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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 is an excellent book. An other good reference is the one by Hale and Verduyn Lunel.

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.

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As mentioned already by Denis Serre, you should consult the 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 is an excellent book. An other good reference is the one by Hale and Verduyn Lunel.

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As mentioned already by Denis Serre, you should consult the literature.

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 is an excellent book. An other good reference is the one by Hale and Verduyn Lunel.