Re: Problem solving a difference equation

*To*: mathgroup at smc.vnet.net*Subject*: [mg68332] Re: Problem solving a difference equation*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Tue, 1 Aug 2006 07:00:04 -0400 (EDT)*Organization*: The University of Western Australia*References*: <eaer8e$5a4$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

In article <eaer8e$5a4$1 at smc.vnet.net>, "aTn" <ayottes at dms.umontreal.ca> wrote: > I am trying to solve the following difference equation: > > s[i] = s[i-1] ( 1 + d ( a/ (b + c s[i-1]) - 1 )) > > , where a,b,c and d are fixed real numbers. > > Here is what happens when I try RSolve: > > In[1] := RSolve[s[i] == s[i-1] ( 1+ d ( a/ (b + c s[i-1]) - 1)), > s[i],i] > Out[1] := RSolve[s[i] == s[i-1] (1+d ( a/ (b + c s[i-1]) - 1)), > s[i],i] > > ,that is, nothing much happens :). > > I have a few questions: > > 1) Why are the input and output of RSolve the same ? Because RSolve cannot solve the difference equation, at least as posed. > 2) Can one solve the difference equation using RSolve ? I do not know if RSolve can solve rational difference equations. Note that as i -> Infinity, the limiting behaviour is Solve[s == s ((a/(b + c s) - 1) d + 1), s] {{s -> 0}, {s -> (a - b)/c}} > 3) If the answer to question (2) is no, then do you have any > suggestions on how to solve the equation. It may be interesting to know how the recurrence arises. Have a look at the literature on (first order) rational difference equations. For example, Yanagihara [1] shows that for any rational function R(y), the difference equation y[z + 1] == R(y[z]) has a non-trivial meromorphic solution. Also, perhaps "generatingfunctionology" by Herbert Wilf, at http://www.math.upenn.edu/~wilf/DownldGF.html will help. [1] Yanagihara N 1980 Meromorphic solutions of some difference equations Funkcial. Ekvac. 23 30926 Cheers, Paul _______________________________________________________________________ Paul Abbott Phone: 61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) AUSTRALIA http://physics.uwa.edu.au/~paul