I have the following system of two equations with two variables $0\leq{}x_c\leq{}1$ and $0\leq{}x_n\leq{}1$:
${\lambda{}}_c\left(-{\tau{}}_cx_c\frac{2C_c-r_cx_c}{C_c{\left(C_c-r_cx_c\right)}^2}+\frac{1}{C_m}.\frac{{\tau{}}_cr_c\left(1-x_c\right)+{\tau{}}_nr_n\left(1-x_n\right)}{C_m-r_c\left(1-x_c\right)-r_n\left(1-x_n\right)}\right)+r_c{\tau{}}_c\frac{\left(1-x_c\right){\lambda{}}_c+\left(1-x_n\right){\lambda{}}_n}{{\left(C_m-r_c\left(1-x_c\right)-r_n\left(1-x_n\right)\right)}^2}=0$
${\lambda{}}_n\left(-{\tau{}}_nx_n\frac{2C_n-r_nx_n}{C_n{\left(C_n-r_nx_n\right)}^2}+\frac{1}{C_m}.\frac{{\tau{}}_cr_c\left(1-x_c\right)+{\tau{}}_nr_n\left(1-x_n\right)}{C_m-r_c\left(1-x_c\right)-r_n\left(1-x_n\right)}\right)+r_n{\tau{}}_n\frac{\left(1-x_c\right){\lambda{}}_c+\left(1-x_n\right){\lambda{}}_n}{{\left(C_m-r_c\left(1-x_c\right)-r_n\left(1-x_n\right)\right)}^2}=0$
The denominators for all fractions are non-zero and positive. All parameters are also positive real numbers. I am trying to solve for these coefficients but I cannot find a way. I tried several difference conversions to convert them to a simpler form but I did not succeed. Is there any way to solve this?
I thought for simplicity, it can be considered it as follows:
$\Rightarrow{}{\lambda{}}_c\left(-{\tau{}}_c\left(1-y_c\right)\frac{2C_c-r_c\left(1-y_c\right)}{C_c{\left(C_c-r_c\left(1-y_c\right)\right)}^2}+\frac{1}{C_m}.\frac{{\tau{}}_cr_cy_c+{\tau{}}_nr_ny_n}{C_m-r_cy_c-r_ny_n}\right)+r_c{\tau{}}_c\frac{y_c{\lambda{}}_c+y_n{\lambda{}}_n}{{\left(C_m-r_cy_c-r_ny_n\right)}^2}=0$
$\Rightarrow{}{\lambda{}}_n\left(-{\tau{}}_n\left(1-y_n\right)\frac{2C_n-r_n\left(1-y_n\right)}{C_n{\left(C_n-r_n\left(1-y_n\right)\right)}^2}+\frac{1}{C_m}.\frac{{\tau{}}_cr_cy_c+{\tau{}}_nr_ny_n}{C_m-r_cy_c-r_ny_n}\right)+r_n{\tau{}}_n\frac{{\lambda{}}_cy_c+{\lambda{}}_ny_n}{{\left(C_m-r_cy_c-r_ny_n\right)}^2}=0$
where $y_c=1-x_c$ and $y_n=1-x_n$. But I don't know how to solve it.
Thanks!