Systems Of First-Order Linear Equations Chapter.7 Test Bank - Complete Test Bank | Differential Equations 12e by William E. Boyce. DOCX document preview.

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Systems Of First-Order Linear Equations Chapter.7 Test Bank

Elementary Differential Equations, 12e (Boyce)

Chapter 7 Systems of First-Order Linear Equations

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

u'' + 5u' - 7u = 0

A) (x) with subscript (1)' = (x) with subscript (1), (x) with subscript (2)' = 5(x) with subscript (2) - 7(x) with subscript (1)

B) (x) with subscript (1)' = (x) with subscript (2), (x) with subscript (2)' = 7(x) with subscript (1) - 5(x) with subscript (2)

C) (x) with subscript (1)' = -(x) with subscript (2), (x) with subscript (2)' = 7(x) with subscript (1) - 5(x) with subscript (2)

D) (x) with subscript (1)' = -(x) with subscript (1), (x) with subscript (2)' = 5(x) with subscript (2) - 7(x) with subscript (1)

Type: MC Var: 1

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

5(t) with superscript (3)u''' + 10(t) with superscript (2)u'' + 2tx' + 8u = 0

A) (x) with subscript (1)' = (x) with subscript (2), (x) with subscript (2)' = (x) with subscript (3), (x) with subscript (3)' = -10(x) with subscript (3) - 2(x) with subscript (2) - 8(x) with subscript (1)

B) (x) with subscript (1)' = (x) with subscript (2), (x) with subscript (2)' = (x) with subscript (3), (x) with subscript (3)' = (10/5t)(x) with subscript (3) + (2/5(t) with superscript (2))(x) with subscript (2) + (8/5(t) with superscript (3))(x) with subscript (1)

C) (x) with subscript (1)' = (x) with subscript (2), (x) with subscript (2)' = (x) with subscript (3), (x) with subscript (3)' = 10(x) with subscript (3) + 2(x) with subscript (2) + 8(x) with subscript (1)

D) (x) with subscript (1)' = (x) with subscript (2), (x) with subscript (2)' = (x) with subscript (3), (x) with subscript (3)' = (-10/5t)(x) with subscript (3) - (2/5(t) with superscript (2))(x) with subscript (2) - (8/5(t) with superscript (3))(x) with subscript (1)

Type: MC Var: 1

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

u'' + 6u' + 7u = (e) with superscript (-2t), u(0) = 3, u'(0) = -8

A) (x) with subscript (1)' = (x) with subscript (2), (x) with subscript (2)' = (e) with superscript (-2t) - 7(x) with subscript (1) - 6(x) with subscript (2)

(x) with subscript (1)(0) = 3, (x) with subscript (2)(0) = -8

B) (x) with subscript (1)' = (x) with subscript (2), (x) with subscript (2)' = (e) with superscript (-2t) + 7(x) with subscript (1) + 6(x) with subscript (2)

(x) with subscript (1)(0) = 3, (x) with subscript (2)(0) = -8

C) (x) with subscript (1)' = (x) with subscript (1), (x) with subscript (2)' = (e) with superscript (-2t) - 7(x) with subscript (1) - 6(x) with subscript (2)

(x) with subscript (1)(0) = 3, (x) with subscript (2)(0) = -8

D) (x) with subscript (1)' = -(x) with subscript (2), (x) with subscript (2)' = 7(x) with subscript (1) + 6(x) with subscript (2) - (e) with superscript (-2t)

(x) with subscript (1)(0) = 3, (x) with subscript (2)(0) = -8

Type: MC Var: 1

4) Consider this system of first-order differential equations:

(x) with subscript (1)' = 3(x) with subscript (2), (x) with subscript (2)' = 4(x) with subscript (1)

(i) Transform this system into a second-order differential equation whose solution is (x) with subscript (1).

A. (x) with subscript (1)'' - (4/3)(x) with subscript (1) = 0

B. (x) with subscript (1)'' + (4/3)(x) with subscript (1) = 0

C. (x) with subscript (1)'' + 12(x) with subscript (1) = 0

D. (x) with subscript (1)'' - 12(x) with subscript (1) = 0

(ii) Find the general solution of the differential equation in part (i).

(iii) Use your solution in (ii) to now find (x) with subscript (2).

(ii) (x) with subscript (1) = (C) with subscript (1)(e) with superscript (-2square root of (3)t) + (C) with subscript (1)(e) with superscript (2square root of (3)t)

(iii) (x) with subscript (2) = -(C) with subscript (1)square root of ((4/3)) ∙ (e) with superscript (-2square root of (3)t) + (C) with subscript (2)square root of ((4/3)) ∙ (e) with superscript (2square root of (3)t)

Type: ES Var: 1

5) Consider this system of first-order differential equations:

(x) with subscript (1)' = 6(x) with subscript (1) + 3(x) with subscript (2), (x) with subscript (2)' = -3(x) with subscript (1) + 6(x) with subscript (1)

Transform this system into a second-order differential equation whose solution is (x) with subscript (2).

A) (x) with subscript (2)'' + 12(x) with subscript (2)' - 45(x) with subscript (2) = 0

B) (x) with subscript (2)'' - 12(x) with subscript (2)' + 45(x) with subscript (2) = 0

C) (x) with subscript (2)'' - 12(x) with subscript (2)' + 2025(x) with subscript (2) = 0

D) (x) with subscript (2)'' + 12(x) with subscript (2)' - 2025(x) with subscript (2) = 0

Type: MC Var: 1

6) Compute: -2matrix ((3 3)(-2 4)) - 2matrix ((-3 -1)(-4 -2))

A) matrix ((0 -4)(12 -4))

B) matrix ((6 4)(2 6))

C) matrix ((0 4)(12 6))

D) matrix ((-12 -8)(-4 -12))

Type: MC Var: 1

7) Compute: matrix ((0 -6)(1 -4)(0 -2)) - 3matrix ((3 2)(6 -5)(-1 0))

A) matrix ((3 -4)(7 -9)(-1 -2))

B) matrix ((9 0)(19 -19)(-3 -2))

C) matrix ((-3 -8)(-5 1)(1 -2))

D) matrix ((-9 -12)(-17 11)(3 -2))

Type: MC Var: 1

8) Consider these matrices:

A = matrix ((-4 3 -10 7)) B = matrix ((-7)(4)(-7)(9)) C = matrix ((-1 -7)(7 8)(8 4)(3 -7)) D = matrix ((-9 0 0 0)(0 8 0 0)(0 0 -10 0)(0 0 0 -5))

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

A) AB

B) (D) with superscript (3)

C) BA

D) AC

E) DC

F) BD

G) A + B

Type: MC Var: 1

9) Consider the matrix A = matrix ((8 -2 -1 -9 7)(3 -5 -5 2 5)). Compute 3(A) with superscript (T).

A) matrix ((24 -6 -3 -27 21)(9 -15 -15 6 15))

B) matrix ((8 3)(-2 -5)(-1 -5)(-9 2)(7 5))

C) matrix ((24 9)(-6 -15)(-3 -15)(-27 6)(21 15))

D) matrix ((21 15)(-27 6)(-3 -15)(-6 -15)(24 9))

E) matrix ((9 24)(-15 -6)(-15 -3)(6 -27)(15 21))

Type: MC Var: 1

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.

Type: MC Var: 1

11) Consider these matrices:

D = matrix ((-2 0 0 0)(0 -9 0 0)(0 0 -8 0)(0 0 0 -4)) E = matrix ((-4 -1 5 8))

Compute (D) with superscript (2)

Type: SA Var: 1

12) Consider these matrices:

D = matrix ((-5 0 0 0)(0 5 0 0)(0 0 5 0)(0 0 0 2)) E = matrix ((10 -1 -8 -1))

Compute (D) with superscript (-1)

Type: SA Var: 1

13) Consider these matrices:

D = matrix ((-4 0 0 0)(0 10 0 0)(0 0 -3 0)(0 0 0 -8)) E = matrix ((-7 2 0 -3))

Compute ED

Type: SA Var: 1

14) Consider the matrix function A(t) = matrix ((sin(2t) cos(5t))(cos(5t) -sin(2t)))

Compute A'(t).

A) matrix ((-2 cos(2t) 5 sin(5t))(5 sin(5t) 2 cos(2t)))

B) matrix ((2 cos(2t) -5 sin(5t))(-5 sin(5t) -2 cos(2t)))

C) matrix ((cos(2t) -sin(5t))(-sin(5t) -cos(2t)))

D) matrix ((-cos(2t) sin(5t))(sin(5t) cos(2t)))

Type: MC Var: 1

15) Consider the matrix function A(t) = matrix ((sin(2t) cos(5t))(cos(5t) -sin(2t)))

Compute ((A'(t))) with superscript (-1).

A) matrix ((-2 cos(2t) -5 sin(5t))(-5 sin(5t) 2 cos(2t)))

B) (1/4 (cos) with superscript (2)(2t) + 25 (sin) with superscript (2)(5t))matrix ((-2 cos(2t) -5 sin(5t))(-5 sin(5t) 2 cos(2t)))

C) matrix ((-2 cos(2t) 5 sin(5t))(5 sin(5t) 2 cos(2t)))

D) (1/4 (cos) with superscript (2)(2t) + 25 (sin) with superscript (2)(5t))matrix ((-2 cos(2t) 5 sin(5t))(5 sin(5t) 2 cos(2t)))

Type: MC Var: 1

16) Consider the matrix function A(t) = matrix (((e) with superscript (-10t) (e) with superscript (-8t))(1 (e) with superscript (-10t))).

Compute integral of (A(t)dt) from (0) to (1)

Type: SA Var: 1

17) Consider the matrix B = matrix ((5 -4)(4 4)). Compute (B) with superscript (-1)

Type: SA Var: 1

18) Consider the following system of linear equations:

4(x) with subscript (1) - 6(x) with subscript (2) - 2(x) with subscript (3) = -7

2(x) with subscript (1) - 2(x) with subscript (2) + 2(x) with subscript (3) = 3

-8(x) with subscript (1) + 10(x) with subscript (2) - 2(x) with subscript (3) = -2

What is the augmented matrix for this system?

Type: SA Var: 1

19) Consider the following system of linear equations:

-8(x) with subscript (1) - 6(x) with subscript (2) + 4(x) with subscript (3) = 5

-4(x) with subscript (1) - 6(x) with subscript (2) + 3(x) with subscript (3) = 7

16(x) with subscript (1) + 18(x) with subscript (2) - 10(x) with subscript (3) = -6

Reduce the augmented matrix of this system to echelon form.

Type: SA Var: 1

20) Consider the following system of linear equations:

-6(x) with subscript (1) - 4(x) with subscript (2) + 6(x) with subscript (3) = 4

-3(x) with subscript (1) - 4(x) with subscript (2) + 4(x) with subscript (3) = -4

12(x) with subscript (1) + 12(x) with subscript (2) - 14(x) with subscript (3) = 4

The system is inconsistent.

Type: TF Var: 1

21) Consider the following system of linear equations:

-8(x) with subscript (1) + 2(x) with subscript (2) - 6(x) with subscript (3) = (b) with subscript (1)

-4(x) with subscript (1) + 4(x) with subscript (2) - 6(x) with subscript (3) = (b) with subscript (2)

16(x) with subscript (1) - 10(x) with subscript (2) + 18(x) with subscript (3) = (b) with subscript (3)

Find a condition involving (b) with subscript (1), (b) with subscript (2), and (b) with subscript (3) that ensures the system has infinitely many solutions.

Type: SA Var: 1

22) Consider this set of vectors: { table ( (4)(2)(-8) ), table ( (6)(6)(-18) ), table ( (-2)(-2)(6) ) }

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 = matrix ((4 6 -2)(2 6 -2)(-8 -18 6)), has only the solution x = 0.

D) The system Ax = 0, where A = matrix ((4 6 -2)(2 6 -2)(-8 -18 6)), is inconsistent.

Type: MC Var: 1

23) Are the vectors (u) with subscript (1), (u) with subscript (2), and (u) with subscript (3) linearly independent or linearly dependent? If they are linearly dependent, identify appropriate constants A, B, and C for which A(u) with subscript (1) + B(u) with subscript (2) + C(u) with subscript (3) = 0 that demonstrates this fact.

(u) with subscript (1) = (table ( (-1)(1)(0) )), (u) with subscript (2) = (table ( (1)(-1)(2) )), (u) with subscript (3) = (table ( (-3)(3)(-2) ))

Use A = 2, B = -1, and C = -1 in the equation A(u) with subscript (1) + B(u) with subscript (2) + C(u) with subscript (3) = 0.

Type: SA Var: 1

24) Are the vectors (u) with subscript (1), (u) with subscript (2), (u) with subscript (3), and (u) with subscript (4) linearly independent or linearly dependent? If they are linearly dependent, identify appropriate constants A, B, C, and D for which A(u) with subscript (1) + B(u) with subscript (2) + C(u) with subscript (3) + D(u) with subscript (4) = 0 that demonstrates this fact.

(u) with subscript (1) = (table ( (8)(2)(-4)(3) )), (u) with subscript (2) = (table ( (6)(0)(0)(0) )), (u) with subscript (3) = (table ( (3)(-7)(2)(0) )), (u) with subscript (4) = (table ( (-6)(-3)(0)(0) ))

Type: SA Var: 1

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

Type: TF Var: 1

26) Given that λ = 1 is an eigenvalue of the matrix B = matrix ((2 -1 1)(1 1 -1)(0 -1 3)), which of the following statements is true regarding the eigenvector of B associated with this eigenvalue λ = 1?

A) (table ( (1)(2)(1) )) is the only eigenvector of B associated with the eigenvalue λ = 1.

B) α (table ( (-1)(2)(1) )) is an eigenvector of B, for any nonzero real constant α, associated with the eigenvalue λ = 1.

C) α (table ( (1)(2)(1) )) is an eigenvector of B, for any nonzero real constant α, associated with the eigenvalue λ = 1.

D) (table ( (-1)(2)(1) )) is the only eigenvector of B associated with the eigenvalue λ = 1.

Type: MC Var: 1

27) Consider the matrix A = matrix ((2 6)(2 -2))

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

A) (λ) with subscript (1) = 4, (ξ) with subscript (1) = (table ( (3)(1) )), (λ) with subscript (2) = -4, (ξ) with subscript (2) = (table ( (-1)(1) ))

B) (λ) with subscript (1) = 4, (ξ) with subscript (1) = (table ( (1)(3) )), (λ) with subscript (2) = -4, (ξ) with subscript (2) = (table ( (1)(-1) ))

C) (λ) with subscript (1) = 2, (ξ) with subscript (1) = (table ( (3)(1) )), (λ) with subscript (2) = -2, (ξ) with subscript (2) = (table ( (-1)(1) ))

D) (λ) with subscript (1) = 2, (ξ) with subscript (1) = (table ( (1)(3) )), (λ) with subscript (2) = -2, (ξ) with subscript (2) = (table ( (1)(-1) ))

Type: MC Var: 1

28) Consider a system of homogeneous first-order linear differential equations of the form x' = Ax, where A is a 2 × 2 constant matrix. If (x) with subscript (1)(t) = (table ( (3(e) with superscript (3t))(2(e) with superscript (3t)) )) and (x) with subscript (2)(t) = (table ( (-5(e) with superscript (-8t))(-5(e) with superscript (-8t)) )) are solutions of this system, which of the following must also be solutions of this system? Select all that apply.

A) -2(x) with subscript (1)(t)

B) -7.2(x) with subscript (1)(t) + 4.4(x) with subscript (2)(t)

C) (x) with subscript (1)(t) ∙ (x) with subscript (2)(t)

D) -6.6t(x) with subscript (1)(t) + 5.8t(x) with subscript (2)(t)

E) (7.2(x) with subscript (1)(t) + 5.4(x) with subscript (2)(t)) - 8((x) with subscript (1)(t) - (x) with subscript (2)(t))

F) 2(x) with subscript (1)(t) - 3.6(x) with subscript (2)(t) - 4.6

Type: MC Var: 1

29) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 4)(3 1) ))x

and the following three vector functions:

(x) with subscript (1)(t) = (table ( ((e) with superscript (4t))((e) with superscript (3t)) )), (x) with subscript (2)(t) = (table ( (4(e) with superscript (-3t))(-3(e) with superscript (-3t)) )), (x) with subscript (3)(t) = (table ( (8(e) with superscript (-3t))(-6(e) with superscript (-3t)) ))

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

A) {(x) with subscript (1), (x) with subscript (2), (x) with subscript (3)} is a fundamental set of solutions for this system.

B) W [(x) with subscript (1)(t), (x) with subscript (2)(t)] ≠ 0 for every real number t.

C) (x) with subscript (1) and (x) with subscript (3) are linearly dependent.

D) 6(x) with subscript (1) + 4(x) with subscript (2) + 3(x) with subscript (3) is a solution of this system.

E) {(x) with subscript (1), (x) with subscript (3)} is a fundamental set of solutions for this system.

Type: MC Var: 1

30) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (8 0)(-10 3) ))x

and the following four vector functions:

(x) with subscript (1)(t) = (table ( (5(e) with superscript (8t))(-10(e) with superscript (8t)) )), (x) with subscript (2)(t) = (table ( (-5(e) with superscript (-8t))(-10(e) with superscript (-8t)) )), (x) with subscript (3)(t) = (table ( (0)((e) with superscript (3t)) )), (x) with subscript (4)(t) = (table ( ((e) with superscript (3t))(0) ))

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

A) (C) with subscript (1)(x) with subscript (1) + (C) with subscript (2)(x) with subscript (2) + (C) with subscript (3)(x) with subscript (3) + (C) with subscript (4)(x) with subscript (4) is a solution of this system, for all real numbers (C) with subscript (1), (C) with subscript (2), (C) with subscript (3), and (C) with subscript (4).

B) W [(x) with subscript (1)(t), (x) with subscript (3)(t)] ≠ 0 for every real number t.

C) 5.5(x) with subscript (1) + 4.5(x) with subscript (3) + C is a solution of this system, for any real number C.

D) 4(x) with subscript (4) is a solution of this system.

E) {(x) with subscript (1), (x) with subscript (3)} is a fundamental set of solutions for this system.

Type: MC Var: 1

31) Consider the first-order homogeneous system of linear differential equations

x'= (table ( (4 16)(8 -4) ))x

Determine the eigenvalue-eigenvector pairs of this system.

(λ) with subscript (2) = -12, (ξ) with subscript (2) = (table ( (1)(-1) ))

Type: SA Var: 1

32) Consider the first-order homogeneous system of linear differential equations

x'= (table ( (3 12)(6 -3) ))x

What is the general solution of this system?

Type: SA Var: 1

33) Consider the first-order homogeneous system of linear differential equations

x'= (table ( (5 20)(10 -5) ))x

If the system were equipped with the initial condition x(0) = (table ( (2)(-6) )), what is the particular solution of the system?

Type: SA Var: 1

34) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 9)(4 5) ))x

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

A) λ = 9, ξ = (table ( (1)(1) ))

B) λ = -9, ξ = (table ( (-1)(1) ))

C) λ = 4, ξ = (table ( (-9)(4) ))

D) λ = 0, ξ = (table ( (1)(0) ))

E) λ = -4, ξ = (table ( (9)(-4) ))

Type: MC Var: 1

35) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 7)(9 -2) ))x

Which of these is the general solution of the system? Here, (C) with subscript (1) and (C) with subscript (2) are arbitrary real constants.

A) x(t) = (C) with subscript (1)(table ( (1)(1) ))(e) with superscript (-7t) + (C) with subscript (2)(table ( (-7)(9) ))(e) with superscript (9t)

B) x(t) = (C) with subscript (1)(table ( (1)(1) ))(e) with superscript (7t) + (C) with subscript (2)(table ( (7)(-9) ))(e) with superscript (-9t)

C) x(t) = (C) with subscript (1)(table ( (1)(1) ))(e) with superscript (7t) + (C) with subscript (2)(table ( (-7)(9) ))(e) with superscript (9t)

D) x(t) = (C) with subscript (1)(table ( (1)(1) ))(e) with superscript (-7t) + (C) with subscript (2)(table ( (7)(-9) ))(e) with superscript (-9t)

E) x(t) = (C) with subscript (1)(table ( (1)(0) )) + (C) with subscript (2)(table ( (7)(-9) ))(e) with superscript (-9t)

Type: MC Var: 1

36) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 5)(9 -4) ))x

The origin is a saddle point.

Type: TF Var: 1

37) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 -9)(-9 0) ))x

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

A) λ = 9, ξ = (table ( (-9)(-9) ))

B) λ = -9, ξ = (table ( (-9)(-9) ))

C) λ = -9, ξ = (table ( (9)(-9) ))

D) λ = 9, ξ = (table ( (3i)(3i) ))

E) λ = 9, ξ = (table ( (9)(-9) ))

F) λ = -9, ξ = (table ( (-3i)(3i) ))

Type: MC Var: 1

38) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 -25)(-16 0) ))x

Which of these is the genreal solution of the system? Here, (C) with subscript (1) and (C) with subscript (2) are arbitrary real constants.

A) x(t) = (C) with subscript (1)(table ( (-25)(-16) ))(e) with superscript (-25t) + (C) with subscript (2)(table ( (16)(-25) ))(e) with superscript (-16t)

B) x(t) = (C) with subscript (1)(table ( (-16)(-25) ))(e) with superscript (25t) + (C) with subscript (2)(table ( (16)(-25) ))(e) with superscript (-16t)

C) x(t) = (C) with subscript (1)(table ( (5i)(4i) ))(e) with superscript (20t) + (C) with subscript (2)(table ( (-5i)(4i) ))(e) with superscript (-20t)

D) x(t) = (C) with subscript (1)(table ( (-16)(-25) ))(e) with superscript (25t) + (C) with subscript (2)(table ( (16)(-25) ))(e) with superscript (16t)

E) x(t) = (C) with subscript (1)(table ( (5i)(4i) ))(e) with superscript (-20t) + (C) with subscript (2)(table ( (-5i)(4i) ))(e) with superscript (20t)

Type: MC Var: 1

39) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 16)(4 0) ))x

The origin is a node.

Type: TF Var: 1

40) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (-8 0)(10 -3) ))x

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

A) λ = 3, ξ = (table ( (0)(-1) ))

B) λ = -3, ξ = (table ( (0)(1) ))

C) λ = 8, ξ = (table ( (10)(5) ))

D) λ = -8, ξ = (table ( (10)(-5) ))

E) λ = -3, ξ = (table ( (1)(0) ))

F) λ = -8, ξ = (table ( (-5)(10) ))

Type: MC Var: 1

41) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (-2 0)(-10 -5) ))x

Which of these is the general solution of the system? Here, (C) with subscript (1) and (C) with subscript (2) are arbitrary real constants.

A) x(t) = (C) with subscript (1)(table ( (3)(-10) ))(e) with superscript (-2t) + (C) with subscript (2)(table ( (0)(1) ))(e) with superscript (-5t)

B) x(t) = (C) with subscript (1)(table ( (3)(-10) ))(e) with superscript (-2t) + (C) with subscript (2)(table ( (0)(-1) ))(e) with superscript (5t)

C) x(t) = (C) with subscript (1)(table ( (3)(-10) ))(e) with superscript (-2t) + (C) with subscript (2)(table ( (-10)(-3) ))(e) with superscript (2t)

D) x(t) = (C) with subscript (1)(table ( (-1)(0) ))(e) with superscript (5t) + (C) with subscript (2)(table ( (3)(-10) ))(e) with superscript (-2t)

E) x(t) = (C) with subscript (1)(table ( (0)(1) ))(e) with superscript (-5t) + (C) with subscript (2)(table ( (-10)(-3) ))(e) with superscript (2t)

Type: MC Var: 1

42) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (-7 0)(9 -10) ))x

The origin is a node.

Type: TF Var: 1

43) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (-3 -3)(0 0) ))x

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

A) λ = 3, ξ = (table ( (-1)(0) ))

B) λ = -3, ξ = (table ( (1)(0) ))

C) λ = 0, ξ = (table ( (-1)(1) ))

D) λ = 0, ξ = (table ( (1)(-1) ))

E) λ = -3, ξ = (table ( (0)(1) ))

F) λ = 3, ξ = (table ( (0)(-1) ))

Type: MC Var: 1

44) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (2 2)(0 0) ))x

Which of these is the general solution of the system? Here, (C) with subscript (1) and (C) with subscript (2) are arbitrary real constants.

A) x(t) = (C) with subscript (1)(table ( (-1)(0) ))(e) with superscript (-2t) + (C) with subscript (2)(table ( (-1)(1) ))

B) x(t) = (C) with subscript (1)(table ( (-1)(0) ))(e) with superscript (-2t) + (C) with subscript (2)(table ( (1)(-1) ))

C) x(t) = (C) with subscript (1)(table ( (1)(0) ))(e) with superscript (2t) + (C) with subscript (2)(table ( (-1)(1) ))

D) x(t) = (C) with subscript (1)(table ( (0)(1) ))(e) with superscript (2t) + (C) with subscript (2)(table ( (1)(-1) ))

Type: MC Var: 1

45) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 1 0)(0 0 1)(2 1 -2) ))x

What is the general solution of this system?

A) x(t) = (C) with subscript (1)(table ( (-1)(2)(-4) ))(e) with superscript (2t) + (C) with subscript (2)(table ( (1)(-1)(1) ))(e) with superscript (-t) + (C) with subscript (3)(table ( (1)(1)(1) ))(e) with superscript (t)

B) x(t) = (C) with subscript (1)(table ( (1)(-2)(4) ))(e) with superscript (-2t) + (C) with subscript (2)(table ( (1)(-1)(1) ))(e) with superscript (-t) + (C) with subscript (3)(table ( (1)(1)(1) ))(e) with superscript (t)

C) x(t) = (C) with subscript (1)(table ( (1)(1)(-1) )) + (C) with subscript (2)(table ( (1)(-1)(1) ))(e) with superscript (-t) + (C) with subscript (3)(table ( (1)(1)(1) ))(e) with superscript (t)

D) x(t) = (C) with subscript (1)(table ( (1)(-2)(4) ))(e) with superscript (-2t) + (C) with subscript (2)(table ( (2)(1)(-1) ))(e) with superscript (-t) + (C) with subscript (3)(table ( (1)(1)(1) ))(e) with superscript (t)

E) x(t) = (C) with subscript (1)(table ( (1)(-2)(4) ))(e) with superscript (-2t) + (C) with subscript (2)(table ( (1)(-1)(1) ))(e) with superscript (-t) + (C) with subscript (3)(table ( (1)(1)(1) ))

Type: MC Var: 1

46) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 1 0)(0 0 1)(2 1 -2) ))x

If the system were equipped with the initial condition x(0) = (table ( (4)(5)(0) )), what is the particular solution of the system?

Type: SA Var: 1

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) with subscript (A)(t), and the amount of salt in tank B, (x) with subscript (B)(t), at any time t.

(x') with subscript (B)(t) = (4.03 + (2.45/100)(x) with subscript (A)(t)) - (1.3/35)(x) with subscript (B)(t)

(x) with subscript (A)(0) = 30

(x) with subscript (B)(0) = 60

Type: SA Var: 1

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.

(I) x(t) = (C) with subscript (1)(table ( (-4)(-2) ))(e) with superscript (-4t) + (C) with subscript (2)(table ( (5)(2) ))(e) with superscript (4t)

(II) x(t) = (C) with subscript (1)(table ( (-2)(2) ))(e) with superscript (-3t) + (C) with subscript (2)(table ( (-5)(3) ))(e) with superscript (-3t)

(III) x(t) = (C) with subscript (1)(table ( (-1)(-1) ))(e) with superscript (6t) + (C) with subscript (2)(table ( (5)(5) ))(e) with superscript (7t)

A) I

B) II

C) III

D) None of them

Type: MC Var: 1

49) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (1 2 1)(6 -1 0)(-1 -2 -1) ))x

Determine a fundamental set of solutions for this system.

Type: SA Var: 1

50) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (1 2 1)(6 -1 0)(-1 -2 -1) ))x

What is the general solution of this system?

Type: SA Var: 1

51) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 8)(-8 0) ))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

Type: MC Var: 1

52) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 6)(-6 0) ))x

Which of these is a fundamental set of solutions for this system?

A) { table ( (-sin(6t))(cos(6t)) ), table ( (cos(6t))(sin(6t)) ) }

B) { table ( (sin(6t))(cos(6t)) ), table ( (-cos(6t))(sin(6t)) ) }

C) { table ( (sin(36t))(cos(36t)) ), table ( (-cos(36t))(sin(36t)) ) }

D) { table ( ((e) with superscript (6t))((e) with superscript (6t)) ), table ( (-(e) with superscript (-6t))((e) with superscript (-6t)) ) }

E) { table ( ((e) with superscript (6t))(-(e) with superscript (6t)) ), table ( ((e) with superscript (-6t))((e) with superscript (-6t)) ) }

Type: MC Var: 1

53) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 6)(-6 0) ))x

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 → ∞.

B) The trajectories are concentric circles centered at the origin.

C) The trajectories spiral away from the origin as t → ∞.

D) The trajectories are line segments that approach the origin as t → ∞.

E) The origin is a saddle point.

Type: MC Var: 1

54) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (10 4)(-4 10) ))x

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

Type: MC Var: 1

55) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (8 3)(-3 8) ))x

What is the general solution of this system? Here, (C) with subscript (1) and (C) with subscript (2) are arbitrary real constants.

A) x(t) = (C) with subscript (1)(e) with superscript (-3t) (table ( (sin(8t))(cos(8t)) )) + (C) with subscript (2)(e) with superscript (-3t) (table ( (-cos(8t))(sin(8t)) ))

B) x(t) = (C) with subscript (1)(e) with superscript (3t) (table ( (sin(8t))(cos(8t)) )) + (C) with subscript (2)(e) with superscript (3t) (table ( (-cos(8t))(sin(8t)) ))

C) x(t) = (C) with subscript (1)(e) with superscript (-8t) (table ( (sin(3t))(cos(3t)) )) + (C) with subscript (2)(e) with superscript (-8t) (table ( (-cos(3t))(sin(3t)) ))

D) x(t) = (C) with subscript (1)(e) with superscript (8t) (table ( (sin(3t))(cos(3t)) )) + (C) with subscript (2)(e) with superscript (8t) (table ( (-cos(3t))(sin(3t)) ))

Type: MC Var: 1

56) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (4 9)(-9 4) ))x

All solution trajectories spiral towards the origin as t → ∞.

Type: TF Var: 1

57) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (-4 -7)(7 -4) ))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

Type: MC Var: 1

58) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (-8 -2)(2 -8) ))x

What is the general solution of this system? Here, (C) with subscript (1) and (C) with subscript (2) are arbitrary real constants.

A) x(t) = (C) with subscript (1)(e) with superscript (-2t) (table ( (sin(-8t))(cos(-8t)) )) + (C) with subscript (2)(e) with superscript (-2t) (table ( (-cos(-8t))(sin(-8t)) ))

B) x(t) = (C) with subscript (1)(e) with superscript (8t) (table ( (sin(2t))(cos(2t)) )) + (C) with subscript (2)(e) with superscript (8t) (table ( (-cos(2t))(sin(2t)) ))

C) x(t) = (C) with subscript (1)(e) with superscript (-8t) (table ( (sin(2t))(cos(2t)) )) + (C) with subscript (2)(e) with superscript (-8t) (table ( (-cos(2t))(sin(2t)) ))

D) x(t) = (C) with subscript (1)(e) with superscript (2t) (table ( (sin(-8t))(cos(-8t)) )) + (C) with subscript (2)(e) with superscript (2t) (table ( (-cos(-8t))(sin(-8t)) ))

Type: MC Var: 1

59) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (-4 -2)(2 -4) ))x

Which of the following is an accurate statement regarding the behavior of the solution trajectories of this system as t → ∞?

A) All trajectories spiral towards the origin as t → ∞.

B) All trajectories spiral away from the origin as t → ∞.

C) The trajectories are concentric circles centered at the origin.

D) The trajectories are line segments that approach the origin as t → ∞.

Type: MC Var: 1

60) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 4 0)(0 0 4)(0 -4 0) ))x

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

Type: MC Var: 1

61) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 6 0)(0 0 6)(0 -6 0) ))x

Select the vectors from this list that, together, constitute a fundamental set of solutions for this system.

A) (table ( (0)(1)(0) ))

B) (table ( (1)(0)(0) ))

C) (table ( (-cos(6t))(sin(6t))(cos(6t)) ))

D) (table ( (cos(6t))(sin(6t))(-cos(6t)) ))

E) (table ( (sin(6t))(cos(6t))(-cos(6t)) ))

F) (table ( (sin(6t))(cos(6t))(-sin(6t)) ))

G) (table ( (-sin(6t))(cos(6t))(sin(6t)) ))

Type: MC Var: 1

62) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (1 -7)(11 1) ))x

Determine the eigenvalues for this system and describe the behavior of the solution trajectories as t → ∞.

A) λ = 1 ± isquare root of (77); all solution trajectories spiral toward the origin as t → ∞.

B) λ = 1 ± isquare root of (77); all solution trajectories spiral away from the origin as t → ∞.

C) λ = 1 ± isquare root of (77); the origin is a saddle.

D) λ = -1 ± isquare root of (77); all solution trajectories spiral toward the origin as t → ∞.

E) λ = -1 ± isquare root of (77); all solution trajectories spiral away from the origin as t → ∞.

Type: MC Var: 1

63) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (8 α)(5 -8) ))x

(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)?

(ii) Concentric circles centered at the origin.

Type: ES Var: 1

64) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (2 α)(5 -2) ))x

What is the bifurcation value of α, if any?

Type: SA Var: 1

65) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (α 4)(-4 0) ))x

Which of these statements is true?

A) For any real number α, the eigenvalues for this system are real numbers.

B) For -8 < α < 0, the trajectories spiral towards the origin as t → ∞.

C) For α < -8, the trajectories spiral away from the origin as t → ∞.

D) For α = 8, the eigenvalues are purely imaginary and the trajectories are concentric circles centered at the origin.

Type: MC Var: 1

66) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (1 5)(-5 1) ))x

The eigenvalues and corresponding eigenvectors for this system are:

(λ) with subscript (1) = 1 + 5i, (ξ) with subscript (1) = (table ( (1)(i) ))

(λ) with subscript (2) = 1 - 5i, (ξ) with subscript (2) = (table ( (1)(-i) ))

Which of these is the general solution for this system?

A) x(t) = (C) with subscript (1)(e) with superscript (t) (table ( (sin(5t))(-cos(5t)) )) + (C) with subscript (2)(e) with superscript (t) (table ( (-sin(5t))(cos(5t)) ))

B) x(t) = (C) with subscript (1)(e) with superscript (t) (table ( (sin(5t))(cos(5t)) )) + (C) with subscript (2)(e) with superscript (t) (table ( (-sin(5t))(cos(5t)) ))

C) x(t) = (C) with subscript (1)(e) with superscript (t) (table ( (-sin(5t))(cos(5t)) )) + (C) with subscript (2)(e) with superscript (t) (table ( (cos(5t))(-sin(5t)) ))

D) x(t) = (C) with subscript (1)(e) with superscript (t) (table ( (sin(5t))(cos(5t)) )) + (C) with subscript (2)(e) with superscript (t) (table ( (cos(5t))(-sin(5t)) ))

Type: MC Var: 1

67) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 4)(5 -1) ))x

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

A) ψ(t) = (table ( ((e) with superscript (4t) -4(e) with superscript (-5t))((e) with superscript (4t) 5(e) with superscript (-5t)) ))

B) ψ(t) = (table ( (-(e) with superscript (4t) -5(e) with superscript (-5t))((e) with superscript (4t) 4(e) with superscript (-5t)) ))

C) ψ(t) = (table ( ((e) with superscript (4t) 4(e) with superscript (-5t))((e) with superscript (4t) -5(e) with superscript (-5t)) ))

D) ψ(t) = (table ( ((e) with superscript (-4t) 4(e) with superscript (-5t))((e) with superscript (-4t) -5(e) with superscript (-5t)) ))

Type: MC Var: 1

68) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (1 0)(-8 -6) ))x

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

A) ψ(t) = (table ( (-7(e) with superscript (-1t) (e) with superscript (-6t))(-8(e) with superscript (-1t) 0) ))

B) ψ(t) = (table ( (-7(e) with superscript (-1t) 0)(-8(e) with superscript (-1t) (e) with superscript (-6t)) ))

C) ψ(t) = (table ( (7(e) with superscript (1t) (e) with superscript (-6t))(-8(e) with superscript (1t) 0) ))

D) ψ(t) = (table ( (7(e) with superscript (1t) 0)(-8(e) with superscript (1t) (e) with superscript (-6t)) ))

Type: MC Var: 1

69) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (-8 0)(10 -2) ))x

Given a fundamental matrix ψ(t) for the system, for what constant vector C = (table ( ((C) with subscript (1))((C) with subscript (2)) )) does ψ(t)C = table ( (-8)(-8) ) ?

Type: SA Var: 1

70) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 16)(9 0) ))x

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

A) ψ(t) = (table ( (4(e) with superscript (12t) - 4(e) with superscript (-12t))(3(e) with superscript (12t) - 3(e) with superscript (-12t)) ))

B) ψ(t) = (table ( (4(e) with superscript (12t) - 4(e) with superscript (-12t))(3(e) with superscript (12t) 3(e) with superscript (-12t)) ))

C) ψ(t) = (table ( (9(e) with superscript (-16t) -9(e) with superscript (9t))(16(e) with superscript (-16t) 16(e) with superscript (9t)) ))

D) ψ(t) = (table ( (9(e) with superscript (-16t) 9(e) with superscript (9t))(-16(e) with superscript (-16t) 16(e) with superscript (9t)) ))

Type: MC Var: 1

71) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 16)(4 0) ))x

The columns of the fundamental matrix of this system, ψ(t), must be linearly independent.

Type: TF Var: 1

72) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 16)(4 0) ))x

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

Type: TF Var: 1

73) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 3)(-3 0) ))x

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

A) ψ(t) = (table ( (-sin(3t) cos(3t))(cos(3t) -sin(3t)) ))

B) ψ(t) = (table ( (sin(3t) -cos(3t))(-cos(3t) sin(3t)) ))

C) ψ(t) = (table ( (sin(3t) cos(3t))(cos(3t) sin(3t)) ))

D) ψ(t) = (table ( (sin(3t) -cos(3t))(cos(3t) sin(3t)) ))

Type: MC Var: 1

74) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 8)(-8 0) ))x

Given a fundamental matrix ψ(t) for the system, which of these is the general solution of this system? Here, C = (table ( ((C) with subscript (1))((C) with subscript (2)) )) is an arbitrary constant vector.

A) x(t) = (ψ) with superscript (-1)(t)C

B) x(t) = ψ(t)C

C) x(t) = ψ(0)C

D) x(t) = ψ(t) + C

Type: MC Var: 1

75) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (4 7)(-7 4) ))x

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

A) ψ(t) = (table ( (-(e) with superscript (-4t)sin(7t) (e) with superscript (-4t)cos(7t))((e) with superscript (-4t)cos(7t) -(e) with superscript (-4t)sin(7t)) ))

B) ψ(t) = (table ( ((e) with superscript (4t)sin(7t) -(e) with superscript (4t)cos(7t))(-(e) with superscript (4t)cos(7t) (e) with superscript (4t)sin(7t)) ))

C) ψ(t) = (table ( ((e) with superscript (4t)sin(7t) -(e) with superscript (4t)cos(7t))((e) with superscript (4t)cos(7t) (e) with superscript (4t)sin(7t)) ))

D) ψ(t) = (table ( ((e) with superscript (-4t)sin(7t) -(e) with superscript (-4t)cos(7t))((e) with superscript (-4t)cos(7t) (e) with superscript (-4t)sin(7t)) ))

Type: MC Var: 1

76) Consider the first-order homogeneous system of linear differential equations

x = (table ( (4 6)(-6 4) ))x

Given a fundamental matrix ψ(t) for the system, which of these is the general solution of this system? Here, C = (table ( ((C) with subscript (1))((C) with subscript (2)) )) is an arbitrary constant vector.

A) x(t) = ψ(t)C

B) x(t) = ψ(t) + C

C) x(t) = (ψ) with superscript (-1)(t)C

D) x(t) = ψ(0)C

Type: MC Var: 1

77) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 4 0)(0 0 4)(0 -4 0) ))x

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

A) ψ(t) = (table ( (1 -cos(4t) sin(4t))(0 sin(4t) cos(4t))(0 cos(4t) -sin(4t)) ))

B) ψ(t) = (table ( (0 cos(4t) -sin(4t))(1 sin(4t) cos(4t))(0 -cos(4t) sin(4t)) ))

C) ψ(t) = (table ( (1 -cos(4t) sin(4t))(0 -sin(4t) cos(4t))(1 cos(4t) -sin(4t)) ))

D) ψ(t) = (table ( (0 cos(4t) sin(4t))(0 -sin(4t) cos(4t))(1 cos(4t) -sin(4t)) ))

Type: MC Var: 1

78) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (0 5 0)(0 0 5)(0 -5 0) ))x

Given a fundamental matrix ψ(t) for the system, which of these is the general solution of this system? Here, C = (table ( ((C) with subscript (1))((C) with subscript (2))((C) with subscript (3)) )) is an arbitrary constant vector.

A) x(t) = ψ(t)C

B) x(t) = ψ(t) + C

C) x(t) = (ψ) with superscript (-1)(t)C

D) x(t) = ψ(0)C

Type: MC Var: 1

79) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (-7 0)(2 -7) ))x

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

A) (table ( ((e) with superscript (-7t))(0) ))

B) (table ( (0)((e) with superscript (-7t)) ))

C) (table ( ((e) with superscript (-7t))(2(e) with superscript (-7t)) ))

D) (table ( ((e) with superscript (-7t))(2(t + 2)(e) with superscript (-7t)) ))

E) (table ( (t(e) with superscript (-7t))(2(e) with superscript (-7t)) ))

F) (table ( ((e) with superscript (-7t))(t(e) with superscript (-7t)) ))

Type: MC Var: 1

80) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (-5 0)(-2 -5) ))x

What is the general solution of this system?

Type: SA Var: 1

81) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (2 0)(-3 2) ))x

The origin is called a(n) ________.

A) improper node

B) center

C) saddle

D) spiral node

Type: MC Var: 1

82) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (-3 8)(0 -3) ))x

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

A) ψ(t) = (table ( (0 (t + 8)(e) with superscript (-3t))((e) with superscript (-3t) (e) with superscript (-3t)) ))

B) ψ(t) = (table ( ((e) with superscript (3t) 8(t + 8)(e) with superscript (3t))(0 (e) with superscript (3t)) ))

C) ψ(t) = (table ( ((e) with superscript (-3t) (t + 8)(e) with superscript (-3t))(0 (e) with superscript (-3t)) ))

D) ψ(t) = (table ( ((e) with superscript (-3t) (e) with superscript (-3t))(0 (t + 8)(e) with superscript (-3t)) ))

Type: MC Var: 1

83) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (-3 10)(0 -3) ))x

Given a fundamental matrix ψ(t) for the system, which of these is the general solution of this system? Here, C = (table ( ((C) with subscript (1))((C) with subscript (2)) )) is an arbitrary constant vector.

A) x(t) = ψ(t)C

B) x(t) = ψ(t) + C

C) x(t) = (ψ) with superscript (-1)(t)C

D) x(t) = ψ(0)C

Type: MC Var: 1

84) Consider the first-order homogeneous system of linear differential equations

x' = (table ( (10 -2 1)(18 -2 0)(0 0 1) ))x

What is the general solution of this system? Here, C = (table ( ((C) with subscript (1))((C) with subscript (2))((C) with subscript (3)) )) is an arbitrary constant vector.

A) x(t) = (table ( ((e) with superscript (t) (e) with superscript (4t) (t - (1/6))(e) with superscript (4t))(6(e) with superscript (t) -3(e) with superscript (4t) -3t(e) with superscript (4t))(3(e) with superscript (t) 0 0) ))C

B) x(t) = (table ( ((e) with superscript (t) (e) with superscript (4t) (t + (1/6))(e) with superscript (4t))(6(e) with superscript (t) 3(e) with superscript (4t) 3t(e) with superscript (4t))(3(e) with superscript (t) 0 0) ))C

C) x(t) = (table ( ((e) with superscript (t) (e) with superscript (4t) (t + (1/6))(e) with superscript (4t))(-3(e) with superscript (t) -3(e) with superscript (4t) 3t(e) with superscript (4t))(6(e) with superscript (t) 0 0) ))C

D) x(t) = (table ( ((e) with superscript (-t) (e) with superscript (-4t) (t + (1/6))(e) with superscript (-4t))(6(e) with superscript (-t) 3(e) with superscript (-4t) 3t(e) with superscript (-4t))(3(e) with superscript (-t) 0 0) ))C

Type: MC Var: 1

85) Consider the first-order nonhomogeneous system of linear differential equations

x' = (table ( (0 3)(1 2) ))x + g(t),

where the components of g(t) are continuous functions.

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

A) ψ(t) = (table ( ((e) with superscript (3t) -3(e) with superscript (-1t))((e) with superscript (3t) 1(e) with superscript (-1t)) ))

B) ψ(t) = (table ( ((e) with superscript (3t) 3(e) with superscript (-1t))((e) with superscript (3t) -1(e) with superscript (-1t)) ))

C) ψ(t) = (table ( (-(e) with superscript (3t) -1(e) with superscript (-1t))((e) with superscript (3t) 3(e) with superscript (-1t)) ))

D) ψ(t) = (table ( ((e) with superscript (-3t) 3(e) with superscript (-1t))((e) with superscript (-3t) -1(e) with superscript (-1t)) ))

Type: MC Var: 1

86) Consider the first-order nonhomogeneous system of linear differential equations

x' = (table ( (0 1)(3 -2) ))x + g(t),

where the components of g(t) are continuous functions.

Given a fundamental matrix ψ(t) for the system, what is the solution of this system if it is equipped with the initial condition x(3.6) = (x) with subscript (0)?

A) x(t) = ψ(0)(x) with subscript (0) + ψ(t)integral of ((ψ) with superscript (-1)(s)g(s)ds) from (0) to (t)

B) x(t) = ψ(t)(ψ) with superscript (-1)(3.6)(x) with subscript (0) + ψ(t)integral of ((ψ) with superscript (-1)(s)g(s)ds) from (3.6) to (t)

C) x(t) = ψ(3.6)(x) with subscript (0) + ψ(t)integral of ((ψ) with superscript (-1)(s)g(s)ds) from (3.6) to (t)

D) x(t) = ψ(t)(ψ) with superscript (-1)(0)(x) with subscript (0) + ψ(t)integral of ((ψ) with superscript (-1)(s)g(s)ds) from (0) to (t)

Type: MC Var: 1

87) Consider the first-order nonhomogeneous initial value problem

x' = (table ( (8 5)(-3 0) ))x + (table ( (-1)(5) )), x(0) = (table ( (-3)(-3) ))

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

A) ψ(t) = (table ( (5(e) with superscript (5t) -(e) with superscript (3t))(-3(e) with superscript (5t) (e) with superscript (3t)) ))

B) ψ(t) = (table ( (-3(e) with superscript (5t) (e) with superscript (3t))(5(e) with superscript (5t) -(e) with superscript (3t)) ))

C) ψ(t) = (table ( (-3(e) with superscript (-5t) (e) with superscript (-3t))(5(e) with superscript (-5t) -(e) with superscript (-3t)) ))

D) ψ(t) = (table ( (5(e) with superscript (-5t) -(e) with superscript (-3t))(-3(e) with superscript (-5t) (e) with superscript (-3t)) ))

Type: MC Var: 1

88) Consider the first-order nonhomogeneous initial value problem

x' = (table ( (3 -3)(-6 0) ))x + (table ( (-5)(-4) )), x(0) = (table ( (1)(2) ))

Compute (ψ) with superscript (-1)(t), the inverse of the fundamental matrix of the system.

Type: SA Var: 1

89) Consider the first-order nonhomogeneous initial value problem

x' = (table ( (-6 -4)(2 0) ))x + (table ( (-4)(-3) )), x(0) = (table ( (-2)(-3) ))

Given a fundamental matrix ψ(t) for the system, what is the solution of this system if it is equipped with the initial condition x(0) = (x) with subscript (0)?

A) x(t) = ψ(t)(table ( (-2)(-3) )) + ψ(t)integral of ((ψ) with superscript (-1)(s)table ( (-4)(-3) )ds) from (0) to (t)

B) x(t) = (ψ) with superscript (-1)(0)(table ( (-2)(-3) )) + ψ(0)integral of ((ψ) with superscript (-1)(s)table ( (-4)(-3) )ds) from (0) to (t)

C) ψ(0)(table ( (-2)(-3) )) + ψ(t)integral of ((ψ) with superscript (-1)(s)table ( (-4)(-3) )ds) from (0) to (t)

D) x(t) = ψ(t)(ψ) with superscript (-1)(0)(table ( (-2)(-3) )) + ψ(t)integral of ((ψ) with superscript (-1)(s)table ( (-4)(-3) )ds) from (0) to (t)

Type: MC Var: 1

90) Consider the first-order nonhomogeneous initial value problem

x' = (table ( (10 6)(-4 0) ))x + (table ( (-1)(-3) )), x(0) = (table ( (-2)(-3) ))

Write simplified formulas for the components of the solution vector x(t) = (table ( ((x) with subscript (1)(t))((x) with subscript (2)(t)) )).

(x) with subscript (2)(t) = (1/2)((68/3)(e) with superscript (6t) - (71/2)(e) with superscript (4t) + (17/6))

Type: SA Var: 1

91) Consider the first-order nonhomogeneous initial value problem

x' = (table ( (5 0)(3 -4) ))x + (table ( (3(t) with superscript (2) + 2)(-3(e) with superscript (-4t)) )), x(1.0) = (table ( (3)(1) ))

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

A) ψ(t) = (table ( (-9(e) with superscript (-5t) (e) with superscript (-4t))(3(e) with superscript (-5t) 0) ))

B) ψ(t) = (table ( (-9(e) with superscript (-5t) 0)(3(e) with superscript (-5t) (e) with superscript (-4t)) ))

C) ψ(t) = (table ( (9(e) with superscript (5t) (e) with superscript (-4t))(3(e) with superscript (5t) 0) ))

D) ψ(t) = (table ( (9(e) with superscript (5t) 0)(3(e) with superscript (5t) (e) with superscript (-4t)) ))

Type: MC Var: 1

92) Consider the first-order nonhomogeneous initial value problem

x' = (table ( (-2 0)(-10 -10) ))x + (table ( (4(t) with superscript (3) + 4)(-5(e) with superscript (-2t)) )), x(3.0) = (table ( (-2)(4) ))

Given a fundamental matrix ψ(t) for the system, what is the solution of this initial value problem?

A) x(t) = ψ(3.0)(table ( (-2)(4) )) + ψ(t)integral of ((ψ) with superscript (-1)(s)table ( (4(s) with superscript (3) + 4)(-5(e) with superscript (-2s)) )ds) from (3.0) to (t)

B) x(t) = ψ(t)(ψ) with superscript (-1)(3.0)(table ( (-2)(4) )) + ψ(t)integral of ((ψ) with superscript (-1)(s)table ( (4(s) with superscript (3) + 4)(-5(e) with superscript (-2s)) )ds) from (3.0) to (t)

C) x(t) = ψ(t)(table ( (-2)(4) )) + ψ(t)integral of ((ψ) with superscript (-1)(s)table ( (4(s) with superscript (3) + 4)(-5(e) with superscript (-2s)) )ds) from (3.0) to (t)

D) x(t) = ψ(t)(ψ) with superscript (-1)(3.0)(table ( (-2)(4) )) + ψ(3.0)integral of ((ψ) with superscript (-1)(s)table ( (4(s) with superscript (3) + 4)(-5(e) with superscript (-2s)) )ds) from (3.0) to (t)

Type: MC Var: 1

93) Consider the first-order nonhomogeneous initial value problem

x' = (table ( (0 3)(-3 0) ))x + (table ( (2)(-4(t) with superscript (2) + 6(t) with superscript (7)) )), x(2.0) = (table ( (-1)(1) ))

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

A) ψ(t) = (table ( (-sin(3t) cos(3t))(cos(3t) -sin(3t)) ))

B) ψ(t) = (table ( (sin(3t) -cos(3t))(-cos(3t) sin(3t)) ))

C) ψ(t) = (table ( (sin(3t) cos(3t))(cos(3t) sin(3t)) ))

D) ψ(t) = (table ( (sin(3t) -cos(3t))(cos(3t) sin(3t)) ))

Type: MC Var: 1

94) Consider the first-order nonhomogeneous initial value problem

x' = (table ( (0 2)(-2 0) ))x + (table ( (-10)(3(t) with superscript (2) + 6(t) with superscript (7)) )), x(2.2) = (table ( (2)(4) ))

Given a fundamental matrix ψ(t) for the system, what is the solution of this initial value problem?

A) x(t) = ψ(t)(ψ) with superscript (-1)(2.2)(table ( (2)(4) )) + ψ(t)integral of ((ψ) with superscript (-1)(s)table ( (-10)(3(s) with superscript (2) + 6(s) with superscript (7)) )ds) from (2.2) to (t)

B) x(t) = ψ(2.2)(table ( (2)(4) )) + ψ(t)integral of ((ψ) with superscript (-1)(s)table ( (-10)(3(s) with superscript (2) + 6(s) with superscript (7)) )ds) from (2.2) to (t)

C) x(t) = (ψ) with superscript (-1)(2.2)(table ( (2)(4) )) + ψ(t)integral of ((ψ) with superscript (-1)(s)table ( (-10)(3(s) with superscript (2) + 6(s) with superscript (7)) )ds) from (2.2) to (t)

D) x(t) = (ψ) with superscript (-1)(2.2)(table ( (2)(4) )) + ψ(2.2)integral of ((ψ) with superscript (-1)(s)table ( (-10)(3(s) with superscript (2) + 6(s) with superscript (7)) )ds) from (2.2) to (t)

Type: MC Var: 1

95) Consider the first-order nonhomogeneous initial value problem

x' = (table ( (3 0)(4 3) ))x + (table ( (4(e) with superscript (2t))(4t) )), x(1.8) = (table ( (-3)(-2) ))

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

A) ψ(t) =(table ( (0 (e) with superscript (3t))((e) with superscript (3t) 4(t + 4)(e) with superscript (3t)) ))

B) ψ(t) =(table ( (0 (e) with superscript (-3t))((e) with superscript (-3t) 4(t + 4)(e) with superscript (-3t)) ))

C) ψ(t) =(table ( ((e) with superscript (3t) (e) with superscript (3t))(0 4(t + 4)(e) with superscript (3t)) ))

D) ψ(t) =(table ( ((e) with superscript (-3t) (e) with superscript (-3t))(0 (t + 4)(e) with superscript (-3t)) ))

Type: MC Var: 1

96) Consider the first-order nonhomogeneous initial value problem

x' = (table ( (7 0)(7 7) ))x + (table ( (2(e) with superscript (5t))(6t) )), x(0.6) = (table ( (-6)(2) ))

Given a fundamental matrix ψ(t) for the system, what is the solution of this initial value problem?

A) x(t) = ψ(t)(ψ) with superscript (-1)(0.6)(table ( (-6)(2) )) + ψ(0.6)integral of ((ψ) with superscript (-1)(s)table ( (2(e) with superscript (5s))(6s) )ds) from (0.6) to (t)

B) x(t) = (ψ) with superscript (-1)(0.6)(table ( (-6)(2) )) + ψ(0.6)integral of ((ψ) with superscript (-1)(s)table ( (2(e) with superscript (5s))(6s) )ds) from (0.6) to (t)

C) x(t) = ψ(t)(ψ) with superscript (-1)(0.6)(table ( (-6)(2) )) + ψ(t)integral of ((ψ) with superscript (-1)(s)table ( (2(e) with superscript (5s))(6s) )ds) from (0.6) to (t)

D) x(t) = (ψ) with superscript (-1)(0.6)(table ( (-6)(2) )) + ψ(t)integral of ((ψ) with superscript (-1)(s)table ( (2(e) with superscript (5s))(6s) )ds) from (0.6) to (t)

Type: MC Var: 1

97) Express the following third-order nonhomogeneous differential equation as a matrix equation of the form x' = Ax + g:

u''' + 3(t) with superscript (3)u' + 5u = (e) with superscript (-4t) - 3

Type: SA Var: 1

© (2022) John Wiley & Sons, Inc. All rights reserved. Instructors who are authorized users of this course are permitted to download these materials and use them in connection with the course. Except as permitted herein or by law, no part of these materials should be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise.

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Document Type:
DOCX
Chapter Number:
7
Created Date:
Aug 21, 2025
Chapter Name:
Chapter 7 Systems Of First-Order Linear Equations
Author:
William E. Boyce

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