Find all solutions of the nonlinear differential equation
on the interval satisfying the initial conditions
 when ;
 when ;
 a finite limit as .

From a previous exercise (Section 8.5, Exercise #13) we know that a function which is never zero on an interval is a solution of the initial value problem
 In the present problem we have the equation
Therefore, to apply the previous exercise we have
Hence, is a solution to the given equation if and only if where is the unique solution to
We can solve this using Theorem 8.3 (page 310 of Apostol), first computing
giving us
Therefore,
 From part (a) we have which implies . Since we then have
 Let where is some finite real number. Then, (using the calculations we already completed in part (a) and just changing the initialvalues and )
Then,
Therefore,
if and only if on where is the unique solution of the initialvalue problem