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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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