Re: Is a solution possible to this exponential equation?

*To*: mathgroup at smc.vnet.net*Subject*: [mg7047] Re: [mg6994] Is a solution possible to this exponential equation?*From*: "w.meeussen" <w.meeussen.vdmcc at vandemoortele.be>*Date*: Sat, 3 May 1997 22:04:41 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

At 14:48 1-05-97 -0400, Michael Hucka wrote: >I have a pair of equations involving exponentials that I'd like to be able to >solve for one of the variables, but I can't seem to find a way to do it. MMA >3.0's Solve operator complains its usual complaint about the equation >involving transcendental functions. Quite possibly there is no analytical >solution, but I'd like to find out from the experts out there about what >approaches one might try. The equations are as follows: > > 1/2 = kc * Exp[ -a * w^2 * x^2 ] - ks * Exp[ -a * b^2 * w^2 * x^2] > >where a and b are constants, and kc, ks, w and x are variables. There is an >additional condition, > > kc - ks = Sqrt[ 1/2 ] > >So there are 2 equations and 4 unknowns. I'd like to solve this for w, or >rather, to express w in terms of the other unknowns. The problem, of course, >is that w is in the exponent of two of the terms. > >What is the right approach to take in cases like this? This is actually >something I've come across before in my work, but haven't had much luck with. > >-- >Mike Hucka hucka at umich.edu http://www.eecs.umich.edu/~hucka University > PhD to be, computational models of human visual processing (AI Lab) of > UNIX systems administrator & programmer/analyst (EECS DCO) Michigan > > > hi, look what happens if you substitute rule1=ks-> kc - Sqrt[ 1/2 ] rule2= Exp[ -a * w^2 * x^2 any_:1 ] -> u^any it= 1/2 ==kc * Exp[ -a * w^2 * x^2 ] - ks * Exp[ -a * b^2 * w^2 * x^2] it/.rule1/.rule2 **************************************************** 1/2 == kc*u - (-(1/Sqrt[2]) + kc)*u^(b^2) **************************************************** so you have in essence alfa x^n + beta x + gamma ==0 no symbolic solution to that I'm afraid. Numerics go ok though. if b and kc are constants with a known numerical value, you can solve for u using NSolve. The rest is simple : you can take the log and find (w * x). To separate w and x needs further conditions. does this help any? Dr. Wouter L. J. MEEUSSEN eu000949 at pophost.eunet.be w.meeussen.vdmcc at vandemoortele.be