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

\[ {\left(t + 5\right)} t {y'''} = -t^{2} - {\left(t - 5\right)} y - 4 \, {y''} - 16 \hspace{2em} x( -1 )= 1 \]