Solve T=2pisqrt{frac{m_{0}+(frac{m_{s}}{3})}{k}}

$$T = 2 \pi \sqrt { \frac { m _ { 0 } + ( \frac { m _ { s } } { 3 } ) } { k } }$$

Solve for T (complex solution)

$$T=\frac{2\pi \sqrt{\frac{3\left(3m_{0}+m_{s}\right)}{k}}}{3}$$

$$k\neq 0$$

Solve for k (complex solution)

$$\left\{\begin{matrix}k=\frac{4\times \left(\frac{\pi }{T}\right)^{2}\left(3m_{0}+m_{s}\right)}{3}\text{, }&m_{0}\neq -\frac{m_{s}}{3}\text{ and }|\frac{arg(T^{2})}{2}-arg(T)|<\pi \text{ and }T\neq 0\\k\neq 0\text{, }&T=0\text{ and }m_{0}=-\frac{m_{s}}{3}\end{matrix}\right.$$

Solve for T

$$T=\frac{2\pi \sqrt{\frac{3\left(3m_{0}+m_{s}\right)}{k}}}{3}$$

$$\left(k>0\text{ or }m_{0}\leq -\frac{m_{s}}{3}\right)\text{ and }\left(k<0\text{ or }m_{0}\geq -\frac{m_{s}}{3}\right)\text{ and }k\neq 0$$

Solve for k

$$\left\{\begin{matrix}k=\frac{4\times \left(\frac{\pi }{T}\right)^{2}\left(3m_{0}+m_{s}\right)}{3}\text{, }&T>0\text{ and }m_{0}\neq -\frac{m_{s}}{3}\\k\neq 0\text{, }&m_{0}=-\frac{m_{s}}{3}\text{ and }T=0\end{matrix}\right.$$