Question 1

For which of the following systems do all solution trajectories converge to the origin as t $\rightarrow$ $\infty$?

Accepted Answer

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

Question 2

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 3

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 4

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 5

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:

Accepted Answer

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

Question 6

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

Accepted Answer

A)  $\lambda$ = 7i 
B)  $\lambda$ = -11i 
C)  $\lambda$= -7i 
D)  $\lambda$ = -2i 
E)  $\lambda$ =  11i 
A)  $\lambda$ = 7i 
B)  $\lambda$ = -11i 
C)  $\lambda$= -7i 
D)  $\lambda$ = -2i 
E)  $\lambda$ =  11i

Question 7

Consider the following nonlinear system:
 
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. 
E)  Both B and C. 
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. 
E)  Both B and C.

Question 8

Suppose ? is a positive real parameter. Consider this competing species model:
 
Which of these are critical points for this system? Select all that apply.

Accepted Answer

A) $ \left(0, \frac{2}{\alpha}\right) $ 
B) $ (0,0) $ 
C) $ \left(\frac{2}{\alpha}, 0\right) $ 
D)  $ \left(0, \frac{\alpha}{2}\right) $ 
E) $ \left(\frac{2}{3}, 0\right) $
F)$ \left(\frac{3}{2}, 0\right) $
G)$ \left(\alpha-2, \frac{-3 \alpha-8}{2}\right) $
H)$ \left(-\alpha+2, \frac{-3 \alpha-8}{2}\right) $
I)$ \left(2-\alpha, \frac{3 \alpha+8}{2}\right) $ 
A) $ \left(0, \frac{2}{\alpha}\right) $ 
B) $ (0,0) $ 
C) $ \left(\frac{2}{\alpha}, 0\right) $ 
D)  $ \left(0, \frac{\alpha}{2}\right) $ 
E) $ \left(\frac{2}{3}, 0\right) $
F)$ \left(\frac{3}{2}, 0\right) $
G)$ \left(\alpha-2, \frac{-3 \alpha-8}{2}\right) $
H)$ \left(-\alpha+2, \frac{-3 \alpha-8}{2}\right) $
I)$ \left(2-\alpha, \frac{3 \alpha+8}{2}\right) $

Question 9

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

Accepted Answer

A)  $\lambda$ = -8 
B)  $\lambda$ = -4 
C)  $\lambda$ = 0 
D)  $\lambda$ = -16 
E)  $\lambda$ = 4
F) $\lambda$= 16
G) $\lambda$ = 8 
A)  $\lambda$ = -8 
B)  $\lambda$ = -4 
C)  $\lambda$ = 0 
D)  $\lambda$ = -16 
E)  $\lambda$ = 4
F) $\lambda$= 16
G) $\lambda$ = 8

Question 10

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

Accepted Answer

A)  $\lambda$ = 10 
B)  $\lambda$ = -10 
C)  $\lambda$ = 0 
D)  $\lambda$ = -9 
E)  $\lambda$ = 9
F) $\lambda$ = -2
G) $\lambda$ = 2 
A)  $\lambda$ = 10 
B)  $\lambda$ = -10 
C)  $\lambda$ = 0 
D)  $\lambda$ = -9 
E)  $\lambda$ = 9
F) $\lambda$ = -2
G) $\lambda$ = 2

Question 11

Suppose α is a positive real parameter. Consider this competing species model:
‪

Accepted Answer

The answer of Suppose α is a positive real parameter....

Question 12

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 13

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
  = -10 x

Accepted Answer

A)  $ y=\frac{3}{5} x $ 
B)  $ 6 y^{2}--10 x^{2}=C $ 
C)  $ 6 y^{2}+-10 x^{2}=C $ 
D)  $ y=\frac{5}{3} x $ 
A)  $ y=\frac{3}{5} x $ 
B)  $ 6 y^{2}--10 x^{2}=C $ 
C)  $ 6 y^{2}+-10 x^{2}=C $ 
D)  $ y=\frac{5}{3} x $

Question 14

For which of the following systems is the origin a saddle point?

Accepted Answer

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

Question 15

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

Accepted Answer

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. 
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.

Question 16

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

Accepted Answer

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

Question 17

Every nonzero solution of this system spirals away from the origin.

Accepted Answer

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

Question 18

For which of the following systems is the origin a degenerate node?

Accepted Answer

A)  $ \mathbf{x}^{\prime}=\left(\begin{array}{ll}2022 & 5 \ 0 & -7\end{array}\right) \mathbf{x} $ 
B)  $ x^{\prime}=\left(\begin{array}{ll}-2022 & 20 \ 0 & -5\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}5 & -4 \ 4 & 0\end{array}\right) \mathbf{x} $ 
E)  $ \mathbf{x}^{\prime}=\left(\begin{array}{cc}0 & 4 \ -1 & 0\end{array}\right) \mathbf{x} $ 
A)  $ \mathbf{x}^{\prime}=\left(\begin{array}{ll}2022 & 5 \ 0 & -7\end{array}\right) \mathbf{x} $ 
B)  $ x^{\prime}=\left(\begin{array}{ll}-2022 & 20 \ 0 & -5\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}5 & -4 \ 4 & 0\end{array}\right) \mathbf{x} $ 
E)  $ \mathbf{x}^{\prime}=\left(\begin{array}{cc}0 & 4 \ -1 & 0\end{array}\right) \mathbf{x} $

Question 19

Consider the linear system   =   x.
Identify which two of the following are fundamental solution vectors for this system.

Accepted Answer

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

Question 20

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

For which of the following systems do all solution trajectories converge to the origin as t $\rightarrow$ $\infty$ ?

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

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

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

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:

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

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

Suppose ? is a positive real parameter. Consider this competing species model: Which of these are critical points for this system? Select all that apply.

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

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

Suppose α is a positive real parameter. Consider this competing species model: ‪

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

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 = -10 x

For which of the following systems is the origin a saddle point?

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

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

Every nonzero solution of this system spirals away from the origin.

For which of the following systems is the origin a degenerate node?

Consider the linear system = x. Identify which two of the following are fundamental solution vectors for this system.

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

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 do all solution trajectories converge to the origin as t →\rightarrow→ ∞\infty∞ ?

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

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

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

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:

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

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

Suppose ? is a positive real parameter. Consider this competing species model: Which of these are critical points for this system? Select all that apply.

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

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

Suppose α is a positive real parameter. Consider this competing species model: ‪

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

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 = -10 x

For which of the following systems is the origin a saddle point?

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

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

Every nonzero solution of this system spirals away from the origin.

For which of the following systems is the origin a degenerate node?

Consider the linear system = x. Identify which two of the following are fundamental solution vectors for this system.

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

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

For which of the following systems do all solution trajectories converge to the origin as t $\rightarrow$ $\infty$ ?