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

\[ 3 \, {y''} = {\left(t - 2\right)} {\left(t - 5\right)} {y'''} + {\left(t + 5\right)} y e^{t} + e^{\left(-2 \, t\right)} \hspace{2em} x( 3 )= 8 \]