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

\[ -{\left(t + 4\right)} {\left(t - 5\right)} {y'''} - {\left(t - 1\right)} y e^{t} = t^{2} + 5 \, {y''} + 25 \hspace{2em} x( 1 )= 7 \]