Mylee Underwood

2022-07-07

I wonder whether the system of equations and inequations below have a solution. If there are solutions, what are they? A numerical solution is also desired.
$\left\{\begin{array}{l}\frac{{c}_{1}}{1-{x}_{1}}+\frac{{c}_{2}}{1-{x}_{2}}+\frac{{c}_{3}}{1-{x}_{3}}=0\\ \frac{{c}_{1}}{1-{x}_{4}}+\frac{{c}_{2}}{1-{x}_{5}}+\frac{{c}_{3}}{1-{x}_{6}}=0\\ {c}_{1}\mathrm{ln}\frac{{x}_{1}}{1-{x}_{1}}+{c}_{2}\mathrm{ln}\frac{{x}_{2}}{1-{x}_{2}}+{c}_{3}\mathrm{ln}\frac{{x}_{3}}{1-{x}_{3}}=0\\ {c}_{1}\mathrm{ln}\frac{{x}_{4}}{1-{x}_{4}}+{c}_{2}\mathrm{ln}\frac{{x}_{5}}{1-{x}_{5}}+{c}_{3}\mathrm{ln}\frac{{x}_{6}}{1-{x}_{6}}=0\\ \frac{{x}_{1}\left(1-{x}_{1}\right)}{{x}_{4}\left(1-{x}_{4}\right)}=\frac{{x}_{2}\left(1-{x}_{2}\right)}{{x}_{5}\left(1-{x}_{5}\right)}=\frac{{x}_{3}\left(1-{x}_{3}\right)}{{x}_{6}\left(1-{x}_{6}\right)}>1\end{array}$
where ${c}_{1},{c}_{2},{c}_{3}$ are constants, and ${x}_{i}\in \left(0,1\right),i=1,2,3,4,5,6$

haingear8v

A simple set of solutions are any $\left({c}_{1},{c}_{2},{c}_{3}\right)$ such that ${c}_{1}+{c}_{2}+{c}_{3}=0$, ${x}_{1}={x}_{2}={x}_{3}=\frac{1}{2}$, ${x}_{4}={x}_{5}={x}_{6}<\frac{1}{2}$