Welcome to MathOverflow

Question

2

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 $y(x)$ ?

edited Apr 28 2012 at 12:51

asked Apr 26 2012 at 14:32

PatrickT
41●3

Are you looking for positive solutions on the entire real line? – fedja Apr 26 2012 at 14:49

Denis Serre · Answer 1 · 2012-04-26 15:59:47Z UTC

1

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).

answered Apr 26 2012 at 15:59

Denis Serre
18.7k●1●31●93

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 2012 at 12:49

Gerald Edgar · Answer 2 · 2012-04-28 17:26:17Z UTC

1

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 $x \ge 5$.

answered Apr 28 2012 at 17:26

Gerald Edgar
15.1k●1●28●67

	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 2012 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 2012 at 5:45
	P.S. I'm too new to vote otherwise I'd give you thumbs up! – PatrickT May 5 2012 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 2012 at 5:48

delay differential equation

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.