Determine the largest interval for which a solution to the initial value problem exists and is unique.

$($a$)(t^2-9)y^{\text{'}\text{'}}-7t{y}^{\text{'}}+y=ln(8-t),y(4)=Y_0,y^{\text{'}}(4)=Y_1.$

$($b$)ln(t+4)y^{\text{'}\text{'}}-y^{\text{'}}+(t+1)y=\sqrt[\phantom{2}]{5-t},y(0)=Y_0,y^{\text{'}}(0)=Y_1.$