System of cross-polynomials of variable degree

[ydydry]

Dear all,

I am stuck with an apparently easy system of 3 simultaneous equations that has to be solved in x,y and z. The system is the following:

$(y+z) / ((x+y+z)²)=ax^{α-1}$
$(x+z) / ((x+y+z)²)=by^{α-1}$
$(x+y) / ((x+y+z)²)=cz^{α-1}$

The parameters (a,b,c) and the variables (x,y,z) are all in ℝ₊. The parameter α is in the open interval (0,1), so that the problem is not trivial. The question is for which α there exists a closed-form solution, and for which α there exists none, given a general vector (a,b,c).

Proving that there always exist a unique solution is trivial as in each equation the right-hand sides are always increasing, whereas the left-hand sides are always decreasing in the corresponding variable (x for the first, y for the second, z for the third).

Hence, my problem is in the existence of a closed-form solution.

Many thanks!

 

[fresh_42]

It looks as if we first should substitute the variables $x,y,z$ by $\frac{x}{n},\frac{y}{n},\frac{z}{n}$ with $n=x+y+z$ which makes the left hand side linear and $n$ should be constant or at least can be put into the parameters on the right hand side. I suppose that the equations are then easier to solve.