6
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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
2
$\begingroup$

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

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.