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I have two nonlinear equations with two unknowns, $x$ and $y$, and many symbolic parameters, $a, b, c, d$. They look someting like:

$\frac{1}{x}(1+x)^a (1-b)(cx+dy)=0$

$(\frac{1}{x^2}+b)(1+x)^d+(1+by)\frac{1}{x}=0$

{
 ((-1 + b) (1 + x)^a (c x + d y))/x == 0, 
 ((1 + x)^d (b + x^(-2))) + (1 + b y)/x == 0
}

I would like to find a solution for this system. But I expect there would be no closed form expression for $x$ and $y$ since my equations are much more complicated than these. In which case, I would like to know, at least, how changes in each of the parameter leads to changes in each of $x$ and $y$, using implicit funciton theorem. For example,

$\frac{\partial x}{\partial a}=-\frac{\frac{\partial F}{\partial a}}{\frac{\partial F}{\partial x}} \lessgtr 0 ?$

where $F$ is an implicit function established from the two nonlinear equations of the system.

I'm not sure what Mathematica command I should use to do these. I know Solve will not be of help for this kind of complicated nonlinear equations.

I'm afraid whether this is an appropriate question for this site; whehter it is too specific without general benefit, etc. But I have been searching for an answer to this problem for a while without success. Any help will be greatly appreciated.

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  • $\begingroup$ Oh, yes, I will clarify that right away. $\endgroup$
    – jim
    Sep 17 '14 at 15:31
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    $\begingroup$ (b + x^2) or (b + x^-2) ? $\endgroup$
    – rhermans
    Sep 17 '14 at 17:15
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    $\begingroup$ @rhermans Oh, the latter one. $\endgroup$
    – jim
    Sep 17 '14 at 17:30
  • $\begingroup$ Take a look here Can mathematica solve this equation? where I demonstrated how you could proceed defining an appropriate function and solving related equations when specified some variables. $\endgroup$
    – Artes
    Sep 18 '14 at 8:02
  • $\begingroup$ @ Artes I really appreciate for this helpful information. I've just went through your post. I'm not sure if I understood completely, which I doubt. But it seems you are saying my two equation system cannot be solved. In fact, I know I cannot obtain closed form solutions. What I am expecting is at least to see how changes in one symbolic parameters lead to changes in each of the two variables, x and y, using implicit function theorem. Do you think this is possible? Thank you so much. $\endgroup$
    – jim
    Sep 20 '14 at 22:52

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