Question 1

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 A

Question 2

Suppose ? is a positive real parameter. Consider this competing species model:
 
What are the nullclines for this system?

Accepted Answer

A)  x = 0 
B)  y = 0 
C)    
D)  4x + 3y = 1 
E)   
F) 
G) 
H)  
A)  x = 0 
B)  y = 0 
C)    
D)  4x + 3y = 1 
E)   
F) 
G) 
H)  A, B, D, G

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 C

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

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

Accepted Answer

A)  $ \left(\frac{5}{4}, \frac{1}{2}\right) $ 
B)  $ \left(-\frac{4}{5},-2\right) $ 
C)  $ \left[-\frac{5}{4}, \frac{1}{2}\right) $ 
D)  $ \left(\frac{5}{4},-\frac{1}{2}\right) $ 
E)  $ \left(\frac{4}{5}, 2\right) $
F) $ \left(\frac{4}{5},-2\right) $
G) $ \left(-\frac{5}{4},-\frac{1}{2}\right) $
H) $ \left(-\frac{4}{5}, 2\right) $
I) $ (0,0) $ 
A)  $ \left(\frac{5}{4}, \frac{1}{2}\right) $ 
B)  $ \left(-\frac{4}{5},-2\right) $ 
C)  $ \left[-\frac{5}{4}, \frac{1}{2}\right) $ 
D)  $ \left(\frac{5}{4},-\frac{1}{2}\right) $ 
E)  $ \left(\frac{4}{5}, 2\right) $
F) $ \left(\frac{4}{5},-2\right) $
G) $ \left(-\frac{5}{4},-\frac{1}{2}\right) $
H) $ \left(-\frac{4}{5}, 2\right) $
I) $ (0,0) $

Question 6

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 7

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 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 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 10

Consider the following Lotka-Volterra system of equations:
   
Determine all critical points for this system.

Accepted Answer

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

Question 11

Consider this competing species model:
 
Which of these are critical points for this system? Select all that apply.

Accepted Answer

A)  $ \left(0, \frac{4}{7}\right) $ 
B)  $ \left(\frac{4}{7}, 0\right) $ 
C)  $ \left(0, \frac{8}{7}\right) $ 
D)  $ \left(0, \frac{7}{8}\right) $ 
E)  $ \left[-\frac{4}{7}, 0\right) $
F) $ \left(0,-\frac{8}{7}\right) $
G) $ (0,0) $
H) $ \left(2,-\frac{10}{7}\right) $ 
A)  $ \left(0, \frac{4}{7}\right) $ 
B)  $ \left(\frac{4}{7}, 0\right) $ 
C)  $ \left(0, \frac{8}{7}\right) $ 
D)  $ \left(0, \frac{7}{8}\right) $ 
E)  $ \left[-\frac{4}{7}, 0\right) $
F) $ \left(0,-\frac{8}{7}\right) $
G) $ (0,0) $
H) $ \left(2,-\frac{10}{7}\right) $

Question 12

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 13

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

Accepted Answer

A)  $\lambda$ = -15i 
B)  $\lambda$ = 3i 
C)  $\lambda$ = 15i 
D)  $\lambda$= -6i 
E)  $\lambda$ = 6i
F) $\lambda$ = -3i 
A)  $\lambda$ = -15i 
B)  $\lambda$ = 3i 
C)  $\lambda$ = 15i 
D)  $\lambda$= -6i 
E)  $\lambda$ = 6i
F) $\lambda$ = -3i

Question 14

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

Accepted Answer

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. 
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.

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

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

Accepted Answer

A)  $\lambda$= -8i 
B)  $\lambda$= 8 
C)  $\lambda$= -8 
D)  $\lambda$ = 8i 
E) $\lambda$ = 0 
A)  $\lambda$= -8i 
B)  $\lambda$= 8 
C)  $\lambda$= -8 
D)  $\lambda$ = 8i 
E) $\lambda$ = 0

Question 17

Consider the function V(x, y) = 7   + $\alpha$xy + 5   , where $\alpha$ is a real number.Which of these statements is true?

Accepted Answer

A) V(x, y) is negative definite, for every nonzero real number $\alpha$. B) V(x, y) is positive definite, for every nonzero real number $\alpha$. C) V(x, y) is negative definite, for every real number $\alpha$ for which > 140. D) V(x, y) is positive definite, for every real number $\alpha$ for which < 140. A) V(x, y) is negative definite, for every nonzero real number $\alpha$. B) V(x, y) is positive definite, for every nonzero real number $\alpha$. C) V(x, y) is negative definite, for every real number $\alpha$ for which > 140. D) V(x, y) is positive definite, for every real number $\alpha$ for which < 140.

Question 18

Consider the following nonlinear system:
 
Which of the following is a complete list of the critical points of this nonlinear system?

Accepted Answer

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

Question 19

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 20

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 =   .

Accepted Answer

The answer of Consider the van der Pol equation ...

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

Suppose ? is a positive real parameter. Consider this competing species model: What are the nullclines for this system?

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

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 linear system = x. The origin is a ________.

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 ________.

Consider the following Lotka-Volterra system of equations: Determine all critical points for this system.

Consider this competing species model: Which of these are critical points for this system? Select all that apply.

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

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

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

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

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

Consider the following nonlinear system: Which of the following is a complete list of the critical points of this nonlinear system?

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

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 = .

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 ________ critical point.

Suppose ? is a positive real parameter. Consider this competing species model: What are the nullclines for this system?

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

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 linear system = x. The origin is a ________.

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 ________.

Consider the following Lotka-Volterra system of equations: Determine all critical points for this system.

Consider this competing species model: Which of these are critical points for this system? Select all that apply.

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

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

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

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

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

Consider the function V(x, y) = 7 + α\alphaα xy + 5 , where α\alphaα is a real number.Which of these statements is true?

Consider the following nonlinear system: Which of the following is a complete list of the critical points of this nonlinear system?

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

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 = .

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