Question 1

For which of the following systems is every solution periodic?

Accepted Answer

A)  $ \mathbf{x}^{\prime}=\left(\begin{array}{ll}2022 & -25 \ 0 & -4\end{array}\right) \mathbf{x} $ 
B)  $ x^{\prime}=\left(\begin{array}{ll}-2022 & -60 \ 0 & -4\end{array}\right) x $ 
C)  $ \mathbf{x}^{\prime}=\left(\begin{array}{ll}2022 & 0 \ 0 & 2022\end{array}\right) \mathbf{x} $ 
D)  $ x^{\prime}=\left(\begin{array}{ll}4 & -3 \ 3 & 0\end{array}\right) x $ 
E)  $ \mathbf{x}^{\prime}=\left(\begin{array}{cc}0 & 1 \ -3 & 0\end{array}\right) \mathbf{x} $ 
A)  $ \mathbf{x}^{\prime}=\left(\begin{array}{ll}2022 & -25 \ 0 & -4\end{array}\right) \mathbf{x} $ 
B)  $ x^{\prime}=\left(\begin{array}{ll}-2022 & -60 \ 0 & -4\end{array}\right) x $ 
C)  $ \mathbf{x}^{\prime}=\left(\begin{array}{ll}2022 & 0 \ 0 & 2022\end{array}\right) \mathbf{x} $ 
D)  $ x^{\prime}=\left(\begin{array}{ll}4 & -3 \ 3 & 0\end{array}\right) x $ 
E)  $ \mathbf{x}^{\prime}=\left(\begin{array}{cc}0 & 1 \ -3 & 0\end{array}\right) \mathbf{x} $

Question 2

Consider the following nonlinear system expressed in polar form:   = r   ( r - 7)
  = -1
Which of these statements is true? Select all that apply.

Accepted Answer

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  7, the corresponding solution trajectories spiral outward away from r = 7 in a counterclockwise fashion. 
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  7, the corresponding solution trajectories spiral outward away from r = 7 in a counterclockwise fashion.

Question 3

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

Accepted Answer

A)  (2, -2) 
B)  (-7, -7) 
C)  (2, 7) 
D)  (-2, 2) 
E)  (-2, 7)
F) (-7, 7)
G) (7, 7)
H) (0, 0) 
A)  (2, -2) 
B)  (-7, -7) 
C)  (2, 7) 
D)  (-2, 2) 
E)  (-2, 7)
F) (-7, 7)
G) (7, 7)
H) (0, 0)

Question 4

Consider the van der Pol equation   - 3.6(1 -   )   + u = 0.If a closed trajectory exists, it must encircle the origin.

Accepted Answer

A) True 
 B)False

Question 5

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 6

Consider the following nonlinear system:
 
What can you conclude about this nonlinear system?

Accepted Answer

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. 
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.

Question 7

Suppose ? is a real parameter. Consider this competing species model:
 
Which of these are true? Select all that apply.

Accepted Answer

A)  The origin is a critical point of this system for all values of $ \alpha $. 
B)  The $ x $-nullcline is the curve $ \frac{6}{3} x^{2}+\frac{4}{3} x $. 
C)  The $ y $-nullcline is the horizontal line $ y=\frac{2}{7} \alpha $. 
D)  The system has no critical points for values of $ \alpha<-\frac{7}{9} $ 
A)  The origin is a critical point of this system for all values of $ \alpha $. 
B)  The $ x $-nullcline is the curve $ \frac{6}{3} x^{2}+\frac{4}{3} x $. 
C)  The $ y $-nullcline is the horizontal line $ y=\frac{2}{7} \alpha $. 
D)  The system has no critical points for values of $ \alpha<-\frac{7}{9} $

Question 8

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.

Accepted Answer

A)  r = 8 is a semistable limit cycle. 
B)  r = 2 is an asymptotically stable limit cycle. 
C)  For 2  8, the corresponding solution trajectories spiral outward away from r = 8 in a counterclockwise fashion. 
A)  r = 8 is a semistable limit cycle. 
B)  r = 2 is an asymptotically stable limit cycle. 
C)  For 2  8, the corresponding solution trajectories spiral outward away from r = 8 in a counterclockwise fashion.

Question 9

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

Accepted Answer

A)  $\lambda$ = -6 
B)  $\lambda$ = -10 
C)  $\lambda$ = 0 
D)  $\lambda$ = 10 
E)  $\lambda$= 6 
A)  $\lambda$ = -6 
B)  $\lambda$ = -10 
C)  $\lambda$ = 0 
D)  $\lambda$ = 10 
E)  $\lambda$= 6

Question 10

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 11

Consider the following nonlinear system:
   
Convert this system into an equivalent system in polar form.

Accepted Answer

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

Question 12

Consider the following Lotka-Volterra system of equations:
 
Which of these statements are true? Select all that apply.

Accepted Answer

A)  The solution trajectories all spiral away from the point $ \left(-\frac{4}{5},-\frac{3}{2}\right) $ as $ t \rightarrow \infty $ 
B)  The solution trajectories are closed curves encircling the point $ \left(\frac{4}{5}, \frac{3}{2}\right) $. 
C)  The point $ \left(\frac{3}{2}, \frac{4}{5}\right) $ 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 $ \frac{2 \pi}{\sqrt{6.0}} $ 
A)  The solution trajectories all spiral away from the point $ \left(-\frac{4}{5},-\frac{3}{2}\right) $ as $ t \rightarrow \infty $ 
B)  The solution trajectories are closed curves encircling the point $ \left(\frac{4}{5}, \frac{3}{2}\right) $. 
C)  The point $ \left(\frac{3}{2}, \frac{4}{5}\right) $ 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 $ \frac{2 \pi}{\sqrt{6.0}} $

Question 13

Consider the van der Pol equation   - 4.2(1 -   )   + u = 0.
The origin is an unstable node.

Accepted Answer

A) True 
 B)False

Question 14

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

Accepted Answer

A)  $\lambda$ = 0 
B)  $\lambda$ = 4 
C)  $\lambda$ = -4 
D) $\lambda$= 3 
E)  $\lambda$ = -3 
A)  $\lambda$ = 0 
B)  $\lambda$ = 4 
C)  $\lambda$ = -4 
D) $\lambda$= 3 
E)  $\lambda$ = -3

Question 15

Consider the following Lotka-Volterra system of equations:

Accepted Answer

The answer of Consider the following Lotka-Volterra system of equations:
...

Question 16

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

Accepted Answer

A)  $\lambda$ = -4 
B)  $\lambda$ = 4 
C)  $\lambda$ = 0 
D)  $\lambda$ = -9 
E) $\lambda$= 9
F) $\lambda$= -10
G)$\lambda$ = 10 
A)  $\lambda$ = -4 
B)  $\lambda$ = 4 
C)  $\lambda$ = 0 
D)  $\lambda$ = -9 
E) $\lambda$= 9
F) $\lambda$= -10
G)$\lambda$ = 10

Question 17

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 18

Consider the following nonlinear system:
  =     +  
Let   Express x and y using polar coordinates and determine which of these statements is true.

Accepted Answer

A)  The system is locally linear near the origin. 
B)  The system is not locally linear near the origin because   
C)  The system is not locally linear near the origin because   
D)  The origin is not an isolated critical point. 
A)  The system is locally linear near the origin. 
B)  The system is not locally linear near the origin because   
C)  The system is not locally linear near the origin because   
D)  The origin is not an isolated critical point.

Question 19

Consider the following nonlinear system:

Accepted Answer

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

Question 20

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

Accepted Answer

A)  $ \left(\frac{\pi}{6}, \frac{5 \pi}{18}\right) $ is an unstable saddle point. 
B)  $ \left(-\frac{\pi}{6}, \frac{\pi}{18}\right) $ is a stable center. 
C)  $ \left(-\frac{\pi}{6}, \frac{\pi}{36}\right) $ is a stable center. 
D)  $ \left(\frac{\pi}{3},-\frac{5 \pi}{18}\right) $ is an asymptotically stable spiral point. 
E)  $ \left(\frac{3 \pi}{6},-\frac{\pi}{18}\right) $ is an improper node.
F) $ \left(\frac{\pi}{3}, \frac{\pi}{36}\right) $ is a stable center. 
A)  $ \left(\frac{\pi}{6}, \frac{5 \pi}{18}\right) $ is an unstable saddle point. 
B)  $ \left(-\frac{\pi}{6}, \frac{\pi}{18}\right) $ is a stable center. 
C)  $ \left(-\frac{\pi}{6}, \frac{\pi}{36}\right) $ is a stable center. 
D)  $ \left(\frac{\pi}{3},-\frac{5 \pi}{18}\right) $ is an asymptotically stable spiral point. 
E)  $ \left(\frac{3 \pi}{6},-\frac{\pi}{18}\right) $ is an improper node.
F) $ \left(\frac{\pi}{3}, \frac{\pi}{36}\right) $ is a stable center.

For which of the following systems is every solution periodic?

Consider the following nonlinear system expressed in polar form: = r ( r - 7) = -1 Which of these statements is true? Select all that apply.

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

Consider the van der Pol equation - 3.6(1 - ) + u = 0.If a closed trajectory exists, it must encircle the origin.

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

Consider the following nonlinear system: What can you conclude about this nonlinear system?

Suppose ? is a real parameter. Consider this competing species model: Which of these are true? Select all that apply.

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.

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

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

Consider the following nonlinear system: Convert this system into an equivalent system in polar form.

Consider the following Lotka-Volterra system of equations: Which of these statements are true? Select all that apply.

Consider the van der Pol equation - 4.2(1 - ) + u = 0. The origin is an unstable node.

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

Consider the following Lotka-Volterra system of equations:

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

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

Consider the following nonlinear system: = + Let Express x and y using polar coordinates and determine which of these statements is true.

Consider the following nonlinear system:

Consider the following nonlinear system: Which of the following statements are true? 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

For which of the following systems is every solution periodic?

Consider the following nonlinear system expressed in polar form: = r ( r - 7) = -1 Which of these statements is true? Select all that apply.

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

Consider the van der Pol equation - 3.6(1 - ) + u = 0.If a closed trajectory exists, it must encircle the origin.

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

Consider the following nonlinear system: What can you conclude about this nonlinear system?

Suppose ? is a real parameter. Consider this competing species model: Which of these are true? Select all that apply.

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.

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

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

Consider the following nonlinear system: Convert this system into an equivalent system in polar form.

Consider the following Lotka-Volterra system of equations: Which of these statements are true? Select all that apply.

Consider the van der Pol equation - 4.2(1 - ) + u = 0. The origin is an unstable node.

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

Consider the following Lotka-Volterra system of equations:

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

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

Consider the following nonlinear system: = + Let Express x and y using polar coordinates and determine which of these statements is true.

Consider the following nonlinear system:

Consider the following nonlinear system: Which of the following statements are true? 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