Find the largest time interval where the system is guaranteed to have a unique solution.

Question

I need to find the largest interval of uniqueness for the system. $$ \vec y'= \begin{bmatrix} 1 & t & t^2 \\ \frac{t}{t+1} & \frac{1}{t} & \frac{1}{t-5} \\ sin(t) & cos(t) & tan(\frac{t}{2}) \\ \end{bmatrix}\vec y + \begin{bmatrix} sin(t) \\ ln(5t)\\ t^2 \\ \end{bmatrix}, \vec y(1)=\begin{bmatrix} 3 \\ 2\\ 5\\ \end {bmatrix} $$ I am struggling to understand differential equations and any advice on the problem would be greatly appreciated. I'm confused as to how to solve the problem with non-constant values in the matrix. Is there a way to answer the question without solving for $\vec y$?