently many first integrals are known globally". By elementary functions I mean rational, exponential, logarithm and functions obtained from then by algebraic operations and composition. (As trigonometric functions, hyperbolic functions and their inverse can be also obtained in this way)

Edit 2: In this wikipedia article is the kind of generalized hypergeometric series I'm thinking about.

Edit 3: Loïc Teyssier help me see that, in the way I originally asked, the answer could be very vague.

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    $\begingroup$ The answer to your "initial question" depends on the precise definition of "integrable" and "elementaty function". There are various definitions. Elliptic functions are not elementary according to most of them. The answer to your second question is negative: most linear differential equations with polynomial coefficients cannot be solved in terms of any "special functions". But again the precise answer depends on what you mean by "generalized hypergeometric functions" (Several definitions exist in the literature). $\endgroup$
    – Alexandre Eremenko
    Commented Nov 10, 2021 at 12:49
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    $\begingroup$ From y point of view, 1-st order equation is $F(y',y,x)=0$, where $F$ is a polynomial. Can you spell EXACTLY what does it mean that THIS equation is integrable? $\endgroup$
    – Alexandre Eremenko
    Commented Nov 10, 2021 at 17:14
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    $\begingroup$ What kind of function $H$? Not even continuous?? $\endgroup$
    – Alexandre Eremenko
    Commented Nov 10, 2021 at 17:34
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    $\begingroup$ According to your definition, EVERY equation $y'=P(x,y)$, where $P$ is a polynomial, or even a continuous function satisfying Lipschitz condition will be integrable. This makes no sense. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 10, 2021 at 17:40
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    $\begingroup$ @Diego Santos: you could simply take the function space spanned by all analytic solutions of polynomial 1st order polynomial ODEs. If you want something more meaningful, you'd probably like to quantify on the kind of "determinacy" of the sets you're after...? $\endgroup$
    – Loïc Teyssier
    Commented Nov 10, 2021 at 20:20

1 Answer 1

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For an account of integrability of linear ODE in the way that you describe, refer section 4 of the 2004 paper

A.G. Khovanskii, "On solvability and unsolvability of equations in explicit form"

Uspekhi Mat. Nauk. 59:4 69-146, English translation in Russian Math Surveys 59:4 661-736.

(Section 1.2 of the paper describes the "classical" classes of functions of a single variable to better define what's meant by "integrable".)

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