Consider the delay differential equation:


$ y\_x(x) = \sqrt{y(x-\bar{x})} $


where $y$ is the unknown function of $x$, and where $\bar{x}$ is a fixed parameter. 


This equation does not seem to have a known closed-form solution.


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


Thanks!



nonlinear delay differential equation
As mentioned already by Denis Serre, there is a rich literature investigating delay equations.


If you make an experiment, and fix $\bar{x}=1$, then you see that you need as an initial value the complete past on $[-1,0]$. To play a bit, tak as an initial function the constant function $y(s)=1$ for $s\in[-1,0]$. Then you can calculate the solution explicitly for $x\in[0,1]$, then using this you can calculate the solutuion on $[1,2]$, etc.
We see that it is far from being analytic. Hence, no chance for a series sepresentation of a solution.


If you are interested in classical stuff, then [Bellman and Cooke](http://books.google.hu/books/about/Differential_difference_equations.html?id=j4LwpwGPXzwC&redir_esc=y) is an excellent book. 
An other good reference is the one by [Hale and Verduyn Lunel](http://books.google.hu/books?id=ZNLjAJQMhqwC&lpg=PP1&hl=de&pg=PP1#v=onepage&q&f=false).


**ADDED:** If it is a delay equation (i.e., $\bar{x}>0$), then the initial condition has to be a function (you have to know the whole past). Then the iteration procedure I described works always. This gives you a possible approximation formula, most numerical methods also work this way.


You are right about analyticity: series representation does it. Smoothness is a consequence. The example I presented to you is only once differentiable at $x=1$, twice at $x=2$, etc.
Hence, cannot be analytic.


