In Exercises $13-16,$ an initial$-$value problem is given.

$($a$)$ Find a formula for the solution.

$($b$)$ State the domain of definition of the solution.

$($c$)$ Describe what happens to the solution as it approachs the limits of its domain of definition. Why can't the solution be extended for more time?

$14.\frac{dy}{dt}=\frac{1}{(y+1)(t-2)},y(0)=0$