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?

  • $\begingroup$ According to E. Kamke book, a sufficient condition is constant Abel invariant. However computing it and equating it to a constant seems to lead to a 2nd order nonlinear ODE, that is a more complicated problem than the initial one. $\endgroup$ – Start wearing purple Jan 3 '16 at 12:00
  • $\begingroup$ And for $f(x)=\lambda x^3$ there seems to be no choice of $\lambda$ that leads to constant invariant. $\endgroup$ – Start wearing purple Jan 3 '16 at 12:08

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

  • $\begingroup$ Thanks for the comment. It should be implied of course that I meant in terms of known elementary or special functions; the alternative is that the answer is always 'yes' provided the DE admits a solution at all. Pity about $f(x) \propto x^3$ not working, that's a shame. Could you summarise which functions do allow for a "solvable" one based on that paper? Thanks $\endgroup$ – Arthur Suvorov Dec 28 '15 at 11:08

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