Determine the largest interval where there is guaranteed to be a unique solution for the following IVP. Justify why this is the largest interval.

$\displaystyle \textbf{X}'=\left(\begin{array}{cc} \ln t & 6 \\ 3t^2 & \sin t\end{array}\right)\textbf{X} + \left(\begin{array}{c} \sqrt{5-t}\\ \cot t \end{array}\right), \textbf{X}(\frac{5\pi}{2})=\left(\begin{array}{c}0 \\1 \end{array}\right)$