@MaPal93 The quintic has all coefficients positive, except for the constant term, which is negative. So if you removed the constant term and plotted it, it goes through the origin and then is monotonically increasing as x increases. Adding the (negative) constant term brings the curve down, and will mean there is exactly one positive root. (Sturm sequences tell something about location of real roots in the more general case; Maple has them only for constant coefficients, but you could apply them for the general case, though I'm not sure this will do much better than solve( , parametric) or the other solvers with inequalities.)
I am trying to formulate your three equations as much as possible in matrix form. Of course you probably got them from some matrices in the first place so I am probably just reverse engineering here. I don't know much about statistics but am reasonably fluent with matrices. I see the matrix origin of the betas. Then in your equations:
the matrix origins, if any, are much harder to see. Would you mind answering some questions that might help me with this?
1. In these equations you introduce new variables Cov_S12, Var_S1, Var_S2, Var_nu1, Var_nu2, Cov_nu12 - is there any relationship between these and the earlier variables?
2. You also have the variables Var_u1, Var_u2, Var_u3 that were used previously. In the earlier equations that you solved, these (and all the other variances and covariances) got multiplied by gamma. Should these also be multiplied by gamma here? Or perhaps gamma multiplied the other new variables as well and then cancelled in these equations?
3. Is theta__12 = theta__21?
4. Since beta__12 is not the same as beta__21, 2*beta__11*beta__12*Cov_S12 is not the same as beta__11*(beta__12+beta__21)*Cov_S12. The latter form seems more natural for a matrix formulation. Is it correct as written or is there perhaps a hidden assumption here?
Here are my manipulations so far: Matrix4.mw
If you think this is a profitable approach, I'd be happy to elaborate on this.