(c) The critical point (0
,
0) is an asymptotically stable spiral point. The critical point (0
,
1)
is a saddle point, therefore, unstable.
The critical point (

2
,

2) is a saddle point,
therefore, unstable. The critical point (3
,

2) is a saddle point, therefore, unstable.
(d) For (0
,
0), the basin of attraction is bounded below by the line
y
=

2, to the right by a
trajectory passing near the point (2
,
0), to the left by a trajectory heading towards (and
then away from) the unstable critical point (0
,
1), and above by a trajectory heading
towards (and then away from) the unstable critical point (0
,
1).
5.
(a) The equation
x
(2

x

y
) = 0 implies
x
= 0 or
x
+
y
= 2.
If
x
= 0, the equation

x
+ 3
y

2
xy
= 0 implies
y
= 0. If
x
+
y
= 2, then the equation

x
+ 3
y

2
xy
= 0
can be reduced to
y
2

1 = 0. Therefore,
y
=
±
1. Now if
y
= 1, then
x
= 1. If
y
=

1,
then
x
= 3. Therefore, the critical points are (0
,
0), (1
,
1) and (3
,

1).
(b)
–6
–4
–2
0
2
4
6
y
–6
–4
–2
2
4
6
x
(c) The critical point (0
,
0) is an unstable node. The critical point (1
,
1) is a saddle point,
therefore, unstable. The critical point (3
,

1) is an asymptotically stable spiral point.
(d) For (3
,

1), the basin of attraction is bounded to the left by the
y

axis and above by a
trajectory heading into (and away from) the unstable critical point (1
,
1).