TITLE: Closed form solutions to Abel equation
QUESTION [6 upvotes]: Consider a (somewhat simplified) Abel equation of the first kind for $\alpha$:
$\left[\alpha(x)\right]^2 \left[1-f(x)\alpha(x)\right] + \alpha'(x) = 0$,
for some smooth function $f$.
Is it known what conditions on $f$ are necessary (and sufficient) to ensure a closed form solution? One particular case I am interested in is $f(x) = \lambda x^3$ for some $\lambda \in \mathbb{R}$; is there any hope of getting a closed form in this case? What other cases have been studied?

REPLY [2 votes]: The question << Is it known what conditions on $f$ are necessary (and sufficient) to ensure a closed form solution? >> is not quite pertinent because the answer depends on the background of special functions allowed. 
A closed form is made of a combination of a finite number of elementary and/or special functions, i.e. functions defined and referenced as "standard". So, if a new special function appears in the specialised litterature, the solutions of an ODE which were previously impossible to write on a closed form, possibly become writen on a closed form, thanks to the new special function.
One can imagine a new set of special functions especially defined and standardized, devoted to the solving of the Abel's ODE. This is not the case today.
In the case of $f(x)=\lambda x^3$ it seems that the ODE isn't of solvable kind in the sens of "solvable" considered for example in this paper : http://arxiv.org/ftp/arxiv/papers/1503/1503.05929.pdf