Re: Equation - problem

*To*: mathgroup at smc.vnet.net*Subject*: [mg49059] Re: Equation - problem*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Tue, 29 Jun 2004 04:50:53 -0400 (EDT)*Organization*: The University of Western Australia*References*: <cbi7ls$lf2$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

In article <cbi7ls$lf2$1 at smc.vnet.net>, rak at dami-rz.pl (rafal) wrote: > I need a good package for computing the equation like below: > > Reduce[r^2*(x^q-1)*((-r*x*(x^q-1))^q-1)==1&&r>=0&&0<=x <=1&&q>0, > x,Reals] Andrzej Kozlowski has answered your question, but I would add that you can determined closed-form solutions for integer q and that you can use ImplicitPlot to visualize the solution curves. Defining f[q_][x_,r_] = r^2 (x^q - 1) ((-r x (x^q - 1))^q - 1) == 1; and g[q_][x_,r_] = f[q][x,r] && r >= 0 && 0 <= x <= 1 && q > 0; here we determine r in terms of x for q = 2, Reduce[g[2][x,r], r] or x in terms of r, Reduce[g[2][x,r], x] After loading the Graphics stubs, <<Graphics` we can visualize the trifurcation at {x,r} = {1/Sqrt[2], 2}. ImplicitPlot[f[2][x,r], {x, 0, 1}, {r, 0, 4}, PlotPoints -> 100, AxesLabel -> {x, r}]]; Animations as q varies Table[ImplicitPlot[f[q][x,r], {x, 0, 1}, {r, 0, 4}, PlotPoints -> 100, AxesLabel -> {x, r}], {q, 0.05, 4, 0.1}]; or as r varies Table[ImplicitPlot[f[q][x, r], {x, 0, 1}, {q, 0.05, 4}, PlotPoints -> 100, AxesLabel -> {x, q}], {r, 1, 4, 0.1}]; are quite illuminating. Cheers, Paul -- Paul Abbott Phone: +61 8 9380 2734 School of Physics, M013 Fax: +61 8 9380 1014 The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling Highway Crawley WA 6009 mailto:paul at physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul