Solving General System of Non-Linear DEs With Many Constant Terms Help (Mathematica)

How can a system of non-linear differential equations be solved using Mathematica?

More specifically, I understand the process that needs to be executed, but I don't understand how to make it work syntactically.

The process as I understand it:

Four expressions are given in four unknowns. Let the unknowns be expressed as x[n]. Each expression has three or less constant terms, which I'll just express as a[n]. These expressions are determined from an ideal, physical system which is being considered, so the constants can all be manipulated through experiment.

dx[2]/dt == a[1]x[3] - a[2]x[2] - a[3]x[2] == 0

dx[3]/dt == a[4]x[4]/(a[5] - x[4]) - a[6]x[3] == 0

dx[1]/dt == a[7]x[4]/(a[8] - x[4]) - a[9]x[1] == 0

x[4]/a[10] == (x[2]/a[11])^n/(1+(x[2]/a[11])^n

To make the situation simpler, the top three differential expressions are set equal to zero to indicate a steady state process, so now I'm interested in what constitutes steady state conditions.

In particular, I would like to get an expression in the form f(x[1]) = g(x[1]), where f(x[1]) is simply x[1]. Then, I can plot both f and g on a graph and find their points of intersection.

My problem is that this is a burdensome calculation and I have no idea how to set a system like this up to be solved in the way described. I've looked up several function definitions on Wolfram help forums, including solve, eliminate, reduce and a few others, but none of the calculations simplified anything.

Is there an alternative method?
 
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Re: Solving General System of Non-Linear DEs With Many Constant Terms Help (Mathemati

How can a system of non-linear differential equations be solved using Mathematica?
...
dx[2]/dt == a[1]x[3] - a[2]x[2] - a[3]x[2] == 0

dx[3]/dt == a[4]x[4]/(a[5] - x[4]) - a[6]x[3] == 0

dx[1]/dt == a[7]x[4]/(a[8] - x[4]) - a[9]x[1] == 0

x[4]/a[10] == (x[2]/a[11])^n/(1+(x[2]/a[11])^n

To make the situation simpler, the top three differential expressions are set equal to zero to indicate a steady state process, so now I'm interested in what constitutes steady state conditions. ...
Unfortunately I can not realize your goal here.

If you interested in steady state conditions, you have to consider the right equations from the top three expressions + the fourth and try to solve this algebraic system. It’s really easy! (There is misprint in fourth expression! Examine parenthesis.)

Otherwise, you have to consider the left equations from the top three expressions + the fourth and try to solve this differential system. Splitting of this system leads to necessity to solve 3rd order nonlinear ODE - it is problematic in general case. (The simplest case - when n=0 leads to analytic solution)
 

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