Solve 0+(1-omegar)r=varepsilon-omegar

$$0 + ( 1 - \omega r ) r = \varepsilon - \omega r$$

Solve for r (complex solution)

$$\left\{\begin{matrix}r=-\frac{\sqrt{1+2\omega +\omega ^{2}-4\epsilon \omega }-\omega -1}{2\omega }\text{; }r=\frac{\sqrt{1+2\omega +\omega ^{2}-4\epsilon \omega }+\omega +1}{2\omega }\text{, }&\omega \neq 0\\r=\epsilon \text{, }&\omega =0\end{matrix}\right.$$

Solve for r

$$\left\{\begin{matrix}r=-\frac{\sqrt{1+2\omega +\omega ^{2}-4\epsilon \omega }-\omega -1}{2\omega }\text{; }r=\frac{\sqrt{1+2\omega +\omega ^{2}-4\epsilon \omega }+\omega +1}{2\omega }\text{, }&\left(\omega \neq 0\text{ and }\epsilon \geq 0\text{ and }\epsilon \leq 1\right)\text{ or }\left(\omega \neq 0\text{ and }\omega \leq -2\sqrt{\epsilon ^{2}-\epsilon }+2\epsilon -1\right)\text{ or }\left(\omega \neq 0\text{ and }\epsilon >0\text{ and }\epsilon <1\right)\text{ or }\left(\omega \neq 0\text{ and }\omega \geq 2\sqrt{\epsilon ^{2}-\epsilon }+2\epsilon -1\right)\\r=\epsilon \text{, }&\omega =0\end{matrix}\right.$$

Solve for ε

$$\epsilon =r\left(1+\omega -r\omega \right)$$