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
- 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 initial-values
and
)
Then,
Therefore,


if and only if on
where
is the unique solution of the initial-value problem