Question 1

Consider the linear system   =   x.
The origin is a ________.

Accepted Answer

A)  spiral sink 
B)  center 
C)  spiral source 
D)  nodal sink 
E)  saddle point
F) degenerate node
G) nodal source 
A)  spiral sink 
B)  center 
C)  spiral source 
D)  nodal sink 
E)  saddle point
F) degenerate node
G) nodal source

Question 2

Which of the following is a critical point of this nonlinear system?
 
Select all that apply.

Accepted Answer

A)  (1, -1) 
B)  (-7, -7) 
C)  (1, 7) 
D)  (-1, 1) 
E)  (-1, 7)
F) (-7, 7)
G) (7, 7)
H) (0, 0) 
A)  (1, -1) 
B)  (-7, -7) 
C)  (1, 7) 
D)  (-1, 1) 
E)  (-1, 7)
F) (-7, 7)
G) (7, 7)
H) (0, 0)

Question 3

Consider the following nonlinear system:

Accepted Answer

The answer of Consider the following nonlinear system:
  ...

Question 4

Consider the following nonlinear system:

Accepted Answer

The answer of Consider the following nonlinear system:
  ...

Question 5

Consider the following nonlinear system:
 
Which of these statements is true? Select all that apply.

Accepted Answer

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. 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.

Question 6

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.

Accepted Answer

A) True 
 B)False

Question 7

The trajectories of some nonzero solutions of this system converge to the origin as t $\rightarrow$ $\infty$ while many other solutions do not.

Accepted Answer

A)   
B)   
C)   
D)   
E)   
A)   
B)   
C)   
D)   
E)

Question 8

Consider the linear system   =   x.
The origin is ________ critical point.

Accepted Answer

A)  a stable 
B)  an unstable 
C)  an asymptotically stable 
A)  a stable 
B)  an unstable 
C)  an asymptotically stable

Question 9

Consider this competing species model:
 
Which of the following statements are true? Select all that apply.

Accepted Answer

A)  The entire first quadrant is the basin of attraction for the critical point $ \left(-\frac{1}{3}, \frac{5}{3}\right) $. 
B)  The origin is an unstable node. 
C)  Both $ \left(-\frac{6}{7}, 0\right) $ and $ \left(-\frac{2}{3}, 0\right) $ are saddle points. 
D)  Both $ \left(\frac{6}{7}, 0\right) $ and $ \left(0, \frac{2}{3}\right) $ are saddle points 
E)  All solution trajectories approach the origin as $ t \rightarrow \infty $.
F) The critical point $ \left(-\frac{1}{3}, \frac{5}{3}\right) $ corresponds to coexistence in this model. 
A)  The entire first quadrant is the basin of attraction for the critical point $ \left(-\frac{1}{3}, \frac{5}{3}\right) $. 
B)  The origin is an unstable node. 
C)  Both $ \left(-\frac{6}{7}, 0\right) $ and $ \left(-\frac{2}{3}, 0\right) $ are saddle points. 
D)  Both $ \left(\frac{6}{7}, 0\right) $ and $ \left(0, \frac{2}{3}\right) $ are saddle points 
E)  All solution trajectories approach the origin as $ t \rightarrow \infty $.
F) The critical point $ \left(-\frac{1}{3}, \frac{5}{3}\right) $ corresponds to coexistence in this model.

Question 10

Consider the linear system   =   x.
Identify all of the eigenvalues.

Accepted Answer

A)  $\lambda$ = -7 
B)  $\lambda$ = 7 
C)  $\lambda$ = 0 
D)  $\lambda$ = 5 
E)  $\lambda$ = -5 
A)  $\lambda$ = -7 
B)  $\lambda$ = 7 
C)  $\lambda$ = 0 
D)  $\lambda$ = 5 
E)  $\lambda$ = -5

Question 11

Suppose ? is a positive real parameter. Consider this competing species model:
 
Which of the following statements is true?

Accepted Answer

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. 
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.

Question 12

Consider the linear system   =   x.
Identify all of the eigenvalues.

Accepted Answer

A)  $\lambda$ = 5 
B)  $\lambda$ = -5 
C)  $\lambda$ = 0 
D)  $\lambda$ = 2 
E)  $\lambda$= -2 
A)  $\lambda$ = 5 
B)  $\lambda$ = -5 
C)  $\lambda$ = 0 
D)  $\lambda$ = 2 
E)  $\lambda$= -2

Question 13

Consider the linear system   =   x.
The origin is ________ critical point.

Accepted Answer

A)  a stable 
B)  an unstable 
C)  an asymptotically stable 
A)  a stable 
B)  an unstable 
C)  an asymptotically stable

Question 14

Consider the linear system   =   x.
The origin is a ________.

Accepted Answer

A)  spiral sink 
B)  center 
C)  spiral source 
D)  nodal sink 
E)  saddle point
F) degenerate node
G) nodal source 
A)  spiral sink 
B)  center 
C)  spiral source 
D)  nodal sink 
E)  saddle point
F) degenerate node
G) nodal source

Question 15

Consider the linear system   =   x.
The origin is a ________.

Accepted Answer

A)  spiral sink 
B)  center 
C)  spiral source 
D)  nodal sink 
E)  saddle point
F) degenerate node
G) nodal source 
A)  spiral sink 
B)  center 
C)  spiral source 
D)  nodal sink 
E)  saddle point
F) degenerate node
G) nodal source

Question 16

Which of the following is a critical point of this nonlinear system?
 
Select all that apply.

Accepted Answer

A)  (7, -4) 
B)  (-7, 4) 
C)    
D)  (0, 0) 
E)   
F)   
A)  (7, -4) 
B)  (-7, 4) 
C)    
D)  (0, 0) 
E)   
F)

Consider the linear system = x. The origin is a ________.

Which of the following is a critical point of this nonlinear system? Select all that apply.

Consider the following nonlinear system:

Consider the following nonlinear system:

Consider the following nonlinear system: Which of these statements is true? Select all that apply.

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.

The trajectories of some nonzero solutions of this system converge to the origin as t $\rightarrow$ $\infty$ while many other solutions do not.

Consider the linear system = x. The origin is ________ critical point.

Consider this competing species model: Which of the following statements are true? Select all that apply.

Consider the linear system = x. Identify all of the eigenvalues.

Suppose ? is a positive real parameter. Consider this competing species model: Which of the following statements is true?

Consider the linear system = x. Identify all of the eigenvalues.

Consider the linear system = x. The origin is ________ critical point.

Consider the linear system = x. The origin is a ________.

Consider the linear system = x. The origin is a ________.

Which of the following is a critical point of this nonlinear system? Select all that apply.

Introduction

First-Order Differential Equations

Second-Order Linear Differential Equations

Higher-Order Linear Differential Equations

Series Solutions of Second-Order Linear Equations

The Laplace Transform

Systems of First-Order Linear Equations

Numerical Methods

Partial Differential Equations and Fourier Series

Boundary Value Problems and Sturm-Liouville Theory

Filters

Exam 9: Nonlinear Differential Equations and Stability

Consider the linear system = x. The origin is a ________.

Which of the following is a critical point of this nonlinear system? Select all that apply.

Consider the following nonlinear system:

Consider the following nonlinear system:

Consider the following nonlinear system: Which of these statements is true? Select all that apply.

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.

The trajectories of some nonzero solutions of this system converge to the origin as t →\rightarrow→ ∞\infty∞ while many other solutions do not.

Consider the linear system = x. The origin is ________ critical point.

Consider this competing species model: Which of the following statements are true? Select all that apply.

Consider the linear system = x. Identify all of the eigenvalues.

Suppose ? is a positive real parameter. Consider this competing species model: Which of the following statements is true?

Consider the linear system = x. Identify all of the eigenvalues.

Consider the linear system = x. The origin is ________ critical point.

Consider the linear system = x. The origin is a ________.

Consider the linear system = x. The origin is a ________.

Which of the following is a critical point of this nonlinear system? Select all that apply.

Introduction

First-Order Differential Equations

Second-Order Linear Differential Equations

Higher-Order Linear Differential Equations

Series Solutions of Second-Order Linear Equations

The Laplace Transform

Systems of First-Order Linear Equations

Numerical Methods

Partial Differential Equations and Fourier Series

Boundary Value Problems and Sturm-Liouville Theory

Filters

The trajectories of some nonzero solutions of this system converge to the origin as t $\rightarrow$ $\infty$ while many other solutions do not.