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I have an equation, that I want to solve numerically. The equation is

$$x(1-cx)-\frac{xy}{1+x^{1+\gamma}}=0$$

The problem here is the $\gamma$. What I want to achieve is a plot of the solutions in the $(x,y,\gamma)$-space. Actually, I want to plot ther borders of the surface, to make the other surfaces visible as well, but this is another issue. I have a fixed value of $c$ and I realized, that symbolically, this would get quite hard. So my first approach was something like this:

$Assumptions = gamma >= 0 && x>=0 && z>=0;
c = 0.1732;
dx := x*(1 - c*x) - x*y/(1 + y^(1 + gamma))
Reap[Do[Sow[NSolve[dx==0,p,Reals]],{gamma,0,1,0.05}]]

I get the following error

NSolve was unable to solve the system with inexact coefficients. 

Isn't there a simple solution to obtain all positive real solutions for $x$, assuming that $y>= 0$?

Thank you!

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  • $\begingroup$ Use Rationalize around c and the step size to get rid of the message. $\endgroup$ – user21 Sep 24 at 8:15
  • $\begingroup$ Reduce[x*(1 - c*x) - x*y/(1 + y^(1 + gamma)) == 0, gamma, Reals] returns what looks like a legitimate solution fairly quickly $\endgroup$ – gpap Sep 24 at 8:43
  • $\begingroup$ @user21 This indeed helps to get rid of the method. But I am searching for a solution of some form $x(y,\gamma)$ and I am only getting ConditionalExpression with this. Any idea? $\endgroup$ – gumpel Sep 24 at 9:01
  • $\begingroup$ @gpap This may be right. But I can't confirm this because I don't know how to plot these expressions.. Any idea on this? $\endgroup$ – gumpel Sep 24 at 9:04
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    $\begingroup$ What's z? Do you mean y? $\endgroup$ – Chris K Sep 24 at 9:13
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If you're only interested in the plot, not the functional form of x[y, gamma], then you can use ContourPlot3D:

ContourPlot3D[dx == 0, {gamma, 0, 1}, {y, 0, 5}, {x, 0, 1/c},
  AxesLabel -> {gamma, y, x}]

Mathematica graphics

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  • $\begingroup$ Perfect, thank you. Do you, by chance, know whether/how it is possible to plot only the borders of the surfaces? $\endgroup$ – gumpel Sep 24 at 9:52
  • $\begingroup$ @gumpel Sorry, I don't know about plotting the borders only. Maybe look at ContourPlot3D options. $\endgroup$ – Chris K Sep 24 at 9:59

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