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MathGroup Archive 2004

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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


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