I'm looking for exact solutions, if such exist, for the following non-linear delay differential equation (DDE):

$ y_x(x) = A y(x-1)^a $

where $ 0 < a < 1 $ and $ A > 0 $ are given constants. Naturally the special case $ a = 1 $ reduces the equation to a linear DDE, whose solution is well known.

Any suggestions will be very welcome: references, impossibility theorems, etc..

Edit: The following is a special case of interest:

$ y_x(x) = \sqrt{y(x-1)} $

Would anyone know how to get a series solution for $y(x)$ ?