Solving “ugly” equations

$k$, $c_1$ and $c_2$ are unkowns, while others are given. How can I solve the equations?

$$\begin{cases} k\left( {c_1 e^{\pi k} + \frac{c_2}{e^{\pi k}}} \right)^{(1 + 2\theta)/\theta} = - \theta^2\tau \\ k(c_1 + c_2)^{(1 + 2\theta)/\theta} = \theta^2\tau \frac{e^ -\theta\eta - 1}{e^{-\theta\eta} + 1}\\ k\left( \frac{c_1}{e^{\pi k}} + c_2 e^{\pi k} \right)^{(1 + 2\theta)/\theta} = \theta^2\tau \end{cases}$$

• If you treat $k$ as a "known" variable, you can easily modify this system to be a system of three linear equations in $c_1$ and $c_2$. Solving this system would yield a solution for $c_1$ and $c_2$ in terms of $k$, as well as a potentially useful consistency criterion in terms of $k$ that must be satisfied. Solving for $k$ is going to be a whole lot less fun, and probably involve the Lambert W function. – Theo Bendit Jun 27 '17 at 4:14
• Is there any restriction on the unkwons and constants? All of them can be any real number or is there any that has to be strictly positive/negative? – AugSB Jun 27 '17 at 9:49
• What is $-1^{ \theta / (1 + 2\theta)}$? – N74 Jun 27 '17 at 22:09
• The constants are all positive real numbers and no restriction on unkwons @AugSB – Frank Jun 28 '17 at 1:43
• $\theta \in \mathbb{R_+}$ @N74 – Frank Jun 28 '17 at 1:43