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 .