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

\[ -{\left(t - 5\right)} {y'''} e^{t} - 5 \, {y''} - e^{t} = {\left(t + 4\right)} t y \hspace{2em} x( 8 )= 4 \]