Consider the first order differential equation y$($prime$)=($x$/($x$^2-25)),$ y$=($e$^$x$/($x$-8))$ For each of the initial conditions below, determine the largest interval on which the existence and uniqueness theorem for first order linear differential equations guarantees the existence of a unique solution.

$($a$)$ y$(-6)=-0\mathrm{.}5$ Interval

$($b$)$ y$(3\mathrm{.}5)=2\mathrm{.}6$ Interval

$($c$)$ y$(0)=0$ Interval

$($d$)$ y$(5\mathrm{.}5)=-3\mathrm{.}14$ Interval

$($e$)$ y$(11)=1\mathrm{.}7$ Interval