Deck 7: Systems of First-Order Linear Equations

Question 1

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

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

Answer 1

= , = 7 - 5

Question 2

Into which of the following systems can this homogeneous third-order differential equation be transformed?

A) $x_{1}^{\prime}=x_{2}, x_{2}^{\prime}=x_{3}, x_{3}^{\prime}=-10 x_{3}-2 x_{2}-8 x_{1}$
B) $x_{1}^{\prime}=x_{2}, x_{2}^{\prime}=x_{3}, x_{3}^{\prime}=\frac{10}{5 t} x_{3}+\frac{2}{5 t^{2}} x_{2}+\frac{8}{5 t^{3}} x_{1}$
C) $x_{1^{\prime}}=x_{2}, x_{2}^{\prime}=x_{3}, x_{3}^{\prime}=10 x_{3}+2 x_{2}+8 x_{1}$
D) $x_{1^{\prime}}^{\prime}=x_{2}, x_{2}^{\prime}=x_{3}, x_{3^{\prime}}^{\prime}=\frac{-10}{5 t} x_{3}-\frac{2}{5 t^{2}} x_{2}-\frac{8}{5 t^{3}} x_{1}$

Answer 2

$x_{1^{\prime}}^{\prime}=x_{2}, x_{2}^{\prime}=x_{3}, x_{3^{\prime}}^{\prime}=\frac{-10}{5 t} x_{3}-\frac{2}{5 t^{2}} x_{2}-\frac{8}{5 t^{3}} x_{1}$

Question 3

Transform this nonhomogeneous second-order initial value problem into an initial value problem comprised of two first-order differential equations:

A)
B)https://d2lvgg3v3hfg70.cloudfront.net/TBW1042/.
C)https://d2lvgg3v3hfg70.cloudfront.net/TBW1042/..
D)https://d2lvgg3v3hfg70.cloudfront.net/TBW1042/11eec281_8310_7efd_8720_6bc18854d758_TBW1042_11...

Answer 3

Question 4

Consider this system of first-order differential equations:

(i) Transform this system into a second-order differential equation whose solution is X₁ .

(ii) Find the general solution of the differential equation in part (i).
(iii) Use your solution in (ii) to now find X₂.

Question 5

Consider this system of first-order differential equations:

Transform this system into a second-order differential equation whose solution is X₂ .

A) $x_{2}^{\prime \prime}+12 x_{2}^{\prime}-45 x_{2}=0$
B) $x_{2}^{\prime \prime}-12 x_{2}^{\prime}+45 x_{2}=0$
C) $x_{2}^{\prime \prime}-12 x_{2}^{\prime}+2025 x_{2}=0$
D) $x_{2}^{\prime \prime}+12 x_{2}^{\prime}-2025 x_{2}=0$

Question 6

Compute:

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 7

Compute:

A) $\left[\begin{array}{ll}3 & -4 \\ 7 & -9 \\ -1 & -2\end{array}\right]$
B) $\left[\begin{array}{cc}9 & 0 \\ 19 & -19 \\ -3 & -2\end{array}\right]$
C) $\left[\begin{array}{ll}-3 & -8 \\ -5 & 1 \\ 1 & -2\end{array}\right]$
D) $\left[\begin{array}{ll}-9 & -12 \\ -17 & 11 \\ 3 & -2\end{array}\right]$

Question 8

Consider these matrices:

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

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

Question 9

Consider the matrix

A) $\left[\begin{array}{lllll}24 & -6 & -3 & -27 & 21 \\ 9 & -15 & -15 & 6 & 15\end{array}\right]$
B) $\left[\begin{array}{cc}8 & 3 \\ -2 & -5 \\ -1 & -5 \\ -9 & 2 \\ 7 & 5\end{array}\right]$
C) $\left[\begin{array}{ll}24 & 9 \\ -6 & -15 \\ -3 & -15 \\ -27 & 6 \\ 21 & 15\end{array}\right]$
D) $\left[\begin{array}{ll}21 & 15 \\ -27 & 6 \\ -3 & -15 \\ -6 & -15 \\ 24 & 9\end{array}\right]$
E) $\left[\begin{array}{ll}9 & 24 \\ -15 & -6 \\ -15 & -3 \\ 6 & -27 \\ 15 & 21\end{array}\right]$

Question 10

If A is an 2 × 4 matrix and B is an 4 × 9 matrix, then:

A) BA is defined and has order 2 × 9.
B) BA is defined and has order 9 × 2.
C) AB is defined and has order 9 × 2.
D) AB is defined and has order 2 × 9.
E) Neither AB nor BA is defined.

Question 11

Consider these matrices:
D = E =
Compute

Question 12

Consider these matrices:

Question 13

Consider these matrices:

Compute ED

Question 14

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

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 15

Consider the matrix function A(t) Compute

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 16

Consider the matrix function .
Compute

Question 17

Consider the matrix Compute B^-1

Question 18

Consider the following system of linear equations:

What is the augmented matrix for this system?

Question 19

Consider the following system of linear equations:

Reduce the augmented matrix of this system to echelon form.

Question 20

Consider the following system of linear equations:

The system is inconsistent.

Question 21

Consider the following system of linear equations:

Find a condition involving that ensures the system has infinitely many solutions.

Question 22

Consider this set of vectors: Which of these statements is true?

A) The vectors in this set are linearly independent.
B) The vectors in this set are linearly dependent.
C) The system Ax = 0, where A = , has only the solution x = 0.
D) The system Ax = 0, where A = , is inconsistent.

Question 23

Are the vectors u₁ , u₂ , and u₃ linearly independent or linearly dependent? If they are linearly dependent, identify appropriate constants A, B, and C for which A u₁ + Bu₂ +Cu₃ = 0
that demonstrates this fact.

Question 24

Are the vectors u₁ , u₂ , u₃ , and u₄ linearly independent or linearly dependent? If they are linearly dependent, identify appropriate constants A, B, C, and D for which A u₁ + Bu₂ +Cu₃ +Du₃ = 0 for which that demonstrates this fact.

Question 25

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

Question 26

Given that $\lambda$ = 1 is an eigenvalue of the matrix B = , which of the following statements is true regarding the eigenvector of B associated with this eigenvalue $\lambda$ = 1?

A) is the only eigenvector of B associated with the eigenvalue $\lambda$ = 1.
B) is an eigenvector of B, for any nonzero real constant $\alpha$ , associated with the eigenvalue $\lambda$ = 1.
C) is an eigenvector of B, for any nonzero real constant $\alpha$ , associated with the eigenvalue $\lambda$ = 1.
D) is the only eigenvector of B associated with the eigenvalue $\lambda$ = 1.

Question 27

Consider the matrix
Which of these is a complete list of eigenvalue-eigenvector pairs of A?

A) $\lambda_{1}=4, \quad \xi_{1}=\left(\begin{array}{l}3 \\ 1\end{array}\right) \quad \lambda_{2}=-4, \quad \xi_{2}=\left(\begin{array}{c}-1 \\ 1\end{array}\right)$
B) $\lambda_{1}=4, \xi_{1}=\left(\begin{array}{l}1 \\ 3\end{array}\right), \lambda_{2}=-4, \xi_{2}=\left(\begin{array}{c}1 \\ -1\end{array}\right)$
C) $\lambda_{1}=2, \xi_{1}=\left(\begin{array}{l}3 \\ 1\end{array}\right) \quad \lambda_{2}=-2, \quad \xi_{2}=\left(\begin{array}{c}-1 \\ 1\end{array}\right)$
D) $\lambda_{1}=2, \quad \xi_{1}=\left(\begin{array}{l}1 \\ 3\end{array}\right) \quad \lambda_{2}=-2, \quad \xi_{2}=\left(\begin{array}{c}1 \\ -1\end{array}\right)$

Question 28

Consider a system of homogeneous first-order linear differential equations of the form = Ax, where A is a 2 × 2 constant matrix. are solutions of this system, which of the following must also be solutions of this system? Select all that apply.

A) $-2 \mathbf{x}_{1}(t)$
B) $-7.2 \mathbf{x}_{1}(t)+4.4 \mathbf{x}_{2}(t)$
C) $\mathbf{x}_{1}(t) \cdot \mathbf{x}_{2}(t)$
D) $-6.6 t \mathbf{x}_{1}(t)+5.8 t x_{2}(t)$
E) $\left.\left(7.2 \mathbf{x}_{1}(t)+5.4 \mathbf{x}_{2}(t)\right)-8\left(\mathbf{x}_{1}(t)-\mathbf{x}_{2} t\right)\right)$
F) $2 \mathbf{x}_{1}(t)-3.6 \mathbf{x}_{2}(t)-4.6$

Question 29

Consider the first-order homogeneous system of linear differential equations and the following three vector functions:

Which of the following statements are true? Select all that apply.

A) {X₁,X₂ ,X₃ } is a fundamental set of solutions for this system.
B) W [X₁(t), X₂(t)] ? 0 for every real number t.
C) X₁ and X₂ are linearly dependent.
D) 6X₁ + 4X₂ + 3X₃ is a solution of this system.
E) {X₁, X₂} is a fundamental set of solutions for this system.

Question 30

Consider the first-order homogeneous system of linear differential equations

And the following four vector functions:

Which of the following statements are true? Select all that apply.

A) is a solution of this system, for all real numbers C₁ ,C₂ ,C₃ , and C₄ .
B) W [ (t), (t)] ? 0 for every real number t.
C) 5.5 + 4.5 + C is a solution of this system, for any real number C.
D) 4 is a solution of this system.
E) { , } is a fundamental set of solutions for this system.

Question 31

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

Question 32

Consider the first-order homogeneous system of linear differential equations

What is the general solution of this system?

Question 33

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?

Question 34

Consider the first-order homogeneous system of linear differential equations
= x
Select all of the correct eigenvalue-eigenvector pairs from the following choices.

A) $\lambda=9, \xi=\left(\begin{array}{l}1 \\ 1\end{array}\right)$
B) $\lambda=-9, \xi=\left(\begin{array}{c}-1 \\ 1\end{array}\right)$
C) $\lambda=4, \xi=\left(\begin{array}{l}-9 \\ 4\end{array}\right)$
D) $\lambda=0, \xi=\left(\begin{array}{l}1 \\ 0\end{array}\right)$
E) $\lambda=-4, \xi=\left(\begin{array}{l}9 \\ -4\end{array}\right)$

Question 35

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.

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 36

Consider the first-order homogeneous system of linear differential equations

The origin is a saddle point.

Question 37

Consider the first-order homogeneous system of linear differential equations

Select all of the correct eigenvalue-eigenvector pairs from the following choices.

A) $\lambda=9, \xi=\left(\begin{array}{l}-9 \\ -9\end{array}\right)$
B) $\lambda=-9, \xi=\left(\begin{array}{l}-9 \\ -9\end{array}\right)$
C) $\lambda=-9, \xi=\left(\begin{array}{l}9 \\ -9\end{array}\right)$
D) $\lambda=9, \xi=\left(\begin{array}{l}3 \mathrm{i} \\ 3 \mathrm{i}\end{array}\right)$
E) $\lambda=9, \xi=\left(\begin{array}{l}9 \\ -9\end{array}\right)$
F) $\lambda=-9, \xi=\left(\begin{array}{c}-3 \mathrm{i} \\ 3 \mathrm{i}\end{array}\right)$

Question 38

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

A) $\mathbf{x}(t)=C_{1}\left(\begin{array}{l}-25 \\ -16\end{array}\right) e^{-25 t}+C_{2}\left(\begin{array}{l}16 \\ -25\end{array}\right) e^{-16 t}$
B) $\mathbf{x}(t)=C_{1}\left(\begin{array}{l}-16 \\ -25\end{array}\right) e^{25 t}+C_{2}\left(\begin{array}{c}16 \\ -25\end{array}\right) e^{-16 t}$
C) $\mathbf{x}(t)=C_{1}\left(\begin{array}{l}5 \mathrm{i} \\ 4 \mathrm{i}\end{array}\right) e^{20 t}+C_{2}\left(\begin{array}{c}-5 \mathrm{i} \\ 4 \mathrm{i}\end{array}\right) e^{-20 t}$
D) $\mathbf{x}(t)=C_{1}\left(\begin{array}{l}-16 \\ -25\end{array}\right) e^{25 t}+C_{2}\left(\begin{array}{c}16 \\ -25\end{array}\right) e^{16 t}$
E) $\mathbf{x}(t)=C_{1}\left(\begin{array}{l}5 \mathrm{i} \\ 4 \mathrm{i}\end{array}\right) e^{-20 t}+C_{2}\left(\begin{array}{c}-5 \mathrm{i} \\ 4 \mathrm{i}\end{array}\right) e^{20 t}$

Question 39

Consider the first-order homogeneous system of linear differential equations

The origin is a node.

Question 40

Consider the first-order homogeneous system of linear differential equations

Select all of the correct eigenvalue-eigenvector pairs from the following choices.

A) $\lambda=3, \xi=\left(\begin{array}{c}0 \\ -1\end{array}\right)$
B) $\lambda=-3, \xi=\left(\begin{array}{l}0 \\ 1\end{array}\right)$
C) $\lambda=8, \xi=\left(\begin{array}{l}10 \\ 5\end{array}\right)$
D) $\lambda=8, \xi=\left(\begin{array}{l}10 \\ -5\end{array}\right)$
E) $\lambda=-3, \xi=\left(\begin{array}{l}1 \\ 0\end{array}\right)$
F) $\lambda=8, \xi=\left(\begin{array}{c}-5 \\ 10\end{array}\right)$

Question 41

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.

A) $\mathbf{x}(t)=C_{1}\left(\begin{array}{l}3 \\ -10\end{array}\right) e^{-2 t}+C_{2}\left(\begin{array}{l}0 \\ 1\end{array}\right) e^{-5 t}$
B) $\mathbf{x}(t)=C_{1}\left(\begin{array}{l}3 \\ -10\end{array}\right) e^{-2 t}+C_{2}\left(\begin{array}{c}0 \\ -1\end{array}\right) e^{5 t}$
C) $\mathbf{x}(t)=C_{1}\left(\begin{array}{l}3 \\ -10\end{array}\right) e^{-2 t}+C_{2}\left(\begin{array}{l}-10 \\ -3\end{array}\right) e^{2 t}$
D) $\mathbf{x}(t)=C_{1}\left(\begin{array}{l}-1 \\ 0\end{array}\right) e^{5 t}+C_{2}\left(\begin{array}{l}3 \\ -10\end{array}\right) e^{-2 t}$
E) $\mathbf{x}(t)=C_{1}\left(\begin{array}{l}0 \\ 1\end{array}\right) e^{-5 t}+C_{2}\left(\begin{array}{l}-10 \\ -3\end{array}\right) e^{2 t}$

Question 42

Consider the first-order homogeneous system of linear differential equations

The origin is a node.

Question 43

Consider the first-order homogeneous system of linear differential equations

Select all of the eigenvalue-eigenvector pairs from the following choices.

A) $\lambda=3, \xi=\left(\begin{array}{c}-1 \\ 0\end{array}\right)$
B) $\lambda=-3, \xi=\left(\begin{array}{l}1 \\ 0\end{array}\right)$
C) $\lambda=0, \xi=\left(\begin{array}{c}-1 \\ 1\end{array}\right)$
D) $\lambda=0, \xi=\left(\begin{array}{c}1 \\ -1\end{array}\right)$
E) $\lambda=-3, \xi=\left(\begin{array}{l}0 \\ 1\end{array}\right)$
F) $\lambda=3, \xi=\left(\begin{array}{l}0 \\ -1\end{array}\right)$

Question 44

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.

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

Question 45

Consider the first-order homogeneous system of linear differential equations

What is the general solution of this system?

A) $\mathbf{x}(t)=C_{1}\left(\begin{array}{l}-1 \\ 2 \\ -4\end{array}\right) e^{2 t}+C_{2}\left(\begin{array}{c}1 \\ -1 \\ 1\end{array}\right) e^{-t}+C_{3}\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right) e^{t}$
B) $\mathbf{x}(t)=C_{1}\left(\begin{array}{c}1 \\ -2 \\ 4\end{array}\right) e^{-2 t}+C_{2}\left(\begin{array}{c}1 \\ -1 \\ 1\end{array}\right) e^{-t}+C_{3}\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right) e^{t}$
C) $\mathbf{x}(t)=C_{1}\left(\begin{array}{l}1 \\ 1 \\ -1\end{array}\right)+C_{2}\left(\begin{array}{c}1 \\ -1 \\ 1\end{array}\right) e^{-t}+C_{3}\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right) e^{t}$
D) $x(t)=C_{1}\left(\begin{array}{c}1 \\ -2 \\ 4\end{array}\right) e^{-2 t}+C_{2}\left(\begin{array}{c}2 \\ 1 \\ -1\end{array}\right) e^{-t}+C_{3}\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right) e^{t}$
E) $\mathbf{x}(t)=C_{1}\left(\begin{array}{l}1 \\ -2 \\ 4\end{array}\right) e^{-2 t}+C_{2}\left(\begin{array}{l}1 \\ -1 \\ 1\end{array}\right) e^{-t}+C_{3}\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right)$

Question 46

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?

Question 47

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.

Question 48

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.

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

Question 49

Consider the first-order homogeneous system of linear differential equations

Determine a fundamental set of solutions for this system.

Question 50

Consider the first-order homogeneous system of linear differential equations

What is the general solution of this system?

Question 51

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

A) -8
B) 8
C) 0
D) 8i
E) -8i
F) 64
G) -64

Question 52

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

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 53

Consider the first-order homogeneous system of linear differential equations

Which of these is an accurate description of the solution trajectories of the phase portrait for this system?

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

Question 54

Consider the first-order homogeneous system of linear differential equations

Which of these are eigenvalues for this system? Select all that apply.

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

Question 55

Consider the first-order homogeneous system of linear differential equations

What is the general solution of this system? Here, and are arbitrary real constants.

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 56

Consider the first-order homogeneous system of linear differential equations

All solution trajectories spiral towards the origin as t $\rightarrow$$\infty$ .

Question 57

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

A) -4 + 7i
B) 7
C) -7i
D) -4
E) 7i
F) 7 + 4i
G) 7 - 4i
H) -4 - 7i

Question 58

Consider the first-order homogeneous system of linear differential equations

What is the general solution of this system? Here, and are arbitrary real constants.

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

Question 59

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$ ?

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 60

Consider the first-order homogeneous system of linear differential equations

Which of these are eigenvalues for this system? Select all that apply.

A) 0
B) -16
C) 4i
D) -4i
E) -4
F) 4
G) 16

Question 61

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.

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 62

Consider the first-order homogeneous system of linear differential equations

Determine the eigenvalues for this system and describe the behavior of the solution trajectories as t $\rightarrow$$\infty$ .

A) $\lambda=1 \pm i \sqrt{77}$ ; all solution trajectories spiral toward the origin as $t \rightarrow \infty$ .
B) $\lambda=1 \pm i \sqrt{77}$ ; all solution trajectories spiral away from the origin as $t \rightarrow \infty$ .
C) $\lambda=1 \pm i \sqrt{77} ;$ the origin is a saddle.
D) $\lambda=-1 \pm i \sqrt{77}$ ; all solution trajectories spiral toward the origin as $t \rightarrow \infty$ .
E) $\lambda=-1 \pm i \sqrt{77}$ ; all solution trajectories spiral away from the origin as $t \rightarrow \infty$ .

Question 63

Consider the first-order homogeneous system of linear differential equations
‪
(i) For what real values of α does this system have complex eigenvalues?
(ii) What do the solution trajectories look like for the values of α found in part (i)?

Question 64

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

Question 65

Consider the first-order homogeneous system of linear differential equations

Which of these statements is true?

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 66

Consider the first-order homogeneous system of linear differential equations

The eigenvalues and corresponding eigenvectors for this system are:

Which of these is the general solution for this system?

A) $\mathbf{x}(t)=C_{1} e^{t}\left[\begin{array}{l}\sin (5 t) \\ -\cos (5 t)\end{array}\right)+C_{2} e^{t}\left[\begin{array}{l}-\sin (5 t) \\ \cos (5 t)\end{array}\right)$
B) $\mathbf{x}(t)=C_{1} e^{t}\left(\begin{array}{l}\sin (5 t) \\ \cos (5 t)\end{array}\right)+C_{2} e^{t}\left(\begin{array}{l}-\sin (5 t) \\ \cos (5 t)\end{array}\right)$
C) $\mathbf{x}(t)=C_{1} e^{t}\left[\begin{array}{l}-\sin (5 t) \\ \cos (5 t)\end{array}\right]+C_{2} e^{t}\left[\begin{array}{l}\cos (5 t) \\ -\sin (5 t)\end{array}\right)$
D) $\mathbf{x}(t)=C_{1} e^{t}\left\{\begin{array}{l}\sin (5 t) \\ \cos (5 t)\end{array}\right]+C_{2} e^{t}\left[\begin{array}{l}\cos (5 t) \\ -\sin (5 t)\end{array}\right]$

Question 67

Consider the first-order homogeneous system of linear differential equations

Which of these is the fundamental matrix ?(t) for this system?

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 68

Consider the first-order homogeneous system of linear differential equations

Which of these is the fundamental matrix (t) for this system?

A) $\psi(t)=\left(\begin{array}{l}-7 e^{-1 t} e^{-6 t} \\ -8 e^{-1 t} 0\end{array}\right)$
B) $\Psi(t)=\left(\begin{array}{l}-7 e^{-1 /} 0 \\ -8 e^{-1 t} e^{-6 t}\end{array}\right)$
C) $\psi(t)=\left(\begin{array}{ll}7 e^{1 t} & e^{-6 t} \\ -8 e^{1 t} & 0\end{array}\right)$
D) $\psi(t)=\left(\begin{array}{ll}7 e^{1 t} & 0 \\ -8 e^{1 t} & e^{-6 t}\end{array}\right)$

Question 69

Consider the first-order homogeneous system of linear differential equations
?
Given a fundamental matrix (t) for the system, for what constant vector does

Question 70

Consider the first-order homogeneous system of linear differential equations

Which of these is the fundamental matrix (t) for this system?

A) $\psi(t)=\left(\begin{array}{l}4 e^{12 t}-4 e^{-12 t} \\ 3 e^{12 t}-3 e^{-12 t}\end{array}\right)$
B) $\Psi(t)=\left(\begin{array}{l}4 e^{12 t}-4 e^{-12 t} \\ 3 e^{12 t} 3 e^{-12 t}\end{array}\right)$
C) $\psi(t)=\left(\begin{array}{ll}9 e^{-16 t} & -9 e^{9 t} \\ 16 e^{-16 t} & 16 e^{9 t}\end{array}\right)$
D) $\psi(t)=\left(\begin{array}{ll}9 e^{-16 t} & 9 e^{9 t} \\ -16 e^{-16 t} & 16 e^{9 t}\end{array}\right)$

Question 71

Consider the first-order homogeneous system of linear differential equations
= x
The columns of the fundamental matrix of this system, (t), must be linearly independent.

Question 72

Consider the first-order homogeneous system of linear differential equations

The fundamental matrix of this system, (t), is invertible.

Question 73

Consider the first-order homogeneous system of linear differential equations

Which of these is the fundamental matrix (t) for this system?

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

Question 74

Consider the first-order homogeneous system of linear differential equations

Given a fundamental matrix (t) for the system, which of these is the general solution of this system?
Here, is an arbitrary constant vector.

A) $\mathbf{x}(t)=\psi^{-1}(t) \mathbf{C}$
B) $x(t)=\psi(t) \mathbf{C}$
C) $\mathbf{x}(t)=\psi(0) \mathbf{C}$
D) $\mathbf{x}(t)=\psi(t)+C$

Question 75

Consider the first-order homogeneous system of linear differential equations

Which of these is the fundamental matrix (t) for this system?

A) $\psi(t)=\left(\begin{array}{ll}-e^{-4 t} \sin (7 t) & e^{-4 t} \cos (7 t) \\ e^{-4 t} \cos (7 t) & -e^{-4 t} \sin (7 t)\end{array}\right)$
B) $\psi(t)=\left\{\begin{array}{ll}e^{4 t} \sin (7 t) & -e^{4 t} \cos (7 t) \\ -e^{4 t} \cos (7 t) & e^{4 t} \sin (7 t)\end{array}\right)$
C) $\psi(t)=\left(\begin{array}{ll}e^{4 t} \sin (7 t) & -e^{4 t} \cos (7 t) \\ e^{4 t} \cos (7 t) & e^{4 t} \sin (7 t)\end{array}\right)$
D) $\psi(t)=\left(\begin{array}{ll}e^{-4 t} \sin (7 t) & -e^{-4 t} \cos (7 t) \\ e^{-4 t} \cos (7 t) & e^{-4 t} \sin (7 t)\end{array}\right)$

Question 76

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.

A) x(t) = 11eec32a_dc1b_6687_8720_5fa3d39c7544_TBW1042_11 (t)C
B) x(t) = 11eec32a_dc1b_6687_8720_5fa3d39c7544_TBW1042_11 (t) + C
C) x(t) = 11eec32a_dc1b_6687_8720_5fa3d39c7544_TBW1042_11 ^-1(t)C
D) x(t) = 11eec32a_dc1b_6687_8720_5fa3d39c7544_TBW1042_11 (0)C

Question 77

Consider the first-order homogeneous system of linear differential equations

Which of these is the fundamental matrix (t) for this system?

A)
B)
C)
D)

Question 78

Consider the first-order homogeneous system of linear differential equations

Given a fundamental matrix (t) for the system, which of these is the general solution of this system? Here, is an arbitrary constant vector.

A) x(t) = 11eec32a_dc1b_6687_8720_5fa3d39c7544_TBW1042_11 (t)C
B) x(t) = 11eec32a_dc1b_6687_8720_5fa3d39c7544_TBW1042_11 (t) + C
C) x(t) = 11eec32a_dc1b_6687_8720_5fa3d39c7544_TBW1042_11 ^-1(t)C
D) x(t) = 11eec32a_dc1b_6687_8720_5fa3d39c7544_TBW1042_11 (0)C

Question 79

Consider the first-order homogeneous system of linear differential equations

Select a pair of vectors from these choices that constitute a fundamental set of solutions for this system.

A) $\left(\begin{array}{l}e^{-7 t} \\ 0\end{array}\right)$
B) $\left(\begin{array}{l}0 \\ e^{-7 t}\end{array}\right)$
C) $\left(\begin{array}{l}e^{-7 t} \\ 2 e^{-7 t}\end{array}\right)$
D) $\left(\begin{array}{l}e^{-7 t} \\ 2(t+2) e^{-7 t}\end{array}\right)$
E) $\left(\begin{array}{l}t e^{-7 t} \\ 2 e^{-7 t}\end{array}\right)$
F) $\left(\begin{array}{l}e^{-7 t} \\ t e^{-7 t}\end{array}\right)$

Question 80

Consider the first-order homogeneous system of linear differential equations

What is the general solution of this system?