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

\[ -{\left(t - 5\right)} {y'''} e^{t} = {\left(t + 4\right)} {\left(t + 2\right)} y + 3 \, {y''} + e^{\left(-5 \, t\right)} \hspace{2em} x( 9 )= 2 \]