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

\[ 0 = -{\left(t + 2\right)} {\left(t - 5\right)} y - {\left(t + 5\right)} {y'''} e^{t} + 5 \, {y''} - e^{\left(2 \, t\right)} \hspace{2em} x( -7 )= -1 \]