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

\[ -{\left(t + 4\right)} {\left(t - 5\right)} y - {\left(t - 2\right)} {y''} - e^{\left(-5 \, t\right)} = -{y'} \hspace{2em} x( 0 )= 5 \]