Homework 8 Solutions
Total points: 16
March 11, 2019
Chapter 7.1
Solution:
When the slope is zero, the equilibrium point can have any stability (stable, unstable, semistable). Thus,
the fact that
dX
0
dX
X
*
= 0
tells us nothing about the stability of
X
*
. In other words, linear stability analysis fails in this case.
Solution:
As you learned in LS 30A, you find the equilibrium points by setting the change equation equal to zero
and solving for
X
:
X
0
=
X
3

X
= 0
X
(
X
2

1) = 0
X
(
X

1)(
X
+ 1) = 0
X
=

1
,
0
,
1
Thus, the equilibrium points are

1
,
0
,
and 1. On the other hand,
dX
0
dX
= 2
X
2

1
.
Thus, linear stability analysis tells us that
dX
0
dX
X
=

1
= 1
>
0 =
⇒ 
1 is unstable
dX
0
dX
X
=0
=

1
<
0 =
⇒
0 is stable
dX
0
dX
X
=1
= 1
>
0 =
⇒
1 is unstable
1
Chapter 7.2
Solution:
By definition, a function is a rule that takes each element of its domain to exactly one element of its
codomain. Hence, for any (
X, Y
)
∈
R
2
, there is single value
f
(
X, Y
). This means that the graph of
f
:
R
2
→
R
has
just a single point above (or below) each (
X, Y
). Otherwise said, the graph of a function cannot have two points