Ch9 Nonlinear Differential Equations And Test Bank + Answers - Complete Test Bank | Differential Equations 12e by William E. Boyce. DOCX document preview.
Elementary Differential Equations, 12e (Boyce)
Chapter 9 Nonlinear Differential Equations and Stability
1) Consider the linear system 
= 
x.
Identify all the eigenvalues.
A) λ = -8i
B) λ = 8
C) λ = -8
D) λ = 8i
E) λ = 0
Type: MC Var: 1
2) Consider the linear system = x.
The origin is a ________.
A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source
Type: MC Var: 1
3) Consider the linear system = 
x.
The origin is ________ critical point.
A) a stable
B) an unstable
C) an asymptotically stable
Type: MC Var: 1
4) Consider the linear system = 
x.
Identify all the eigenvalues.
A) λ = -15i
B) λ = 3i
C) λ = 15i
D) λ = -6i
E) λ = 6i
F) λ = -3i
Type: MC Var: 1
5) Consider the linear system = 
x.
The origin is a ________.
A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source
Type: MC Var: 1
6) Consider the linear system = 
x.
The origin is ________ critical point.
A) a stable
B) an unstable
C) an asymptotically stable
Type: MC Var: 1
7) Consider the linear system = 
x.
Identify all the eigenvalues.
A) λ = 7i
B) λ = -11i
C) λ = -7i
D) λ = -2i
E) λ = 2i
F) λ = 11i
Type: MC Var: 1
8) Consider the linear system = 
x.
The origin is a ________.
A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source
Type: MC Var: 1
9) Consider the linear system = 
x.
The origin is ________ critical point.
A) a stable
B) an unstable
C) an asymptotically stable
Type: MC Var: 1
10) Consider the linear system = 
x.
Identify all of the eigenvalues.
A) λ = -7
B) λ = 7
C) λ = 0
D) λ = 5
E) λ = -5
Type: MC Var: 1
11) Consider the linear system = 
x.
Identify which two of the following are fundamental solution vectors for this system.
A) 
B) 
C) 
D) 
E) 
F) 
Type: MC Var: 1
12) Consider the linear system = 
x.
The origin is a ________.
A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source
Type: MC Var: 1
13) Consider the linear system = 
x.
The origin is ________ critical point.
A) a stable
B) an unstable
C) an asymptotically stable
Type: MC Var: 1
14) Consider the linear system = 
x.
Identify all of the eigenvalues.
A) λ = 5
B) λ = -5
C) λ = 0
D) λ = 2
E) λ = -2
Type: MC Var: 1
15) Consider the linear system = 
x.
Identify which two of the following are fundamental solution vectors for this system.
A)
B)
C)
D) 
E) 
F) 
Type: MC Var: 1
16) Consider the linear system = 
x.
The origin is a ________.
A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source
Type: MC Var: 1
17) Consider the linear system = 
x.
The origin is ________ critical point.
A) a stable
B) an unstable
C) an asymptotically stable
Type: MC Var: 1
18) Consider the linear system = 
x.
Identify all of the eigenvalues.
A) λ = -6
B) λ = -10
C) λ = 0
D) λ = 10
E) λ = 6
Type: MC Var: 1
19) Consider the linear system = 
x.
The origin is a ________.
A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source
Type: MC Var: 1
20) Consider the linear system = 
x.
The origin is ________ critical point.
A) a stable
B) an unstable
C) an asymptotically stable
Type: MC Var: 1
21) Consider the linear system = 
x.
Identify all of the eigenvalues.
A) λ = -8
B) λ = -4
C) λ = 0
D) λ = -16
E) λ = 4
F) λ = 16
G) λ = 8
Type: MC Var: 1
22) Consider the linear system = 
x.
The origin is a ________.
A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source
Type: MC Var: 1
23) Consider the linear system = 
x.
The origin is ________ critical point.
A) a stable
B) an unstable
C) an asymptotically stable
Type: MC Var: 1
24) Consider the linear system = 
x.
Identify all of the eigenvalues.
A) λ = -4
B) λ = 4
C) λ = 0
D) λ = -9
E) λ = 9
F) λ = -10
G) λ = 10
Type: MC Var: 1
25) Consider the linear system = 
x.
The origin is a ________.
A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source
Type: MC Var: 1
26) Consider the linear system = 
x.
The origin is ________ critical point.
A) a stable
B) an unstable
C) an asymptotically stable
Type: MC Var: 1
27) Consider the linear system = 
x.
Identify all of the eigenvalues.
A) λ = 10
B) λ = -10
C) λ = 0
D) λ = -9
E) λ = 9
F) λ = -2
G) λ = 2
Type: MC Var: 1
28) Consider the linear system = 
x.
The origin is a ________.
A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source
Type: MC Var: 1
29) Consider the linear system = 
x.
The origin is ________ critical point.
A) a stable
B) an unstable
C) an asymptotically stable
Type: MC Var: 1
30) Consider the linear system = 
x.
Identify all of the eigenvalues.
A) λ = 0
B) λ = 4
C) λ = -4
D) λ = 3
E) λ = -3
Type: MC Var: 1
31) Consider the linear system = 
x.
The origin is a ________.
A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source
Type: MC Var: 1
32) Consider the linear system = 
x.
The origin is ________ critical point.
A) a stable
B) an unstable
C) an asymptotically stable
Type: MC Var: 1
33) Consider the linear system = 
x.
Identify all of the eigenvalues.
A) λ = 0
B) λ = -3
C) λ = 3
D) λ = -4
E) λ = 4
Type: MC Var: 1
34) Consider the linear system = 
x.
The origin is a ________.
A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source
Type: MC Var: 1
35) Consider the linear system = x.
The origin is ________ critical point.
A) a stable
B) an unstable
C) an asymptotically stable
Type: MC Var: 1
36) For which of the following systems is the origin a saddle point?
A) = 
x
B) = 
x
C) = 
x
D) = 
x
E) = 
x
Type: MC Var: 1
37) The trajectories of some nonzero solutions of this system converge to the origin as t → ∞ while many other solutions do not.
A) = 
x
B) = 
x
C) = x
D) = 
x
E) = 
x
Type: MC Var: 1
38) Every nonzero solution of this system spirals away from the origin.
A) = 
x
B) = 
x
C) = x
D) = 
x
E) = 
x
Type: MC Var: 1
39) For which of the following systems do all solution trajectories converge to the origin as t → ∞?
A) = 
x
B) = 
x
C) = x
D) = 
x
E) = 
x
Type: MC Var: 1
40) For which of the following systems is the origin a degenerate node?
A) = 
x
B) = 
x
C) = x
D) = 
x
E) = 
x
Type: MC Var: 1
41) For which of the following systems is every solution periodic?
A) = 
x
B) = 
x
C) = x
D) = 
x
E) = 
x
Type: MC Var: 1
42) Which of the following is a critical point of this nonlinear system?
= x - 4xy
= y + 7xy
Select all that apply.
A) (7, -4)
B) (-7, 4)
C) 
D) (0, 0)
E) 
F) 
Type: MC Var: 1
43) Which of the following is a critical point of this nonlinear system?
= 16 - 4
= 16 - 25
Select all that apply.
A) 
B) 
C) 
D) 
E) 
F) 
G) 
H) 
J) (0, 0)
Type: MC Var: 1
44) Which of the following is a critical point of this nonlinear system?
= (1 + x)(y - x)
= (7 - y)(y + x)
Select all that apply.
A) (1, -1)
B) (-7, -7)
C) (1, 7)
D) (-1, 1)
E) (-1, 7)
F) (-7, 7)
G) (7, 7)
H) (0, 0)
Type: MC Var: 1
45) Find an equation of the form H(x, y) = C, where C is an arbitrary real constant, satisfied by the trajectories of the following nonlinear system:
= 6y
= -10x
A) y = 
x
B) 6 - -10 = C
C) 6 + -10 = C
D) y = 
x
Type: MC Var: 1
46) Find an equation of the form H(x, y) = C, where C is an arbitrary real constant, satisfied by the trajectories of the following nonlinear system:
= 5
+ 3
= sin(5x) + 4
A) 
+ 3
- cos(5x) - 4x = C
B) 
+ 
- 
cos(5x) - 4x = C
C) + + cos(5x) - 4x = C
D) + 3 + cos(5x) - 4x = C
Type: MC Var: 1
47) Consider the following nonlinear system:
= 
+ 
Let 
(x, y) = -4 + -4xy and 
(x, y) = 3xy + 4. Express x and y using polar coordinates and determine which of these statements is true.
A) The system is locally linear near the origin.
B) The system is not locally linear near the origin because 
≠ 0.
C) The system is not locally linear near the origin because 
≠ 0.
D) The origin is not an isolated critical point.
Type: MC Var: 1
48) Consider the following nonlinear system:
= 
+ 
Let (x, y) = -5 + 9y and (x, y) = -3xy + 10. Express x and y using polar coordinates and determine which of these statements is true.
A) The system is locally linear near the origin.
B) The system is not locally linear near the origin because ≠ 0.
C) The system is not locally linear near the origin because ≠ 0.
D) The origin is not an isolated critical point.
E) Both B and C.
Type: MC Var: 1
49) Consider the following nonlinear system:
= (2 + x)(y - x)
= (7 - y)(y + x)
Which of the following is a critical point of this nonlinear system? Select all that apply.
A) (2, -2)
B) (-7, -7)
C) (2, 7)
D) (-2, 2)
E) (-2, 7)
F) (-7, 7)
G) (7, 7)
H) (0, 0)
Type: MC Var: 1
50) Consider the following nonlinear system:
= (3 + x)(y - x)
= (9 - y)(y + x)
Let f(x, y) = 
. Compute the Jacobian matrix J(x, y) of f.
Type: SA Var: 1
51) Consider the following nonlinear system:
= (3 + x)(y - x)
= (6 - y)(y + x)
Which of the following statements are true? Select all that apply.
A) (-3, 3) is an asymptotically stable node.
B) (3, -3) is an unstable spiral node.
C) (-3, 3) is an unstable node.
D) (6, 6) is an unstable node.
E) (6, -6) is an unstable node.
F) (6, 6) is an asymptotically stable node.
G) (0, 0) is a saddle point.
H) (-6, -6) is an asymptotically stable node.
Type: MC Var: 1
52) Consider the following nonlinear system:
= cos(5y) - 
= -4 cos(6x)
Which of the following is a complete list of the critical points of this nonlinear system?
A) All ordered pairs (x, y) such that x ∈ 
and 
.
B) All ordered pairs (x, y) such that x ∈ 
and 
.
C) All ordered pairs (x, y) such that x ∈ 
and .
D) All ordered pairs (x, y) such that x ∈ and 
.
E) All ordered pairs (x, y) such that x ∈ and .
Type: MC Var: 1
53) Consider the following nonlinear system:
= cos(4y) -
= -6 cos(6x)
Let f(x, y) = 
. Compute the Jacobian matrix J(x, y) of f.
Type: SA Var: 1
54) Consider the following nonlinear system:
= cos(6y) -
= -2 cos(3x)
Which of the following statements are true? Select all that apply.
A) 
is an unstable saddle point.
B) 
is a stable center.
C) 
is a stable center.
D) 
is an asymptotically stable spiral point.
E) 
is an improper node.
F) 
is a stable center.
Type: MC Var: 1
55) Consider the following nonlinear system:
= -9x + -3y + 
= -5y - 
Which of the following statements is true?
A) The origin is an unstable spiral point.
B) The origin is an asymptotically stable node.
C) The origin is an asymptotically stable spiral point.
D) The origin is a stable sink.
E) The origin is a stable improper node.
Type: MC Var: 1
56) Consider this competing species model:
= 4x(4 - 7x - 7y)
= 3y(8 - 9x - 7y)
Which of these are critical points for this system? Select all that apply.
A) 
B) 
C) 
D) 
E) 
F) 
G) (0, 0)
H) 
Type: MC Var: 1
57) Consider this competing species model:
= 3x(6 - 7x - 5y)
= 5y(2 - 9x - 3y)
Which of the following statements are true? Select all that apply.
A) The entire first quadrant is the basin of attraction for the critical point 
.
B) The origin is an unstable node.
C) Both 
and 
are saddle points.
D) Both 
and 
are saddle points.
E) All solution trajectories approach the origin as t → ∞.
F) The critical point corresponds to coexistence in this model.
Type: MC Var: 1
58) Suppose α is a real parameter. Consider this competing species model:
= -4x + 3y - 6
= 2α - 7y
Which of these are true? Select all that apply.
A) The origin is a critical point of this system for all values of α.
B) The x-nullcline is the curve 
+ 
x.
C) The y-nullcline is the horizontal line y = 
α.
D) The system has no critical points for values of α < - 
Type: MC Var: 1
59) Suppose α is a positive real parameter. Consider this competing species model:
= 5x(1 - 4x - 3y)
= 2y(α - 3x - 5y)
What are the nullclines for this system?
A) x = 0
B) y = 0
C) x =
D) 4x + 3y = 1
E) y =
F) 4x - 3y = 1
G) 5x + 3y = α
H) 5x - 3y = α
Type: MC Var: 1
60) Suppose α is a positive real parameter. Consider this competing species model:
= 4x(2 - 3x - 2y)
= 5y(α - 2x - 4y)
Which of these are critical points for this system? Select all that apply.
A) 
B) (0, 0)
C) 
D) 
E) 
F) 
G) 
H) 
J) 
Type: MC Var: 1
61) Suppose α is a positive real parameter. Consider this competing species model:
= 2x(3 - 4x - 5y)
= 5y(α - 5x - 5y)
Let f(x, y) = 
. Compute the Jacobian matrix J(x, y) of f.
Type: SA Var: 1
62) Suppose α is a positive real parameter. Consider this competing species model:
= 3x(1 - 5x - 3y)
= 4y(α - 3x - 6y)
Which of the following statements is true?
A) The origin is an unstable node.
B) The origin is an unstable saddle point.
C) The origin is an asymptotically stable node.
D) The origin is a stable center.
Type: MC Var: 1
63) Consider the following Lotka-Volterra system of equations:
= x(5 - 1.5y)
= y(-3.0 + 1.5x)
Determine all critical points for this system.
Type: SA Var: 1
64) Consider the following Lotka-Volterra system of equations:
= x(4 - 1.5y)
= y(-2.5 + 2.5x)
Let f(x, y) = 
. Compute the Jacobian matrix J(x, y) of f.
Type: SA Var: 1
65) Consider the following Lotka-Volterra system of equations:
= x(3 - 2.0y)
= y(-2.0 + 2.5x)
Which of these statements are true? Select all that apply.
A) The solution trajectories all spiral away from the point 
as t → ∞
B) The solution trajectories are closed curves encircling the point 
.
C) The point 
is an unstable node.
D) The predator and prey populations exhibit a cyclic variation.
E) The origin is a saddle point.
F) The period of the solution trajectories is 
Type: MC Var: 1
66) Consider the function V(x, y) = 7 + αxy + 5, where α is a real number.
Which of these statements is true?
A) V(x, y) is negative definite, for every nonzero real number α.
B) V(x, y) is positive definite, for every nonzero real number α.
C) V(x, y) is negative definite, for every real number α for which 
> 140.
D) V(x, y) is positive definite, for every real number α for which < 140.
Type: MC Var: 1
67) Consider the following nonlinear system:
= -8x
= -8y
Consider the Lyapunov function V(x, y) = at the critical point (0, 0). Compute 
.
Type: SA Var: 1
68) Consider the following nonlinear system:
= -4x
= -4y
Consider the Lyapunov function V(x, y) = at the critical point (0, 0). Using , what can you conclude about this nonlinear system?
A) The origin is asymptotically stable.
B) The origin is an unstable node.
C) The basin of attraction for the origin is the entire xy-plane.
D) The origin is a center and all solution trajectories encircle it.
Type: MC Var: 1
69) Consider the following nonlinear system:
= -y + x( + - 1)
= x + y( + - 1)
Convert this system into an equivalent system in polar form.
= 1
Type: SA Var: 1
70) Consider the following nonlinear system:
= -y + x( + - 1)
= x + y( + - 1)
Which of these statements is true? Select all that apply.
A) There are no isolated closed trajectories for this system.
B) For r > 0, the corresponding solution trajectories spiral outward away from r = 1 in a counterclockwise fashion.
C) For 0 < r < 1, the corresponding solution trajectories spiral toward the origin in a counterclockwise fashion.
D) The unit circle is a semistable limit cycle.
E) The unit circle is an unstable limit cycle.
Type: MC Var: 1
71) Consider the following nonlinear system expressed in polar form:
= r(r - 2)(r - 8)
= 1
Which of these statements is true? Select all that apply.
A) r = 8 is a semistable limit cycle.
B) r = 2 is an asymptotically stable limit cycle.
C) For 2 < r < 8, the corresponding solution trajectories spiral outward away from r = 2 in a clockwise fashion.
D) For 0 < r < 2, the corresponding solution trajectories spiral toward the origin in a clockwise fashion.
E) For r > 8, the corresponding solution trajectories spiral outward away from r = 8 in a counterclockwise fashion.
Type: MC Var: 1
72) Consider the following nonlinear system expressed in polar form:
= r
(r - 7)
= -1
Which of these statements is true? Select all that apply.
A) All solution trajectories approach either r = 5 or r = 7 in a counterclockwise fashion.
B) r = 5 is a semistable limit cycle.
C) r = 7 is an unstable limit cycle.
D) For 0 < r < 5, the corresponding solution trajectories spiral toward r = 5 in a clockwise fashion.
E) For r > 7, the corresponding solution trajectories spiral outward away from r = 7 in a counterclockwise fashion.
Type: MC Var: 1
73) Consider the van der Pol equation 
- 2.8(1 - 
)
+ u = 0.
Write this equation as a nonlinear system in x and y, where x = u and y = .
= -x + 2.8(1 - )y
Type: SA Var: 1
74) Consider the van der Pol equation - 4.2(1 - ) + u = 0.
The origin is an unstable node.
Type: TF Var: 1
75) Consider the van der Pol equation - 3.6(1 - ) + u = 0.
If a closed trajectory exists, it must encircle the origin.
Type: TF Var: 1
76) Consider the van der Pol equation - 2.4(1 - ) + u = 0.
If a closed trajectory exists, then it must be contained within the vertical strip |x| < 1.
Type: TF Var: 1
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