Question 1

Consider the first-order homogeneous system of linear differential equations
 
Which of these are eigenvalues for this system? Select all that apply.

Accepted Answer

A)  10 
B)  4 
C)  -4i 
D)  10 + 4i 
E)  4i
F) 10 - 4i
G) 4 + 10i
H) 4 - 10i 
A)  10 
B)  4 
C)  -4i 
D)  10 + 4i 
E)  4i
F) 10 - 4i
G) 4 + 10i
H) 4 - 10i

Question 2

Consider the first-order nonhomogeneous initial value problem
 
Given a fundamental matrix   (t) for the system, what is the solution of this initial value problem?

Accepted Answer

A)  $ \mathbf{x}(t)=\psi(3.0)\left(\begin{array}{l}-2 \ 4\end{array}\right)+\psi(t) \int_{3.0}^{t} \psi^{-1}(s){ }_{-5 e^{-2 s}}^{4 s^{3}+4} d s $ 
B)  $ \mathbf{x}(t)=\psi(t) \psi^{-1}(3.0)\left(\begin{array}{c}-2 \ 4\end{array}\right)+\psi(t) \int_{3.0}^{t} \psi^{-1}(s){ }_{-5 e^{-2 s}}^{4 s^{3}+4} d s $ 
C)  $ \mathbf{x}(t)=\psi(t)\left(\begin{array}{l}-2 \ 4\end{array}\right)+\psi(t) \int_{3.0}^{t} \psi^{-1}(s) \begin{array}{l}4 s^{3}+4 \ -5 e^{-2 s}\end{array} s $ 
D)  $ \mathbf{x}(t)=\psi(t) \psi^{-1}(3.0)\left(\begin{array}{c}-2 \ 4\end{array}\right)+\psi(3.0) \int_{3.0}^{t} \psi^{-1}(s) s^{3}+4 e^{-2 s} d s $ 
A)  $ \mathbf{x}(t)=\psi(3.0)\left(\begin{array}{l}-2 \ 4\end{array}\right)+\psi(t) \int_{3.0}^{t} \psi^{-1}(s){ }_{-5 e^{-2 s}}^{4 s^{3}+4} d s $ 
B)  $ \mathbf{x}(t)=\psi(t) \psi^{-1}(3.0)\left(\begin{array}{c}-2 \ 4\end{array}\right)+\psi(t) \int_{3.0}^{t} \psi^{-1}(s){ }_{-5 e^{-2 s}}^{4 s^{3}+4} d s $ 
C)  $ \mathbf{x}(t)=\psi(t)\left(\begin{array}{l}-2 \ 4\end{array}\right)+\psi(t) \int_{3.0}^{t} \psi^{-1}(s) \begin{array}{l}4 s^{3}+4 \ -5 e^{-2 s}\end{array} s $ 
D)  $ \mathbf{x}(t)=\psi(t) \psi^{-1}(3.0)\left(\begin{array}{c}-2 \ 4\end{array}\right)+\psi(3.0) \int_{3.0}^{t} \psi^{-1}(s) s^{3}+4 e^{-2 s} d s $

Question 3

Consider the first-order homogeneous system of linear differential equations
  
Which of these is the fundamental matrix ?(t) for this system?

Accepted Answer

A)  $ \psi(t)=\left(\begin{array}{ll}e^{4 t}-4 e^{-5 t} \ e^{4 t} 5 e^{-5 t}\end{array}\right) $ 
B)  $ \Psi(t)=\left(\begin{array}{ll}-e^{4 t} & -5 e^{-5 t} \ e^{4 t} & 4 e^{-5 t}\end{array}\right) $ 
C)  $ \Psi(t)=\left(\begin{array}{l}e^{4 t} 4 e^{-5 t} \ e^{4 t}-5 e^{-5 t}\end{array}\right) $ 
D)  $ \psi(t)=\left(\begin{array}{l}e^{-4 t} 4 e^{-5 t} \ e^{-4 t}-5 e^{-5 t}\end{array}\right) $ 
A)  $ \psi(t)=\left(\begin{array}{ll}e^{4 t}-4 e^{-5 t} \ e^{4 t} 5 e^{-5 t}\end{array}\right) $ 
B)  $ \Psi(t)=\left(\begin{array}{ll}-e^{4 t} & -5 e^{-5 t} \ e^{4 t} & 4 e^{-5 t}\end{array}\right) $ 
C)  $ \Psi(t)=\left(\begin{array}{l}e^{4 t} 4 e^{-5 t} \ e^{4 t}-5 e^{-5 t}\end{array}\right) $ 
D)  $ \psi(t)=\left(\begin{array}{l}e^{-4 t} 4 e^{-5 t} \ e^{-4 t}-5 e^{-5 t}\end{array}\right) $

Question 4

Consider the first-order homogeneous system of linear differential equations
 
What is the general solution of this system? Here,   and   are arbitrary real constants.

Accepted Answer

A)  $ \mathbf{x}(t)=C_{1} e^{-3 t}\left(\begin{array}{l}\sin (8 t) \ \cos (8 t)\end{array}\right)+C_{2} e^{-3 t}\left[\begin{array}{l}-\cos (8 t) \ \sin (8 t)\end{array}\right) $ 
B)  $ \mathbf{x}(t)=C_{1} e^{3 t}\left(\begin{array}{l}\sin (8 t) \ \cos (8 t)\end{array}\right)+C_{2} e^{3 t}\left(\begin{array}{l}-\cos (8 t) \ \sin (8 t)\end{array}\right) $ 
C)  $ \mathbf{x}(t)=C_{1} e^{-8 t}\left(\begin{array}{l}\sin (3 t) \ \cos (3 t)\end{array}\right]+C_{2} e^{-8 t}\left(\begin{array}{l}-\cos (3 t) \ \sin (3 t)\end{array}\right) $ 
D)  $ \mathbf{x}(t)=C_{1} e^{8 t}\left(\begin{array}{l}\sin (3 t) \ \cos (3 t)\end{array}\right)+C_{2} e^{8 t}\left\{\begin{array}{l}-\cos (3 t) \ \sin (3 t)\end{array}\right) $ 
A)  $ \mathbf{x}(t)=C_{1} e^{-3 t}\left(\begin{array}{l}\sin (8 t) \ \cos (8 t)\end{array}\right)+C_{2} e^{-3 t}\left[\begin{array}{l}-\cos (8 t) \ \sin (8 t)\end{array}\right) $ 
B)  $ \mathbf{x}(t)=C_{1} e^{3 t}\left(\begin{array}{l}\sin (8 t) \ \cos (8 t)\end{array}\right)+C_{2} e^{3 t}\left(\begin{array}{l}-\cos (8 t) \ \sin (8 t)\end{array}\right) $ 
C)  $ \mathbf{x}(t)=C_{1} e^{-8 t}\left(\begin{array}{l}\sin (3 t) \ \cos (3 t)\end{array}\right]+C_{2} e^{-8 t}\left(\begin{array}{l}-\cos (3 t) \ \sin (3 t)\end{array}\right) $ 
D)  $ \mathbf{x}(t)=C_{1} e^{8 t}\left(\begin{array}{l}\sin (3 t) \ \cos (3 t)\end{array}\right)+C_{2} e^{8 t}\left\{\begin{array}{l}-\cos (3 t) \ \sin (3 t)\end{array}\right) $

Question 5

Consider the first-order nonhomogeneous initial value problem
 
Which of these is the fundamental matrix   (t) for this system?

Accepted Answer

A)  $ \psi(t)=\left(\begin{array}{ll}-9 e^{-5 t} & e^{-4 t} \ 3 e^{-5 t} & 0\end{array}\right) $ 
B)  $ \psi(t)=\left(\begin{array}{ll}-9 e^{-5 t} & 0 \ 3 e^{-5 t} & e^{-4 t}\end{array}\right) $ 
C)  $ \psi(t)=\left(\begin{array}{l}9 e^{5 t} e^{-4 t} \ 3 e^{5 t} 0\end{array}\right) $ 
D)  $ \psi(t)=\left(\begin{array}{ll}9 e^{5 t} & 0 \ 3 e^{5 t} & e^{-4 t}\end{array}\right) $ 
A)  $ \psi(t)=\left(\begin{array}{ll}-9 e^{-5 t} & e^{-4 t} \ 3 e^{-5 t} & 0\end{array}\right) $ 
B)  $ \psi(t)=\left(\begin{array}{ll}-9 e^{-5 t} & 0 \ 3 e^{-5 t} & e^{-4 t}\end{array}\right) $ 
C)  $ \psi(t)=\left(\begin{array}{l}9 e^{5 t} e^{-4 t} \ 3 e^{5 t} 0\end{array}\right) $ 
D)  $ \psi(t)=\left(\begin{array}{ll}9 e^{5 t} & 0 \ 3 e^{5 t} & e^{-4 t}\end{array}\right) $

Question 6

Consider the first-order homogeneous system of linear differential equations
  
Which of the following is an accurate statement regarding the behavior of the solution trajectories of this system as t $\rightarrow$$\infty$?

Accepted Answer

A)  All trajectories spiral towards the origin as t$\rightarrow$$\infty$. 
B)  All trajectories spiral away from the origin as t  $\rightarrow$$\infty$. 
C)  The trajectories are concentric circles centered at the origin. 
D)  The trajectories are line segments that approach the origin as t $\rightarrow$$\infty$. 
A)  All trajectories spiral towards the origin as t$\rightarrow$$\infty$. 
B)  All trajectories spiral away from the origin as t  $\rightarrow$$\infty$. 
C)  The trajectories are concentric circles centered at the origin. 
D)  The trajectories are line segments that approach the origin as t $\rightarrow$$\infty$.

Question 7

Consider these matrices:

Accepted Answer

The answer of Consider these matrices:
  ...

Question 8

Consider the first-order homogeneous system of linear differential equations 
   =   x
What is the bifurcation value of α, if any?

Accepted Answer

The answer of Consider the first-order homogeneous system of linear...

Question 9

Consider the matrix function A(t)   Compute

Accepted Answer

A)  $ \left[\begin{array}{ll}-2 \cos (2 t) & -5 \sin (5 t) \ -5 \sin (5 t) & 2 \cos (2 t)\end{array}\right] $ 
B)  $ \frac{1}{4 \cos ^{2}(2 t)+25 \sin ^{2}(5 t)}\left[\begin{array}{ll}-2 \cos (2 t) & -5 \sin (5 t) \ -5 \sin (5 t) & 2 \cos (2 t)\end{array}\right] $ 
C)  $ \left[\begin{array}{ccc}-2 \cos (2 t) & 5 \sin (5 t) \ 5 \sin (5 t) & 2 \cos (2 t)\end{array}\right] $ 
D)  $ \frac{1}{4 \cos ^{2}(2 t)+25 \sin ^{2}(5 t)}\left[\begin{array}{ll}-2 \cos (2 t) & 5 \sin (5 t) \ 5 \sin (5 t) & 2 \cos (2 t)\end{array}\right] $ 
A)  $ \left[\begin{array}{ll}-2 \cos (2 t) & -5 \sin (5 t) \ -5 \sin (5 t) & 2 \cos (2 t)\end{array}\right] $ 
B)  $ \frac{1}{4 \cos ^{2}(2 t)+25 \sin ^{2}(5 t)}\left[\begin{array}{ll}-2 \cos (2 t) & -5 \sin (5 t) \ -5 \sin (5 t) & 2 \cos (2 t)\end{array}\right] $ 
C)  $ \left[\begin{array}{ccc}-2 \cos (2 t) & 5 \sin (5 t) \ 5 \sin (5 t) & 2 \cos (2 t)\end{array}\right] $ 
D)  $ \frac{1}{4 \cos ^{2}(2 t)+25 \sin ^{2}(5 t)}\left[\begin{array}{ll}-2 \cos (2 t) & 5 \sin (5 t) \ 5 \sin (5 t) & 2 \cos (2 t)\end{array}\right] $

Question 10

Consider the first-order homogeneous system of linear differential equations
X =   x
Given a fundamental matrix   (t) for the system, which of these is the general solution of this system? Here, C =   is an arbitrary constant vector.

Accepted Answer

A)  x(t) =   (t)C 
B)  x(t) =    (t) + C 
C)  x(t) =    -1(t)C 
D)  x(t) =   (0)C 
A)  x(t) =   (t)C 
B)  x(t) =    (t) + C 
C)  x(t) =    -1(t)C 
D)  x(t) =   (0)C

Question 11

Consider the first-order nonhomogeneous initial value problem
   
 Compute ψ-1(t) the inverse of the fundamental matrix of the system.

Accepted Answer

Question 12

Consider the first-order nonhomogeneous system of linear differential equations
 
Where the components of g(t) are continuous functions.
Which of these is the fundamental matrix   (t) for this system?

Accepted Answer

A)  $ \psi(t)=\left(\begin{array}{l}e^{3 t}-3 e^{-1 t} \ e^{3 t} 1 e^{-1 t}\end{array}\right) $ 
B)  $ \Psi(t)=\left(\begin{array}{l}e^{3 t} 3 e^{-1 t} \ e^{3 t}-1 e^{-1 t}\end{array}\right) $ 
C)  $ \psi(t)=\left(\begin{array}{ll}-e^{3 t} & -1 e^{-1 t} \ e^{3 t} & 3 e^{-1 t}\end{array}\right) $ 
D)  $ \psi(t)=\left(\begin{array}{l}e^{-3 t} 3 e^{-1 t} \ e^{-3 t}-1 e^{-1 t}\end{array}\right) $ 
A)  $ \psi(t)=\left(\begin{array}{l}e^{3 t}-3 e^{-1 t} \ e^{3 t} 1 e^{-1 t}\end{array}\right) $ 
B)  $ \Psi(t)=\left(\begin{array}{l}e^{3 t} 3 e^{-1 t} \ e^{3 t}-1 e^{-1 t}\end{array}\right) $ 
C)  $ \psi(t)=\left(\begin{array}{ll}-e^{3 t} & -1 e^{-1 t} \ e^{3 t} & 3 e^{-1 t}\end{array}\right) $ 
D)  $ \psi(t)=\left(\begin{array}{l}e^{-3 t} 3 e^{-1 t} \ e^{-3 t}-1 e^{-1 t}\end{array}\right) $

Question 13

Consider the first-order nonhomogeneous initial value problem
 
Which of these is the fundamental   (t) for this system?

Accepted Answer

A)  $ \psi(t)=\left(\begin{array}{ll}5 e^{5 t} & -e^{3 t} \ -3 e^{5 t} & e^{3 t}\end{array}\right) $ 
B)  $ \Psi(t)=\left(\begin{array}{cc}-3 e^{5 t} & e^{3 t} \ 5 e^{5 t} & -e^{3 t}\end{array}\right) $ 
C)  $ \psi(t)=\left(\begin{array}{ll}-3 e^{-5 t} & e^{-3 t} \ 5 e^{-5 t} & -e^{-3 t}\end{array}\right) $ 
D)  $ \psi(t)=\left(\begin{array}{ll}5 e^{-5 t} & -e^{-3 t} \ -3 e^{-5 t} & e^{-3 t}\end{array}\right) $ 
A)  $ \psi(t)=\left(\begin{array}{ll}5 e^{5 t} & -e^{3 t} \ -3 e^{5 t} & e^{3 t}\end{array}\right) $ 
B)  $ \Psi(t)=\left(\begin{array}{cc}-3 e^{5 t} & e^{3 t} \ 5 e^{5 t} & -e^{3 t}\end{array}\right) $ 
C)  $ \psi(t)=\left(\begin{array}{ll}-3 e^{-5 t} & e^{-3 t} \ 5 e^{-5 t} & -e^{-3 t}\end{array}\right) $ 
D)  $ \psi(t)=\left(\begin{array}{ll}5 e^{-5 t} & -e^{-3 t} \ -3 e^{-5 t} & e^{-3 t}\end{array}\right) $

Question 14

If $\lambda$ = 0 is an eigenvalue of a 5 × 5 matrix A, then A is not invertible.

Accepted Answer

A) True 
 B)False

Question 15

Each of the following is the general solution of a system of differential equations. For which one(s) is the origin a node? Select all that apply.

Accepted Answer

A)  I 
B)  II 
C)  III 
D)  None of them 
A)  I 
B)  II 
C)  III 
D)  None of them

Question 16

Consider the following system of linear equations:
   
What is the augmented matrix for this system?

Accepted Answer

The answer of Consider the following system of linear equations:
...

Question 17

Consider these matrices:
 
Which of the following matrices are defined? Select all that apply.

Accepted Answer

A)  AB 
B)    
C)  BA 
D)  AC 
E)  DC
F) BD
G) A + B 
A)  AB 
B)    
C)  BA 
D)  AC 
E)  DC
F) BD
G) A + B

Consider the first-order homogeneous system of linear differential equations Which of these are eigenvalues for this system? Select all that apply.

Consider the first-order nonhomogeneous initial value problem Given a fundamental matrix (t) for the system, what is the solution of this initial value problem?

Consider the first-order homogeneous system of linear differential equations Which of these is the fundamental matrix ?(t) for this system?

Consider the first-order homogeneous system of linear differential equations What is the general solution of this system? Here, and are arbitrary real constants.

Consider the first-order nonhomogeneous initial value problem Which of these is the fundamental matrix (t) for this system?

Consider these matrices:

Consider the first-order homogeneous system of linear differential equations = x What is the bifurcation value of α, if any?

Consider the matrix function A(t) Compute

Consider the first-order homogeneous system of linear differential equations X = x Given a fundamental matrix (t) for the system, which of these is the general solution of this system? Here, C = is an arbitrary constant vector.

Consider the first-order nonhomogeneous initial value problem Compute ψ^-1(t) the inverse of the fundamental matrix of the system.

Consider the first-order nonhomogeneous system of linear differential equations Where the components of g(t) are continuous functions. Which of these is the fundamental matrix (t) for this system?

Consider the first-order nonhomogeneous initial value problem Which of these is the fundamental matrix (t) for this system?

If $\lambda$ = 0 is an eigenvalue of a 5 × 5 matrix A, then A is not invertible.

Each of the following is the general solution of a system of differential equations. For which one(s) is the origin a node? Select all that apply.

Consider the following system of linear equations: What is the augmented matrix for this system?

Consider these matrices: Which of the following matrices are defined? 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

Numerical Methods

Nonlinear Differential Equations and Stability

Partial Differential Equations and Fourier Series

Boundary Value Problems and Sturm-Liouville Theory

Filters

Exam 7: Systems of First-Order Linear Equations

Consider the first-order homogeneous system of linear differential equations Which of these are eigenvalues for this system? Select all that apply.

Consider the first-order nonhomogeneous initial value problem Given a fundamental matrix (t) for the system, what is the solution of this initial value problem?

Consider the first-order homogeneous system of linear differential equations Which of these is the fundamental matrix ?(t) for this system?

Consider the first-order homogeneous system of linear differential equations What is the general solution of this system? Here, and are arbitrary real constants.

Consider the first-order nonhomogeneous initial value problem Which of these is the fundamental matrix (t) for this system?

Consider the first-order homogeneous system of linear differential equations Which of the following is an accurate statement regarding the behavior of the solution trajectories of this system as t →\rightarrow→∞\infty∞ ?

Consider these matrices:

Consider the first-order homogeneous system of linear differential equations = x What is the bifurcation value of α, if any?

Consider the matrix function A(t) Compute

Consider the first-order homogeneous system of linear differential equations X = x Given a fundamental matrix (t) for the system, which of these is the general solution of this system? Here, C = is an arbitrary constant vector.

Consider the first-order nonhomogeneous initial value problem Compute ψ-1(t) the inverse of the fundamental matrix of the system.

Consider the first-order nonhomogeneous system of linear differential equations Where the components of g(t) are continuous functions. Which of these is the fundamental matrix (t) for this system?

Consider the first-order nonhomogeneous initial value problem Which of these is the fundamental matrix (t) for this system?

If λ\lambdaλ = 0 is an eigenvalue of a 5 × 5 matrix A, then A is not invertible.

Each of the following is the general solution of a system of differential equations. For which one(s) is the origin a node? Select all that apply.

Consider the following system of linear equations: What is the augmented matrix for this system?

Consider these matrices: Which of the following matrices are defined? 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

Numerical Methods

Nonlinear Differential Equations and Stability

Partial Differential Equations and Fourier Series

Boundary Value Problems and Sturm-Liouville Theory

Filters

Consider the first-order nonhomogeneous initial value problem Compute ψ^-1(t) the inverse of the fundamental matrix of the system.

If $\lambda$ = 0 is an eigenvalue of a 5 × 5 matrix A, then A is not invertible.