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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
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2 Answers

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
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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
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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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