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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 $y(x)$ ?

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Are you looking for positive solutions on the entire real line? – fedja Apr 26 '12 at 14:49

There is an entire book on differential delay equations, by Jack Hale and Lunel Verduyn : Introduction to functional-differential equations. Applied Mathematical Sciences, 99. Springer-Verlag (1993).

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Thanks a lot. I've now looked at the book. It's a great resource. Much of it is about stability, with a particular emphasis on the linear case. There are some non-linear examples, but I didn't see any trick I could apply to my case. I was rather hoping that there might be a trick similar to the change of variable that linearizes Bernoulli equations. Now perhaps there is no such trick? – PatrickT Apr 28 '12 at 12:49
The original edition by Hale only is better. But you should not expect to find ways of solving equations explicitly, this is not the usual approach to the subject. Hale had always some special interest for delay equations, which he usually considered as a playground for trying new things. – John B Dec 14 '15 at 2:52

Series solution? That is what I like to do with transseries.

Here are a few terms for an expansion as $x \to +\infty$: $$\begin{align} Y(x) &= \frac{x^2}{2} -\frac{x\log x}{2} +\left(\frac{\log x}{2}+\frac{(\log x)^2}{4}\right) +\frac{1}{x}\left(-\frac{1}{8}-\frac{\log x}{2}-\frac{(\log x)^2}{4}\right) \cr &\qquad +\frac{1}{x^2}\left(\frac{17}{72}+\frac{3\log x}{8} -\frac{(\log x)^3}{12}\right) +\dots \end{align}$$ Even truncating here, we see that $Y'(x)$ and $\sqrt{Y(x-1)}$ agree quite well when $x \ge 5$.

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Thank you Gerald. I googled for "transseries" and the first hit was "Transseries for beginners" by one G.A.Edgar. So I'll read it up and get back to you if I have follow-up questions! Thanks. – PatrickT Apr 30 '12 at 3:28
How did you go about computing these? Does it generalize easily to $Y'(x)=Y(x-1)^a$? Did you compute that by hand or did you use Maple (I'm a Maple user)? Please point me to a related example, I've had difficulty in finding a reference to transseries and delay differential equations. Many thanks! – PatrickT May 5 '12 at 5:45
P.S. I'm too new to vote otherwise I'd give you thumbs up! – PatrickT May 5 '12 at 5:46
forgot to say that I'm interested in behaviour around some given finite value of x rather than infinity. – PatrickT May 5 '12 at 5:48

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