# delay differential equation

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
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 $x \ge 5$.
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