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

\[ -{\left(t + 5\right)} {\left(t - 2\right)} y + 2 \, {y''} - e^{\left(3 \, t\right)} = {\left(t - 5\right)} {y'''} e^{t} \hspace{2em} x( 6 )= 3 \]