Question 1

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

Question 2

Consider the first-order homogeneous system of linear differential equations
 
The origin is called a(n) ________.

Accepted Answer

A)  improper node 
B)  center 
C)  saddle 
D)  spiral node 
A)  improper node 
B)  center 
C)  saddle 
D)  spiral node

Question 3

Consider the first-order homogeneous system of linear differential equations
   
The origin is a node.

Accepted Answer

A) True 
 B)False

Question 4

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

Accepted Answer

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

Question 5

Compute:

Accepted Answer

A)  $ \left[\begin{array}{ll}0 & -4 \ 12 & -4\end{array}\right] $ 
B)  $ \left[\begin{array}{ll}6 & 4 \ 2 & 6\end{array}\right] $ 
C)  $ \left[\begin{array}{ll}0 & 4 \ 12 & 6\end{array}\right] $ 
D)  $ \left[\begin{array}{ll}-12 & -8 \ -4 & -12\end{array}\right] $ 
A)  $ \left[\begin{array}{ll}0 & -4 \ 12 & -4\end{array}\right] $ 
B)  $ \left[\begin{array}{ll}6 & 4 \ 2 & 6\end{array}\right] $ 
C)  $ \left[\begin{array}{ll}0 & 4 \ 12 & 6\end{array}\right] $ 
D)  $ \left[\begin{array}{ll}-12 & -8 \ -4 & -12\end{array}\right] $

Question 6

Consider the first-order homogeneous system of linear differential equations
   
If the system were equipped with the initial condition   what is the particular solution of the system?

Accepted Answer

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

Question 7

Consider the first-order homogeneous system of linear differential equations
  =   x
Which of these is a fundamental set of solutions for this system?

Accepted Answer

A)  $ \left\{\begin{array}{ll}-\sin (6 t) & \cos (6 t) \ \cos (6 t) & \sin (6 t)\end{array}\right\} $ 
B)  $ \left\{\begin{array}{ll}\sin (6 t) & -\cos (6 t) \ \cos (6 t) & \sin (6 t)\end{array}\right\} $ 
C)  $ \left\{\begin{array}{ll}\sin (36 t) & -\cos (36 t) \ \cos (36 t) & \sin (36 t)\end{array}\right\} $ 
D)  $ \left\{\begin{array}{ll}e^{6 t} & -e^{-6 t} \ e^{6 t} & e^{-6 t}\end{array}\right\} $ 
E)  $ \left\{\begin{array}{ll}e^{6 t} & e^{-6 t} \ -e^{6 t}, & e^{-6 t}\end{array}\right\} $ 
A)  $ \left\{\begin{array}{ll}-\sin (6 t) & \cos (6 t) \ \cos (6 t) & \sin (6 t)\end{array}\right\} $ 
B)  $ \left\{\begin{array}{ll}\sin (6 t) & -\cos (6 t) \ \cos (6 t) & \sin (6 t)\end{array}\right\} $ 
C)  $ \left\{\begin{array}{ll}\sin (36 t) & -\cos (36 t) \ \cos (36 t) & \sin (36 t)\end{array}\right\} $ 
D)  $ \left\{\begin{array}{ll}e^{6 t} & -e^{-6 t} \ e^{6 t} & e^{-6 t}\end{array}\right\} $ 
E)  $ \left\{\begin{array}{ll}e^{6 t} & e^{-6 t} \ -e^{6 t}, & e^{-6 t}\end{array}\right\} $

Question 8

Into which of the following systems can this homogeneous second-order differential equation be transformed? 
  + 5   - 7 u = 0

Accepted Answer

A)    =   ,   = 5   - 7   
B)    =   ,   = 7   - 5   
C)    = -   ,   = 7   - 5   
D)    = -   ,   = 5   - 7   
A)    =   ,   = 5   - 7   
B)    =   ,   = 7   - 5   
C)    = -   ,   = 7   - 5   
D)    = -   ,   = 5   - 7

Question 9

Consider the first-order homogeneous system of linear differential equations
   
The origin is a saddle point.

Accepted Answer

A) True 
 B)False

Question 10

Consider the matrix function   .
Compute

Accepted Answer

The answer of Consider the matrix function   .
Compute...

Question 11

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(t) \psi^{-1}(0.6)\left(\begin{array}{l}-6 \ 2\end{array}\right)+\psi(0.6) \int_{0.6}^{t} \psi^{-1}(s)_{6 s}^{2 e^{5 s}} d s $ 
B)  $ x(t)=\psi^{-1}(0.6)\left(\begin{array}{l}-6 \ 2\end{array}\right)+\psi(0.6) \int_{0.6}^{t} \psi^{-1}(s){ }_{6 s}^{2 e^{5 s}} d s $ 
C)  $ \mathbf{x}(t)=\psi(t) \psi^{-1}(0.6)\left(\begin{array}{c}-6 \ 2\end{array}\right)+\psi(t) \int_{0.6}^{t} \psi^{-1}(s)_{6 s}^{2 e^{5 s}} d s $ 
D)  $ \mathbf{x}(t)=\psi^{-1}(0.6)\left(\begin{array}{c}-6 \ 2\end{array}\right)+\psi(t) \int_{0.6}^{t} \psi^{-1}(s) e^{2 e^{5 s}} d s $ 
A)  $ \mathbf{x}(t)=\psi(t) \psi^{-1}(0.6)\left(\begin{array}{l}-6 \ 2\end{array}\right)+\psi(0.6) \int_{0.6}^{t} \psi^{-1}(s)_{6 s}^{2 e^{5 s}} d s $ 
B)  $ x(t)=\psi^{-1}(0.6)\left(\begin{array}{l}-6 \ 2\end{array}\right)+\psi(0.6) \int_{0.6}^{t} \psi^{-1}(s){ }_{6 s}^{2 e^{5 s}} d s $ 
C)  $ \mathbf{x}(t)=\psi(t) \psi^{-1}(0.6)\left(\begin{array}{c}-6 \ 2\end{array}\right)+\psi(t) \int_{0.6}^{t} \psi^{-1}(s)_{6 s}^{2 e^{5 s}} d s $ 
D)  $ \mathbf{x}(t)=\psi^{-1}(0.6)\left(\begin{array}{c}-6 \ 2\end{array}\right)+\psi(t) \int_{0.6}^{t} \psi^{-1}(s) e^{2 e^{5 s}} d s $

Question 12

Consider the matrix function
A(t) =  
Compute   (t).

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)  $ \left[\begin{array}{l}2 \cos (2 t)-5 \sin (5 t) \ -5 \sin (5 t)-2 \cos (2 t)\end{array}\right] $ 
C)  $ \left[\begin{array}{ll}\cos (2 t) & -\sin (5 t) \ -\sin (5 t) & -\cos (2 t)\end{array}\right] $ 
D)  $ \left[\begin{array}{ll}-\cos (2 t) & \sin (5 t) \ \sin (5 t) & \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)  $ \left[\begin{array}{l}2 \cos (2 t)-5 \sin (5 t) \ -5 \sin (5 t)-2 \cos (2 t)\end{array}\right] $ 
C)  $ \left[\begin{array}{ll}\cos (2 t) & -\sin (5 t) \ -\sin (5 t) & -\cos (2 t)\end{array}\right] $ 
D)  $ \left[\begin{array}{ll}-\cos (2 t) & \sin (5 t) \ \sin (5 t) & \cos (2 t)\end{array}\right] $

Question 13

Consider the first-order homogeneous system of linear differential equations
  
Select the vectors from this list that, together, constitute a fundamental set of solutions for this system.

Accepted Answer

A)  $ \left(\begin{array}{l}0 \ 1 \ 0\end{array}\right) $ 
B)  $ \left(\begin{array}{l}1 \ 0 \ 0\end{array}\right) $ 
C)  $ \left[\begin{array}{l}-\cos (6 t) \ \sin (6 t) \ \cos (6 t)\end{array}\right) $ 
D)  $ \left(\begin{array}{l}\cos (6 t) \ \sin (6 t) \ -\cos (6 t)\end{array}\right) $ 
E)  $ \left(\begin{array}{l}\sin (6 t) \ \cos (6 t) \ -\cos (6 t)\end{array}\right) $
F) $ \left(\begin{array}{l}\sin (6 t) \ \cos (6 t) \ -\sin (6 t)\end{array}\right) $
G) $ \left[\begin{array}{l}-\sin (6 t) \ \cos (6 t) \ \sin (6 t)\end{array}\right) $ 
A)  $ \left(\begin{array}{l}0 \ 1 \ 0\end{array}\right) $ 
B)  $ \left(\begin{array}{l}1 \ 0 \ 0\end{array}\right) $ 
C)  $ \left[\begin{array}{l}-\cos (6 t) \ \sin (6 t) \ \cos (6 t)\end{array}\right) $ 
D)  $ \left(\begin{array}{l}\cos (6 t) \ \sin (6 t) \ -\cos (6 t)\end{array}\right) $ 
E)  $ \left(\begin{array}{l}\sin (6 t) \ \cos (6 t) \ -\cos (6 t)\end{array}\right) $
F) $ \left(\begin{array}{l}\sin (6 t) \ \cos (6 t) \ -\sin (6 t)\end{array}\right) $
G) $ \left[\begin{array}{l}-\sin (6 t) \ \cos (6 t) \ \sin (6 t)\end{array}\right) $

Question 14

Consider the first-order homogeneous system of linear differential equations
   
Determine a fundamental set of solutions for this system.

Accepted Answer

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

Question 15

Consider the first-order homogeneous system of linear differential equations
 
What is the general solution of this system? Here,   is an arbitrary constant vector.

Accepted Answer

A)  $ \mathbf{x}(t)=\left(\begin{array}{lll}e^{t} & e^{4 t} & \left(t-\frac{1}{6}\right) e^{4 t} \ 6 e^{t} & -3 e^{4 t} & -3 t e^{4 t} \ 3 e^{t} & 0 & 0\end{array}\right) \mathbf{C} $ 
B)  $ \mathbf{x}(t)=\left(\begin{array}{lll}e^{t} & e^{4 t} & \left(t+\frac{1}{6}\right) e^{4 t} \ 6 e^{t} & 3 e^{4 t} & 3 t e^{4 t} \ 3 e^{t} & 0 & 0\end{array}\right) $ C 
C)  $ \mathbf{x}(t)=\left[\begin{array}{lll}e^{t} & e^{4 t} & \left(t+\frac{1}{6}\right) e^{4 t} \ -3 e^{t} & -3 e^{4 t} & 3 t e^{4 t} \ 6 e^{t} & 0 & 0\end{array}\right] $ C 
D)  $ \mathbf{x}(t)=\left[\begin{array}{lll}e^{-t} & e^{-4 t} & \left(t+\frac{1}{6}\right) e^{-4 t} \ 6 e^{-t} & 3 e^{-4 t} & 3 t e^{-4 t} \ 3 e^{-t} & 0 & 0\end{array}\right] $ C 
A)  $ \mathbf{x}(t)=\left(\begin{array}{lll}e^{t} & e^{4 t} & \left(t-\frac{1}{6}\right) e^{4 t} \ 6 e^{t} & -3 e^{4 t} & -3 t e^{4 t} \ 3 e^{t} & 0 & 0\end{array}\right) \mathbf{C} $ 
B)  $ \mathbf{x}(t)=\left(\begin{array}{lll}e^{t} & e^{4 t} & \left(t+\frac{1}{6}\right) e^{4 t} \ 6 e^{t} & 3 e^{4 t} & 3 t e^{4 t} \ 3 e^{t} & 0 & 0\end{array}\right) $ C 
C)  $ \mathbf{x}(t)=\left[\begin{array}{lll}e^{t} & e^{4 t} & \left(t+\frac{1}{6}\right) e^{4 t} \ -3 e^{t} & -3 e^{4 t} & 3 t e^{4 t} \ 6 e^{t} & 0 & 0\end{array}\right] $ C 
D)  $ \mathbf{x}(t)=\left[\begin{array}{lll}e^{-t} & e^{-4 t} & \left(t+\frac{1}{6}\right) e^{-4 t} \ 6 e^{-t} & 3 e^{-4 t} & 3 t e^{-4 t} \ 3 e^{-t} & 0 & 0\end{array}\right] $ C

Question 16

Consider the first-order homogeneous system of linear differential equations
 
Which of these is the general solution of the system? Here, C1 and C2 are arbitrary real constants.

Accepted Answer

A)  $ \mathbf{x}(t)=C_{1}\left(\begin{array}{l}1 \ 1\end{array}\right) e^{-7 t}+C_{2}\left(\begin{array}{c}-7 \ 9\end{array}\right) e^{9 t} $ 
B)  $ x(t)=C_{1}\left(\begin{array}{l}1 \ 1\end{array}\right) e^{7 t}+C_{2}\left(\begin{array}{c}7 \ -9\end{array}\right) e^{-9 t} $ 
C)  $ \mathbf{x}(t)=C_{1}\left(\begin{array}{l}1 \ 1\end{array}\right) e^{7 t}+C_{2}\left(\begin{array}{c}-7 \ 9\end{array}\right) e^{9 t} $ 
D)  $ \mathbf{x}(t)=C_{1}\left(\begin{array}{l}1 \ 1\end{array}\right) e^{-7 t}+C_{2}\left(\begin{array}{c}7 \ -9\end{array}\right) e^{-9 t} $ 
E)  $ \mathbf{x}(t)=C_{1}\left(\begin{array}{l}1 \ 0\end{array}\right)+C_{2}\left(\begin{array}{c}7 \ -9\end{array}\right) e^{-9 t} $ 
A)  $ \mathbf{x}(t)=C_{1}\left(\begin{array}{l}1 \ 1\end{array}\right) e^{-7 t}+C_{2}\left(\begin{array}{c}-7 \ 9\end{array}\right) e^{9 t} $ 
B)  $ x(t)=C_{1}\left(\begin{array}{l}1 \ 1\end{array}\right) e^{7 t}+C_{2}\left(\begin{array}{c}7 \ -9\end{array}\right) e^{-9 t} $ 
C)  $ \mathbf{x}(t)=C_{1}\left(\begin{array}{l}1 \ 1\end{array}\right) e^{7 t}+C_{2}\left(\begin{array}{c}-7 \ 9\end{array}\right) e^{9 t} $ 
D)  $ \mathbf{x}(t)=C_{1}\left(\begin{array}{l}1 \ 1\end{array}\right) e^{-7 t}+C_{2}\left(\begin{array}{c}7 \ -9\end{array}\right) e^{-9 t} $ 
E)  $ \mathbf{x}(t)=C_{1}\left(\begin{array}{l}1 \ 0\end{array}\right)+C_{2}\left(\begin{array}{c}7 \ -9\end{array}\right) e^{-9 t} $

Question 17

Consider the first-order homogeneous system of linear differential equations
  
Which of these statements is true?

Accepted Answer

A)  For any real number $\alpha$, the eigenvalues for this system are real numbers. 
B)  For -8 < $\alpha$ < 0, the trajectories spiral towards the origin as t $\rightarrow$$\infty$. 
C)  For $\alpha$ < -8, the trajectories spiral away from the origin as t $\rightarrow$$\infty$. 
D)  For $\alpha$= 8, the eigenvalues are purely imaginary and the trajectories are concentric circles centered at the origin. 
A)  For any real number $\alpha$, the eigenvalues for this system are real numbers. 
B)  For -8 < $\alpha$ < 0, the trajectories spiral towards the origin as t $\rightarrow$$\infty$. 
C)  For $\alpha$ < -8, the trajectories spiral away from the origin as t $\rightarrow$$\infty$. 
D)  For $\alpha$= 8, the eigenvalues are purely imaginary and the trajectories are concentric circles centered at the origin.

Question 18

Suppose Tank A contains 50 gallons of water in which 30 ounces of salt are dissolved, and tank B contains 35 gallons of water in which 60 ounces of salt are dissolved. The following conditions also hold:
• Water with salt concentration of 1.6 ounces per gallon flows into Tank A at a rate of 1.8 gallons per minute.
• Water with salt concentration of 3.1 ounces per gallon flows into Tank B at a rate of 1.3 gallons per minute.
• Water flows from Tank A to Tank B at a rate of 1.8 gallons per minute.
• Water flows from Tank B to Tank A at a rate of 0.65 gallons per minute.
• Water drains from Tank B at a rate of 0.65 gallons per minute.
Set up a system of equations governing the amount of salt in Tank A,X A (t), and the amount of salt in tank B, X B (t), at any time t.

Accepted Answer

Question 19

Consider the first-order homogeneous system of linear differential equations 
   =   x
Determine the eigenvalue-eigenvector pairs of this system.

Accepted Answer

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

Question 20

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}-\sin (3 t) & \cos (3 t) \ \cos (3 t) & -\sin (3 t)\end{array}\right) $ 
B)  $ \Psi(t)=\left(\begin{array}{ll}\sin (3 t) & -\cos (3 t) \ -\cos (3 t) & \sin (3 t)\end{array}\right) $ 
C)  $ \psi(t)=\left(\begin{array}{ll}\sin (3 t) & \cos (3 t) \ \cos (3 t) & \sin (3 t)\end{array}\right) $ 
D)  $ \psi(t)=\left(\begin{array}{ll}\sin (3 t) & -\cos (3 t) \ \cos (3 t) & \sin (3 t)\end{array}\right] $ 
A) $ \psi(t)=\left(\begin{array}{ll}-\sin (3 t) & \cos (3 t) \ \cos (3 t) & -\sin (3 t)\end{array}\right) $ 
B)  $ \Psi(t)=\left(\begin{array}{ll}\sin (3 t) & -\cos (3 t) \ -\cos (3 t) & \sin (3 t)\end{array}\right) $ 
C)  $ \psi(t)=\left(\begin{array}{ll}\sin (3 t) & \cos (3 t) \ \cos (3 t) & \sin (3 t)\end{array}\right) $ 
D)  $ \psi(t)=\left(\begin{array}{ll}\sin (3 t) & -\cos (3 t) \ \cos (3 t) & \sin (3 t)\end{array}\right] $

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 The origin is called a(n) ________.

Consider the first-order homogeneous system of linear differential equations The origin is a node.

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

Compute:

Consider the first-order homogeneous system of linear differential equations If the system were equipped with the initial condition what is the particular solution of the system?

Consider the first-order homogeneous system of linear differential equations = x Which of these is a fundamental set of solutions for this system?

Into which of the following systems can this homogeneous second-order differential equation be transformed? + 5 - 7 u = 0

Consider the first-order homogeneous system of linear differential equations The origin is a saddle point.

Consider the matrix function . Compute

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 matrix function A(t) = Compute (t).

Consider the first-order homogeneous system of linear differential equations Select the vectors from this list that, together, constitute a fundamental set of solutions for this system.

Consider the first-order homogeneous system of linear differential equations Determine a fundamental set of solutions for this system.

Consider the first-order homogeneous system of linear differential equations What is the general solution of this system? Here, is an arbitrary constant vector.

Consider the first-order homogeneous system of linear differential equations Which of these is the general solution of the system? Here, C₁ and C₂ are arbitrary real constants.

Consider the first-order homogeneous system of linear differential equations Which of these statements is true?

Consider the first-order homogeneous system of linear differential equations = x Determine the eigenvalue-eigenvector pairs of this system.

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

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 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 The origin is called a(n) ________.

Consider the first-order homogeneous system of linear differential equations The origin is a node.

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

Compute:

Consider the first-order homogeneous system of linear differential equations If the system were equipped with the initial condition what is the particular solution of the system?

Consider the first-order homogeneous system of linear differential equations = x Which of these is a fundamental set of solutions for this system?

Into which of the following systems can this homogeneous second-order differential equation be transformed? + 5 - 7 u = 0

Consider the first-order homogeneous system of linear differential equations The origin is a saddle point.

Consider the matrix function . Compute

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 matrix function A(t) = Compute (t).

Consider the first-order homogeneous system of linear differential equations Select the vectors from this list that, together, constitute a fundamental set of solutions for this system.

Consider the first-order homogeneous system of linear differential equations Determine a fundamental set of solutions for this system.

Consider the first-order homogeneous system of linear differential equations What is the general solution of this system? Here, is an arbitrary constant vector.

Consider the first-order homogeneous system of linear differential equations Which of these is the general solution of the system? Here, C1 and C2 are arbitrary real constants.

Consider the first-order homogeneous system of linear differential equations Which of these statements is true?

Consider the first-order homogeneous system of linear differential equations = x Determine the eigenvalue-eigenvector pairs of this system.

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

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 homogeneous system of linear differential equations Which of these is the general solution of the system? Here, C₁ and C₂ are arbitrary real constants.