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Math 3301
Homework Set 1 – Solutions
10 Points
1. (2 pts)
From the differential equation we can see
that the derivative will be zero at :
2,2,4
y
= −
.
A sketch of the direction field and a few solutions is
shown to the right.
From this we can see that the
long term behaviors are,
( ) ( )
( ) ( )
( ) ( )
( ) ( )
04
4
2
2
02
2
y
yt
y
y
y
>
→∞
=
=
−<
<
→
≤−
→−
4. (3 pts)
( ) ( )
( )
( )
( )
( )
2
3
36
3 2
2 3
2
23
3
3
1
3
26
1
3
ln
33
3
t
dt
t
tt
t
t
t
t
y
t
t
t
y dt
t
dt
t
y
t
c
y t
t
t
ct
µ
−−
− −
∫
′
−+ =
=
=
′
=
→
=
−+
=
⌠
⌡
∫
e
ee
e
e
e
( )
( )
( )
( )
6
3
3
23 3
4
4
14
3
3
01
y
c
c
yt t
t
t
−
−
= −=
→
=
⇒
=
e
e
e
5. (2 pts)
We know from Calc I that relative extrema occur at critical points and critical points are those
points where the derivative is zero or doesn’t exist.
However, we’ve been told that the derivative exists
and is continuous everywhere so that means that at the critical point that gives the relative maximum,
let’s call it
1
2
c
t
=
,
we
must have
( )
1
2
0
y
′
=
and we want to determine
( )
1
2
y
so all we need to do is
“plug”
1
2
c
t
=

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