From: israel@math.ubc.ca (Robert Israel)
Subject: Re: How To Solve a D.E. Like This?
Date: 14 Jun 1999 23:16:28 GMT
Newsgroups: sci.math
Keywords: delay differential equation

In article <7k1ohi$d4k$1@nnrp1.deja.com>,
 t_n_southton@hotmail.com writes:
> F(X - 10) = 4 * F'(X)
> 
> Are there general methods to solve differential equations that calls
> upon itself "in some other point in time?"

It's called a "delay differential equation", and in general these are
quite a bit more trickier than differential equations.  In
the case of linear constant-coefficient equations such as this, you might
start by "rounding up the usual suspects", which are exponentials (and then
perhaps polynomials * exponentials).  So you look for solutions of the
form F(X) = exp(a x), and you find that this works if exp(-10 a) = 4 a.
That equation has one real root: in Maple's notation a = LambertW(5/2)/10,
which is approximately .095858635672870291217.  There are also complex 
roots, obtained by using non-principal branches of LambertW, and these
also produce solutions (their real and imaginary parts are solutions
involving exponentials and sines or cosines).  Thus another root is
approximately -.061411863562923902066 + .45790702365466800001 i, corresponding
to solutions of your equation exp(b x) cos(c x) and exp(b x) sin(c x) where
b = -.061411863562923902066 and c = .45790702365466800001 (approximately).
One of the symptoms of the fact that delay differential equations are
trickier than differential equations is the fact that you get infinitely
many linearly independent solutions.

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2