(a) We're given that
. We have four cases to consider: (1)
; (2)
; (3)
; and (4)
.
In case (1), both
and
, so
, which means
is increasing and converging to
as
.
In case (2),
and
again, so
as well, so
is increasing again and converging to
from below.
In case (3),
and
, so
, which means
is decreasing and converging to
from above as
.
Finally, in case (4), both
and
, so tha t
, and so
is increasing and diverging as
.
So a simplified phase portrait for the solution
might look something like
where the vertical axis represents
.
(b) If
, then
, which has the phase portrait
so that
when
;
from below when
; and
from below as
in either case of
or
.
(c) With
and
, we have the separable ODE
With the initial value
, we have
With
, we would get
In both cases, we're assuming that
, since we are given that
from the start. In part (b), we found that
as
(regardless of whether
is smaller or larger than
, in fact), so we only need to verify that the solution we found also conforms to this trend. We have
which confirms the conclusion of the phase portrait.