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This question was asked here in MO by someone seeking for the solution of the functional -differential:$f'=e^{f^{-1}}$ not exactly an O.D.E, and again here seeking for the growth rate of it solution . this lead me to ask the following question which include some addition in the side of it's behavior which it related to the existence of divergent solution of Gevrey type and the Borel summation .then my question here is :

Question: Does the divergent solution of this equation :$f'=e^{f^{-1}}$ of Gevrey type and could be Borel summation applied for it ?.or any reference discussed that ?

Note:The motivation of this question is to seek if the Borel sum is analytic solution for the equation .

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    $\begingroup$ Borel summation could be applied if the solution is an asymptotic series $\endgroup$
    – zeraoulia rafik
    Commented Mar 30, 2017 at 23:58
  • $\begingroup$ I think to start a bounty for this question it's very interesting $\endgroup$
    – zeraoulia rafik
    Commented Apr 2, 2017 at 20:01
  • $\begingroup$ I don't have enough reputation for that , could you help me for that ? $\endgroup$
    – user99666
    Commented Apr 2, 2017 at 20:18
  • $\begingroup$ ok, no problem since it's interesting and it's include my question $\endgroup$
    – zeraoulia rafik
    Commented Apr 2, 2017 at 20:21

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