Disclaimer: I have posted this question on mathoverflow.net following the instructions of 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?
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 set of functions such that all solutions to linear second order ODE, with polynomial coeficients, can be expressed with? (may be: rational functions, exponentials and generalized hypergeometric series) If yes, where can I find a comprehensive list?