The equation...
(1)
... in the complex domain is equilvalent to a couple of equations in real domain. Setting the (1) became...
(2)
If is a solution of (2) then the proposed D.E. becomes...
(3)
... and its solution is...
(4)
The question is now: does (2) have solutions?...
Kind regards
The answers is done by the follwing lemma of complex variable functions theory...
Given the complex variable function...
, (1)
... holomorphic in a circle , its inverse function is done by the Taylor expansion...
, (2)
Now we use this lemma to find the solution of the trascendental equation...
(3)
With the formulas (2) we obtain the Taylor expansion...
(4)
... which converges only for . But our equation was...
(5)
... and then it has been demonstrated it has no solutions...
Kind regards