Solve for O_2
\left\{\begin{matrix}O_{2}=\frac{O_{3}^{\frac{2}{3}}}{\sqrt[3]{k_{0}}}\text{, }&O_{3}\neq 0\text{ and }k_{0}\neq 0\text{ and }S\neq 0\\O_{2}\neq 0\text{, }&S\neq 0\text{ and }k_{0}=0\text{ and }O_{3}=0\end{matrix}\right.
Solve for O_3
O_{3}=\sqrt{k_{0}O_{2}^{3}}
O_{3}=-\sqrt{k_{0}O_{2}^{3}}\text{, }\left(k_{0}\geq 0\text{ and }S\neq 0\text{ and }O_{2}>0\right)\text{ or }\left(k_{0}\leq 0\text{ and }S\neq 0\text{ and }O_{2}<0\right)
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