454
Chapter 9
7.
The first equation
3
40
xx
−=
gives
x
=

2,
x
=
0,
or
x
=
2
at a critical point.
Then the second equation
20
xy
gives
y
=

1,
y
=
0,
or
y
=
1,
respectively.
The only figure among Figs. 9.1.11 through 9.1.18 showing three critical points at
(–2, –1),
(0, 0), and (2, 1) is Fig. 9.1.14.
Thus the critical points of the given
autonomous system are the spiral point (0, 0) and the saddle points (

2, 1) and (2, 1)
shown in Figure 9.1.14 in the text.
8.
The second
2
0
yx
−− =
equation gives
y
=

x
2
at a critical point.
Substitution of this
in the first equation
2
0
xyx x
y
−− + =
then gives
x

x
3
=
0,
so
x
=

1,
x
=
0,
or
x
=
1.
The only figure among Figs. 9.1.11 through 9.1.18 showing three critical points
at
(–1, –1),
(0, 0), and (1, –1) is Fig. 9.1.16.
Thus the critical points of the given
autonomous system are the spiral point (

1,

1), the saddle point (0, 0), and the node
(1,

1) shown in Figure 9.1.16 in the text.