Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP.

\[ -{\left(t + 6\right)} {\left(t - 1\right)} {y} - {\left(t^{2} + 1\right)} {y'} = {\left(t - 6\right)} e^{t} \hspace{2em} x( 5 )= -1 \]