Ch8 Full Test Bank Numerical Methods - Complete Test Bank | Differential Equations 12e by William E. Boyce. DOCX document preview.

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Ch8 Full Test Bank Numerical Methods

Elementary Differential Equations, 12e (Boyce)

Chapter 8 Numerical Methods

1) The velocity (measured in meters per second) of an air-dropped container of food and supplies affixed to a parachute is described by the function v(t) = 66.8 tanh(0.20t), t ≥ 2.4, where t is measured in seconds. You are interested in approximating the vertical distance s(t) traveled by the package by time t = 2.50 seconds.

Using a step size of h = 0.10 seconds, compute the vertical distance traveled in the first 2.50 seconds.

(s) with subscript (1) = (s) with subscript (0) + (1/6)((k) with subscript (1) + 2(k) with subscript (2) + 2(k) with subscript (3) + (k) with subscript (4))h

(k) with subscript (1) = ________

A) 66.8 tanh(0.20)

B) 66.8 tanh(0.20 × 2.4)

C) (66.8/2)tanh(0.20 × 2.4)

D) (66.8/2)tanh((0.20/2))

E) 66.8 tanh((0.20 × 2.4/2))

Type: MC Var: 1

2) The velocity (measured in meters per second) of an air-dropped container of food and supplies affixed to a parachute is described by the function v(t) = 57.2 tanh(0.12t), t ≥ 3.0, where t is measured in seconds. You are interested in approximating the vertical distance s(t) traveled by the package by time t = 3.05 seconds.

Using a step size of h = 0.05 seconds, compute the vertical distance traveled in the first 3.05 seconds.

(s) with subscript (1) = (s) with subscript (0) + (1/6)((k) with subscript (1) + 2(k) with subscript (2) + 2(k) with subscript (3) + (k) with subscript (4))h

(k) with subscript (2) = ________

A) 57.2 tanh(0.12 × (3.0 + 0.05))

B) 57.2 tanh(3.0 + (0.05/2))

C) (57.2/2) tanh(3.0 + 0.05)

D) 57.2 tanh(0.12 × 3.0 + (0.05/2) )

E) (57.2/2) tanh(3.0 + (0.05/2))

Type: MC Var: 1

3) The velocity (measured in meters per second) of an air-dropped container of food and supplies affixed to a parachute is described by the function v(t) = 65.6 tanh(0.19t), t ≥ 3.0, where t is measured in seconds. You are interested in approximating the vertical distance s(t) traveled by the package by time t = 3.10 seconds.

Using a step size of h = 0.10 seconds, compute the vertical distance traveled in the first 3.10 seconds.

(s) with subscript (1) = (s) with subscript (0) + (1/6)((k) with subscript (1) + 2(k) with subscript (2) + 2(k) with subscript (3) + (k) with subscript (4))h

(k) with subscript (3) = ________

A) 65.6 tanh(3.0 + (0.10/2))

B) 65.6 tanh(0.19 × 3.0 + (0.10/2) )

C) (65.6/2) tanh(3.0 + (0.10/2))

D) (65.6/2) tanh(3.0 + 0.10)

E) 65.6 tanh(0.19 × (3.0 + 0.10))

Type: MC Var: 1

4) The velocity (measured in meters per second) of an air-dropped container of food and supplies affixed to a parachute is described by the function v(t) = 56 tanh(0.23t), t ≥ 3.2, where t is measured in seconds. You are interested in approximating the vertical distance s(t) traveled by the package by time t = 3.45 seconds.

Using a step size of h = 0.25 seconds, compute the vertical distance traveled in the first 3.45 seconds.

(s) with subscript (1) = (s) with subscript (0) + (1/6)((k) with subscript (1) + 2(k) with subscript (2) + 2(k) with subscript (3) + (k) with subscript (4))h

(k) with subscript (4) = ________

A) 56 tanh(0.23 × (3.2 + 0.25))

B) tanh(0.23 × (3.2 + 0.25))

C) 56 tanh(0.23 × 3.2 + (0.25/2) )

D) (56/2) tanh(0.23 × (3.2+ 0.25))

E) (56/2) tanh(0.23 × 3.2 + (0.25/2) )

Type: MC Var: 1

5) Consider the following initial value problem

3(dy/dx) + (x) with superscript (3)(y) with superscript (3) = sin(3πx), y(0) = 4

How would you need to rewrite this problem in order to apply the Runge-Kutta 4th order method to approximate the solution of this problem at a value of x?

A) (dy/dx) = (1/3)(sin(3πx) - (x) with superscript (3)(y) with superscript (3)), y(0) = 4

B) (dy/dx) = (1/3)sin(3πx), y(0) = 4

C) (dy/dx) = sin(3πx) - (x) with superscript (3)(y) with superscript (3), y(0) = 4

D) (dy/dx) = (1/3)(- (sin(3πx)/3π) - (x) with superscript (3)((y) with superscript (4)/4)), y(0) = 4

Type: MC Var: 1

6) Consider the following initial value problem

13(dy/dx) + 17(y) with superscript (4) = sin x, y(0) = 3

How would you need to rewrite this problem in order to apply Euler's method to approximate the solution of this problem at a value of x?

A) (dy/dx) = sin x - 17(y) with superscript (4), y(0) = 3

B) (dy/dx) = (1/13)(sin x - 17(y) with superscript (4)), y(0) = 3

C) (dy/dx) = (1/13)(- cos x - (4(y) with superscript (5)/5)), y(0) = 3

D) (dy/dx) = (1/13)sin x, y(0) = 3

Type: MC Var: 1

7) Consider the following initial value problem

(dy/dx) = (1/4)(cos(4x) - 4(y) with superscript (3)), y(0.5) = 5

The following question pertains to the various computational steps and identifications involved in applying Runge-Kutta's 4th order method to evaluate y(0.7).

To compute (x) with subscript (1) = (x) with subscript (0) + 0.20 and (y) with subscript (1) = (y) with subscript (0) + (1/6)((k) with subscript (1) + 2(k) with subscript (2) + 2(k) with subscript (3) + (k) with subscript (4))0.20, identify the parameter below. In what follows, f (x, y) = (1/4)(cos(4x) - 4(y) with superscript (3)).

(y) with subscript (0) = ________

Type: SA Var: 1

8) Consider the following initial value problem

(dy/dx) = (1/4)(cos(5x) - 8(y) with superscript (2)), y(0.4) = 2

The following question pertains to the various computational steps and identifications involved in applying Runge-Kutta's 4th order method to evaluate y(0.55).

To compute (x) with subscript (1) = (x) with subscript (0) + 0.15 and (y) with subscript (1) = (y) with subscript (0) + (1/6)((k) with subscript (1) + 2(k) with subscript (2) + 2(k) with subscript (3) + (k) with subscript (4))0.15, identify the parameter below. In what follows, f (x, y) = (1/4)(cos(5x) - 8(y) with superscript (2)).

(x) with subscript (0) = ________

Type: SA Var: 1

9) Consider the following initial value problem

(dy/dx) = (1/4)(cos(3x) - 5(y) with superscript (2)), y(0.1) = 5

The following question pertains to the various computational steps and identifications involved in applying Runge-Kutta's 4th order method to evaluate y(0.25).

To compute (x) with subscript (1) = (x) with subscript (0) + 0.15 and (y) with subscript (1) = (y) with subscript (0) + (1/6)((k) with subscript (1) + 2(k) with subscript (2) + 2(k) with subscript (3) + (k) with subscript (4))0.15, identify the parameter below. In what follows, f (x, y) = (1/4)(cos(3x) - 5(y) with superscript (2)).

(k) with subscript (1) = ________

A) f (0, (y) with subscript (0))

B) f ((y) with subscript (0), (x) with subscript (0))

C) f ((x) with subscript (0), (y) with subscript (0))

D) f ((x) with subscript (0) + 0.15, (y) with subscript (0) + 0.15)

Type: MC Var: 1

10) Consider the following initial value problem

(dy/dx) = (1/6)(cos(4x) - 7(y) with superscript (4)), y(0.1) = 4

The following question pertains to the various computational steps and identifications involved in applying Runge-Kutta's 4th order method to evaluate y(0.15).

To compute (x) with subscript (1) = (x) with subscript (0) + 0.05 and (y) with subscript (1) = (y) with subscript (0) + (1/6)((k) with subscript (1) + 2(k) with subscript (2) + 2(k) with subscript (3) + (k) with subscript (4))0.05, identify the parameter below. In what follows, f (x, y) = (1/6)(cos(4x) - 7(y) with superscript (4)).

(k) with subscript (2) = ________

A) f ((x) with subscript (0) + (0.05/2), (y) with subscript (0) + (0.05/2)(k) with subscript (1))

B) f ((x) with subscript (0) + 0.05, (y) with subscript (0) + 0.05(k) with subscript (1))

C) f ((x) with subscript (0) + (0.05/2), (y) with subscript (0) + (0.05/2))

D) f ((x) with subscript (0) + 0.05, (y) with subscript (0) + (0.05/2)(k) with subscript (1))

Type: MC Var: 1

11) Consider the following initial value problem

(dy/dx) = (1/4)(cos(5x) - 4(y) with superscript (3)), y(0.4) = 2

The following question pertains to the various computational steps and identifications involved in applying Runge-Kutta's 4th order method to evaluate y(0.6).

To compute (x) with subscript (1) = (x) with subscript (0) + 0.20 and (y) with subscript (1) = (y) with subscript (0) + (1/6)((k) with subscript (1) + 2(k) with subscript (2) + 2(k) with subscript (3) + (k) with subscript (4))0.20, identify the parameter below. In what follows, f (x, y) = (1/4)(cos(5x) - 4(y) with superscript (3)).

(k) with subscript (3) = ________

A) f ((x) with subscript (0) + (0.20/2), (y) with subscript (0) + (0.20/2))

B) f ((x) with subscript (0) + 0.20, (y) with subscript (0) + 0.20)

C) f ((x) with subscript (0) + 0.20, (y) with subscript (0) + 0.20(k) with subscript (1))

D) f ((x) with subscript (0) + (0.20/2), (y) with subscript (0) + (0.20/2)(k) with subscript (2))

Type: MC Var: 1

12) Consider the following initial value problem

(dy/dx) = (1/6)(cos(4x) - 6(y) with superscript (2)), y(0.5) = 1

The following question pertains to the various computational steps and identifications involved in applying Runge-Kutta's 4th order method to evaluate y(0.7).

To compute (x) with subscript (1) = (x) with subscript (0) + 0.20 and (y) with subscript (1) = (y) with subscript (0) + (1/6)((k) with subscript (1) + 2(k) with subscript (2) + 2(k) with subscript (3) + (k) with subscript (4))0.20, identify the parameter below. In what follows, f (x, y) = (1/6)(cos(4x) - 6(y) with superscript (2)).

(k) with subscript (4) = ________

A) f ((x) with subscript (0) + (0.20/2), (y) with subscript (0) + (0.20/2))

B) f ((x) with subscript (0) + (0.20/2), (y) with subscript (0) + (0.20/2)(k) with subscript (3))

C) f ((x) with subscript (0) + 0.20, (y) with subscript (0) + 0.20(k) with subscript (3))

D) f ((x) with subscript (0) + 0.20, (y) with subscript (0) + 0.20)

Type: MC Var: 1

13) Given the initial value problem y' = 9y + 6t, y(0) = 5, how many steps n are needed for the Euler method to find an approximation for y(1.77) using a step size of h = 0.03?

n = ________

Type: SA Var: 1

14) Given the initial value problem y' = (8 - 4y/3 + t)cos y, y(1) = 2, how many steps n are needed for the Euler method to find an approximation for y(4.6) using a step size of h = 0.06?

n = ________

Type: SA Var: 1

15) Given the initial value problem y' = (e) with superscript (4x) - 2y, y(0) = 4.0, how many steps n are needed for the RK method to find an approximation for y(1.8) using a step size of h = 0.010?

n = ________

A) n = 100

B) n = 180

C) n = 90

D) n = 360

Type: MC Var: 1

16) Consider the initial value problem y' = 9(t) with superscript (2) - (y) with superscript (2), y(0) = 1. This question relates to using the Euler method to approximate the solution of this problem at t = 0.4 using a step size of h = (1/50).

Which of the following identifications are correct when setting up Euler's method? Select all that apply.

A) f (t, y) = -(y) with superscript (2)

B) f (t, y) = 9(t) with superscript (2) - (y) with superscript (2)

C) (t) with subscript (0) = 0

D) (t) with subscript (0) = 1

E) (y) with subscript (0) = 0

F) (y) with subscript (0) = 1

Type: MC Var: 1

17) Consider the initial value problem y' = 4(t) with superscript (2) - (y) with superscript (2), y(0) = 1. This question relates to using the Euler method to approximate the solution of this problem at t = 0.5 using a step size of h = (1/40).

What are the correct values of (t) with subscript (1) and (t) with subscript (2)?

A) (t) with subscript (1) = (1/20), (t) with subscript (2) = (1/10)

B) (t) with subscript (1) = (0.5/10), (t) with subscript (2) = (0.5/5)

C) (t) with subscript (1) = (0.5/20), (t) with subscript (2) = (0.5/10)

D) (t) with subscript (1) = (1/10), (t) with subscript (2) = (1/5)

Type: MC Var: 1

18) Consider the initial value problem y' = 4(t) with superscript (2) - (y) with superscript (2), y(0) = 1. This question relates to using the Euler method to approximate the solution of this problem at t = 0.2 using a step size of h = (1/100).

(y) with subscript (1) = ________

Type: SA Var: 1

19) Consider the initial value problem y' = 9(t) with superscript (2) - (y) with superscript (2), y(0) = 1. This question relates to using the Euler method to approximate the solution of this problem at t = 0.1 using a step size of h = (1/200).

Which of the following equals (y) with subscript (2)?

A) (y) with subscript (2) = (1/200)((9/(200) with superscript (2)) - (1 - (1/200)) with superscript (2))

B) (y) with subscript (2) = ((9/(200) with superscript (2)) - (1 - (1/200)) with superscript (2))

C) (y) with subscript (2) = 1 - (1/200) + ((9/(200) with superscript (2)) - (1 - (1/200)) with superscript (2))

D) (y) with subscript (2) = 1 - (1/200) + (1/200)((9/(200) with superscript (2)) - (1 - (1/200)) with superscript (2))

Type: MC Var: 1

20) Consider the initial value problem y' = 25(t) with superscript (2) - (y) with superscript (2), y(0) = 1. This question relates to using the backward Euler method to approximate the solution at t = 0.2, namely (y) with subscript (1) = y(0.2), using a step size of h = 0.02.

Which of these identifications are correct when setting up the backward Euler method? Select all that apply.

A) (t) with subscript (0) = 0

B) (y) with subscript (0) = 0

C) (y) with subscript (0) = 1

D) (t) with subscript (1) = 1

E) (t) with subscript (1) = 0.2

Type: MC Var: 1

21) Consider the initial value problem y' = 25(t) with superscript (2) - (y) with superscript (2), y(0) = 1. This question relates to using the backward Euler method to approximate the solution at t = 0.4, namely (y) with subscript (1) = y(0.4), using a step size of h = 0.02.

Which of these equations is the result of applying the backward Euler method to solve for (y) with subscript (1) = y(0.4)?

A) ((y) with subscript (1)) with superscript (2) - (y) with subscript (1) + 25 × (0.4) with superscript (2) = 0

B) 0.02((y) with subscript (1)) with superscript (2) + (y) with subscript (1) - (1 + 25 × (0.4) with superscript (2) × 0.02) = 0

C) 0.02((y) with subscript (1)) with superscript (2) - (y) with subscript (1) + (1 + 25 × (0.4) with superscript (2) × 0.02) = 0

D) ((y) with subscript (1)) with superscript (2) + (y) with subscript (1) - 25 × (0.4) with superscript (2) = 0

Type: MC Var: 1

22) Consider the initial value problem y' = (y) with superscript (2)sin(2t), y(0) = 4. Use the Runge-Kutta method to approximate the solution at t = 0.12, namely (y) with subscript (1) = y(0.12), using a step size of h = 0.12.

(a) Find the following constants needed for the Runge-Kutta method.

(i) (k) with subscript (1) = ________

(ii) (k) with subscript (2) = ________

(iii) (k) with subscript (3) = ________

A. (sin) with superscript (3) 0.12 + 4 (sin) with superscript (2) 0.12 + 4 sin 0.12

B. 4 (sin) with superscript (3) 0.12 + 16 (sin) with superscript (2) 0.12 + 16 sin 0.12

C. 16 sin 0.12 ((sin) with superscript (2) 0.12 + sin 0.12 + 1)

D. 4 sin 0.12 ((sin) with superscript (2) 0.12 + sin 0.12 + 1)

(iv) (k) with subscript (4) = ________

A. (4 + 0.12 × ((k) with subscript (3)) with superscript (2))sin(2 × 0.12)

B. 4 + 0.12 × (k) with subscript (3) × sin(0.12)

C. 4 + 0.12 × ((k) with subscript (3)) with superscript (2) × sin(0.12)

D. ((4 + 0.12 × (k) with subscript (3))) with superscript (2)sin(2 × 0.12)

E. (4 + 0.12 × (k) with subscript (3)) sin(0.12)

(b) Now estimate (y) with subscript (1) = y(0.12) using the Runge-Kutta method.

A. 4 + (0.12/6)[8 sin 0.12 (8 + 4 sin 0.12 + (sin) with superscript (2) 0.12) + ((4 + 0.12 × (k) with subscript (3))) with superscript (2)sin(2 × 0.12)]

B. 4 + (0.12/6)[8 sin 0.12 (8 + 4 sin 0.12 + (sin) with superscript (2) 0.12) + 4 + 0.12 × ((k) with subscript (3)) with superscript (2)sin(2 × 0.12)]

C. 4 + (1/6)[sin 0.12 (16 + 8 sin 0.12 + 8 (sin) with superscript (2) 0.12) + ((4 + 0.12 × (k) with subscript (3))) with superscript (2)sin(0.12)]

D. (4 + 0.12/6)[sin 0.12 (1 + sin 0.12 + (sin) with superscript (2) 0.12) +( 4 + 0.12 × (k) with subscript (3))sin(2 × 0.12)]

(ii) 16 × sin(0.12)

(iii) B

(iv) D

(b) A

Type: ES Var: 1

23) Consider the initial value problem y' = -3t + 7y, y(0) = 1. This question is related to using the predictor-corrector method to estimate the solution y(0.2) using a step size of h = 0.05.

Which of the following is the correct formula for (t) with subscript (i), i = 1, 2, 3, 4?

A) (t) with subscript (i) + (t) with subscript (i-1) = 0.05

B) (t) with subscript (i) - 0.05(t) with subscript (i-1) = 0

C) (t) with subscript (i) - (t) with subscript (i-1) = 0.05

D) (t) with subscript (i) = 0.05

Type: MC Var: 1

24) Consider the initial value problem y' = -3t + 7y, y(0) = 1. This question is related to using the predictor-corrector method to estimate the solution y(0.2) using a step size of h = 0.05.

For this problem, you will need these values to carry out the computations:

(y) with subscript (0) = 1

(y) with subscript (1) = 1.4147953

(y) with subscript (2) = 1.9944196

(y) with subscript (3) = 2.8079396

Which of the following is the correct formula for (f) with subscript (i), i = 1, 2, 3?

A) (f) with subscript (i) = 0.05i + 7(y) with subscript (i)

B) (f) with subscript (i) = -3 × 0.05i + 7(y) with subscript (i-1)

C) (f) with subscript (i) = -3 × 0.05(i - 1) + 7(y) with subscript (i-1)

D) (f) with subscript (i) = -3 × 0.05i + 7(y) with subscript (i)

Type: MC Var: 1

25) Consider the initial value problem y' = -3t + 7y, y(0) = 1. This question is related to using the predictor-corrector method to estimate the solution y(0.2) using a step size of h = 0.05.

For this problem, you will need these values to carry out the computations:

(y) with subscript (0) = 1

(y) with subscript (1) = 1.4147953

(y) with subscript (2) = 1.9944196

(y) with subscript (3) = 2.8079396

Which of the following is the predicted value for (y) with subscript (4)?

A) (y) with subscript (3) + (0.05/24)[55(-3 × 0.15 + 7(y) with subscript (3)) - 59(-3 × 0.10 + 7(y) with subscript (2)) + 37(-3 × 0.05 + 7(y) with subscript (1)) - 9 × 5]

B) (y) with subscript (3) + (0.05/24)[59(-3 × 0.15 + 7(y) with subscript (3)) - 55(-3 × 0.10 + 7(y) with subscript (2)) - 37(-3 × 0.05 + 7(y) with subscript (1)) + 9 × 5]

C) (y) with subscript (3) + (1/24)[59(-3 × 0.15 + 7(y) with subscript (3)) - 55(-3 × 0.10 + 7(y) with subscript (2)) - 37(-3 × 0.05 + 7(y) with subscript (1)) + 9 × 5]

D) (y) with subscript (3) + (1/24)[55(-3 × 0.15 + 7(y) with subscript (3)) - 59(-3 × 0.10 + 7(y) with subscript (2)) + 37(-3 × 0.05 + 7(y) with subscript (1)) - 9 × 5]

Type: MC Var: 1

26) Consider the initial value problem y' = -3t + 7y, y(0) = 1. This question is related to using the predictor-corrector method to estimate the solution y(0.2) using a step size of h = 0.05.

For this problem, you will need these values to carry out the computations:

(y) with subscript (0) = 1

(y) with subscript (1) = 1.4147953

(y) with subscript (2) = 1.9944196

(y) with subscript (3) = 2.8079396

To use the corrector formula, you need (f) with subscript (4). Which of the following is the correct expression for (f) with subscript (4)?

A) 0.2×(-3) - 7×((y) with subscript (3) + (0.05/24)[55(-3×0.15 + 7(y) with subscript (3)) - 59(-3×0.10 + 7(y) with subscript (2)) + 37(-3×0.05 + 7(y) with subscript (1)) - 9×5])

B) 7 × ((y) with subscript (3) + (0.05/24)[55(-3 × 0.15 + 7(y) with subscript (3)) - 59(-3 × 0.10 + 7(y) with subscript (2)) + 37(-3 × 0.05 + 7(y) with subscript (1)) - 9 × 5])

C) 0.2×(-3)+7×((y) with subscript (3) + (0.05/24)[55(-3×0.15 + 7(y) with subscript (3)) - 59(-3×0.10 + 7(y) with subscript (2)) + 37(-3×0.05 + 7(y) with subscript (1)) - 9×5])

D) -7 × ((y) with subscript (3) + (0.05/24)[55(-3 × 0.15 + 7(y) with subscript (3)) - 59(-3 × 0.10 + 7(y) with subscript (2)) + 37(-3 × 0.05 + 7(y) with subscript (1)) - 9 × 5])

Type: MC Var: 1

27) Consider the initial value problem y' = -3t + 7y, y(0) = 1. This question is related to using the predictor-corrector method to estimate the solution y(0.2) using a step size of h = 0.05.

For this problem, you will need these values to carry out the computations:

(y) with subscript (0) = 1

(y) with subscript (1) = 1.4147953

(y) with subscript (2) = 1.9944196

(y) with subscript (3) = 2.8079396

Which of the following show a portion of the formula for the corrected value of (y) with subscript (4)?

A) (y) with subscript (3) + (9×0.05/24)(0.2×(-3) + 7×((y) with subscript (3) + (0.05/24)[55(-3×0.15 + 7(y) with subscript (3)) - 59(-3×0.10 + 7(y) with subscript (2)) + 37(-3×0.05 + 7(y) with subscript (1)) - 9×5]) )

B) (y) with subscript (3) - (9×0.05/24)(0.2×(-3) + 7×((y) with subscript (3) + (0.05/24)[55(-3×0.15 + 7(y) with subscript (3)) - 59(-3×0.10 + 7(y) with subscript (2)) + 37(-3×0.05 + 7(y) with subscript (1)) - 9×5]) )

C) (y) with subscript (3) + (37×0.05/24)(7×((y) with subscript (3) + (0.05/24)[55(-3×0.15 + 7(y) with subscript (3)) - 59(-3×0.10 + 7(y) with subscript (2)) + 37(-3×0.05 + 7(y) with subscript (1)) - 9×5]) )

D) (y) with subscript (3) - (37×0.05/24)(7×((y) with subscript (3) + (0.05/24)[55(-3×0.15 + 7(y) with subscript (3)) - 59(-3×0.10 + 7(y) with subscript (2)) + 37(-3×0.05 + 7(y) with subscript (1)) - 9×5]) )

Type: MC Var: 1

28) Consider the initial value problem y' = 2t + 5y, y(0) = 1. This question is related to using the predictor-corrector method to estimate the solution y(0.2) using a step size of h = 0.05.

Which of the following is the correct formula for (t) with subscript (i), i = 1, 2, 3, 4?

A) (t) with subscript (i) + (t) with subscript (i-1) = 0.05

B) (t) with subscript (i) - 0.05(t) with subscript (i-1) = 0

C) (t) with subscript (i) - (t) with subscript (i-1) = 0.05

D) (t) with subscript (i) = 0.05

Type: MC Var: 1

29) Consider the initial value problem y' = 2t + 5y, y(0) = 1. This question is related to using the predictor-corrector method to estimate the solution y(0.2) using a step size of h = 0.05.

For this problem, you will need these values to carry out the computations:

(y) with subscript (0) = 1

(y) with subscript (1) = 1.2867383

(y) with subscript (2) = 1.6605954

(y) with subscript (3) = 2.1463147

Which of the following is the correct formula for (f) with subscript (i), i = 1, 2, 3?

A) (f) with subscript (i) = 0.05i + 5(y) with subscript (i)

B) (f) with subscript (i) = 2 × 0.05i + 5(y) with subscript (i - 1)

C) (f) with subscript (i) = 2 × 0.05(i - 1) + 5(y) with subscript (i - 1)

D) (f) with subscript (i) = 2 × 0.05i + 5(y) with subscript (i)

Type: MC Var: 1

30) Consider the initial value problem y' = 2t + 5y, y(0) = 1. This question is related to using the predictor-corrector method to estimate the solution y(0.2) using a step size of h = 0.05.

For this problem, you will need these values to carry out the computations:

(y) with subscript (0) = 1

(y) with subscript (1) = 1.2867383

(y) with subscript (2) = 1.6605954

(y) with subscript (3) = 2.1463147

Which of the following is the predicted value for (y) with subscript (4)?

A) (y) with subscript (3) + (0.05/24)[55(2 × 0.15 + 5(y) with subscript (3)) - 59(2 × 0.10 + 5(y) with subscript (2)) + 37(2 × 0.05 + 5(y) with subscript (1)) - 9 × 5]

B) (y) with subscript (3) + (0.05/24)[55(2 × 0.15 + 5(y) with subscript (3)) - 37(2 × 0.10 + 5(y) with subscript (2)) + 9(2 × 0.05 + 5(y) with subscript (1)) - 1 × 5]

C) (y) with subscript (3) + (1/24)[55(2 × 0.15 + 5(y) with subscript (3)) - 37(2 × 0.10 + 5(y) with subscript (2)) + 9(2 × 0.05 + 5(y) with subscript (1)) - 1 × 5]

D) (y) with subscript (3) + (1/24)[55(2 × 0.15 + 5(y) with subscript (3)) - 59(2 × 0.10 + 5(y) with subscript (2)) + 37(2 × 0.05 + 5(y) with subscript (1)) - 9 × 5]

Type: MC Var: 1

31) Consider the initial value problem y' = 2t + 5y, y(0) = 1. This question is related to using the predictor-corrector method to estimate the solution y(0.2) using a step size of h = 0.05.

For this problem, you will need these values to carry out the computations:

(y) with subscript (0) = 1

(y) with subscript (1) = 1.2867383

(y) with subscript (2) = 1.6605954

(y) with subscript (3) = 2.1463147

To use the corrector formula, you need (f) with subscript (4). Which of the following is the correct expression for (f) with subscript (4)?

A) 0.2×2 - 5×((y) with subscript (3) + (0.05/24)[55(2×0.15 + 5(y) with subscript (3)) - 59(2×0.10 + 5(y) with subscript (2)) + 37(2×0.05 + 5(y) with subscript (1)) - 9×5])

B) 5 × ((y) with subscript (3) + (0.05/24)[55(2 × 0.15 + 5(y) with subscript (3)) - 59(2 × 0.10 + 5(y) with subscript (2)) + 37(2 × 0.05 + 5(y) with subscript (1)) - 9 × 5])

C) 0.2×2 + 5×((y) with subscript (3) + (0.05/24)[55(2×0.15 + 5(y) with subscript (3)) - 59(2×0.10 + 5(y) with subscript (2)) + 37(2×0.05 + 5(y) with subscript (1)) - 9×5])

D) -5 × ((y) with subscript (3) + (0.05/24)[55(2 × 0.15 + 5(y) with subscript (3)) - 59(2 × 0.10 + 5(y) with subscript (2)) + 37(2 × 0.05 + 5(y) with subscript (1)) - 9 × 5])

Type: MC Var: 1

32) Consider the initial value problem y' = 2t + 5y, y(0) = 1. This question is related to using the predictor-corrector method to estimate the solution y(0.2) using a step size of h = 0.05.

For this problem, you will need these values to carry out the computations:

(y) with subscript (0) = 1

(y) with subscript (1) = 1.2867383

(y) with subscript (2) = 1.6605954

(y) with subscript (3) = 2.1463147

Which of the following show a portion of the formula for the corrected value of (y) with subscript (4)?

A) (y) with subscript (3) + (9×0.05/24)(0.2×2 + 5×((y) with subscript (3) + (0.05/24)[55(2×0.15 + 5(y) with subscript (3)) - 59(2×0.10 + 5(y) with subscript (2)) + 37(2×0.05 + 5(y) with subscript (1)) - 9×5]) )

B) (y) with subscript (3) - (9×0.05/24)(0.2×2 + 5×((y) with subscript (3) + (0.05/24)[55(2×0.15 + 5(y) with subscript (3)) - 59(2×0.10 + 5(y) with subscript (2)) + 37(2×0.05 + 5(y) with subscript (1)) - 9×5]) )

C) (y) with subscript (3) + (37×0.05/24)(5×((y) with subscript (3) + (0.05/24)[55(2×0.15 + 5(y) with subscript (3)) - 59(2×0.10 + 5(y) with subscript (2)) + 37(2×0.05 + 5(y) with subscript (1)) - 9×5]) )

D) (y) with subscript (3) - (37×0.05/24)(5×((y) with subscript (3) + (0.05/24)[55(2×0.15 + 5(y) with subscript (3)) - 59(2×0.10 + 5(y) with subscript (2)) + 37(2×0.05 + 5(y) with subscript (1)) - 9×5]) )

Type: MC Var: 1

33) Consider the initial value problem y' = 2t - 5y, y(0) = 1. This question is related to using the fourth-order backward differentiation formula to estimate the solution y(0.2) using a step size of h = 0.05.

Which of these is the correct formula for (t) with subscript (i), i = 1, 2, 3, 4?

A) (t) with subscript (i) + (t) with subscript (i - 1) = 0.05

B) (t) with subscript (i) - 0.05(t) with subscript (i - 1) = 0

C) (t) with subscript (i) - (t) with subscript (i - 1) = 0.05

D) (t) with subscript (i) = 0.05

Type: MC Var: 1

34) Consider the initial value problem y' = 2t - 5y, y(0) = 1. This question is related to using the fourth-order backward differentiation formula to estimate the solution y(0.2) using a step size of h = 0.05.

For the following problem, you will need these values to carry out the computation:

(y) with subscript (0) = 1

(y) with subscript (1) = 0.7811133

(y) with subscript (2) = 0.6150663

(y) with subscript (3) = 0.4901712

Which of the following is the value of (y) with subscript (4)?

A) (y) with subscript (4) = (48 × 0.4901712/25 - 12(0.05)(5)) - (36 × 0.6150663/25 - 12(0.05)(5)) + (16 × 0.4901712/25 - 12(0.05)(5)) - (3 × 1/25 - 12(0.05)(5)) + (12 × (0.05) × (2) × (0.2)/25 - 12(0.05)(5))

B) (y) with subscript (4) = (48 × 0.4901712/25 + 12(0.05)(5)) + (36 × 0.6150663/25 + 12(0.05)(5)) - (16 × 0.4901712/25 + 12(0.05)(5)) + (3 × 1/25 + 12(0.05)(5)) - (12 × (0.05) × (2) × (0.2)/25 + 12(0.05)(5))

C) (y) with subscript (4) = (36 × 0.4901712/25 + 12(0.05)(5)) - (16 × 0.6150663/25 + 12(0.05)(5)) + (3 × 0.4901712/25 + 12(0.05)(5)) - (2 × 1/25 + 12(0.05)(5)) + (12 × (0.05) × (2) × (0.2)/25 + 12(0.05)(5))

D) (y) with subscript (4) = (48 × 0.4901712/25 + 12(0.05)(5)) - (36 × 0.6150663/25 + 12(0.05)(5)) + (16 × 0.4901712/25 + 12(0.05)(5)) - (3 × 1/25 + 12(0.05)(5)) + (12 × (0.05) × (2) × (0.2)/25 + 12(0.05)(5))

Type: MC Var: 1

35) Consider the initial value problem y' = 2t - 5y, y(0) = 1. This question is related to using the fourth-order backward differentiation formula to estimate the solution y(0.2) using a step size of h = 0.05.

For the following problem, you will need these values to carry out the computation:

(y) with subscript (0) = 1

(y) with subscript (1) = 0.7811133

(y) with subscript (2) = 0.6150663

(y) with subscript (3) = 0.4901712

Which of the following expressions represents the error (E) with subscript (4) incurred in using this method to estimate y(0.2)?

A) y(0.2) + (y) with subscript (4)

B) y(0.2) - (y) with subscript (4)

C) (y) with subscript (4) - 0.2

D) (y) with subscript (4) + 0.2

Type: MC Var: 1

36) Consider the initial value problem y' = -3t + 7y, y(0) = 1. This question is related to using the fourth-order backward differentiation formula to estimate the solution y(0.2) using a step size of h = 0.05.

Which of these is the correct formula for (t) with subscript (i), i = 1, 2, 3, 4?

A) (t) with subscript (i) + (t) with subscript (i - 1) = 0.05

B) (t) with subscript (i) - 0.05(t) with subscript (i - 1) = 0

C) (t) with subscript (i) - (t) with subscript (i - 1) = 0.05

D) (t) with subscript (i) = 0.05

Type: MC Var: 1

37) Consider the initial value problem y' = -3t + 7y, y(0) = 1. This question is related to using the fourth-order backward differentiation formula to estimate the solution y(0.2) using a step size of h = 0.05.

For the following problem, you will need these values to carry out the computation:

(y) with subscript (0) = 1

(y) with subscript (1) = 1.4147953

(y) with subscript (2) = 1.9944196

(y) with subscript (3) = 2.8079396

Which of the following is the value of (y) with subscript (4)?

A) (y) with subscript (4) = (48 × 2.8079396/25×(1+12(0.05)(7))) - (36 × 1.9944196/25×(1+12(0.05)(7))) + (16 × 1.4147953/25×(1+12(0.05)(7))) - (3 × 1/25×(1+12(0.05)(7))) + (12×(0.05)×(-3)×(0.2)/25×(1+12(0.05)(7)))

B) (y) with subscript (4) = (48 × 2.8079396/25×(1+12(0.05)(7))) + (36 × 1.9944196/25×(1+12(0.05)(7))) - (16 × 1.4147953/25×(1+12(0.05)(7))) + (3 × 1/25×(1+12(0.05)(7))) - (12×(0.05)×(-3)×(0.2)/25×(1+12(0.05)(7)))

C) (y) with subscript (4) = (36 × 2.8079396/25×(1+12(0.05)(7))) - (16 × 1.9944196/25×(1+12(0.05)(7))) + (3 × 1.4147953/25×(1+12(0.05)(7))) - (1 × 1/25×(1+12(0.05)(7))) + (12×(0.05)×(-3)×(0.2)/25×(1+12(0.05)(7)))

D) (y) with subscript (4) = (48 × 2.8079396/25×(1-12(0.05)(7))) - (36 × 1.9944196/25×(1-12(0.05)(7))) + (16 × 1.4147953/25×(1-12(0.05)(7))) - (3 × 1/25×(1-12(0.05)(7))) + (12×(0.05)×(-3)×(0.2)/25×(1-12(0.05)(7)))

Type: MC Var: 1

38) Consider the initial value problem y' = -3t + 7y, y(0) = 1. This question is related to using the fourth-order backward differentiation formula to estimate the solution y(0.2) using a step size of h = 0.05.

For the following problem, you will need these values to carry out the computation:

(y) with subscript (0) = 1

(y) with subscript (1) = 1.4147953

(y) with subscript (2) = 1.9944196

(y) with subscript (3) = 2.8079396

Which of the following expressions represents the error (E) with subscript (4) incurred in using this method to estimate y(0.2)?

A) y(0.2) + (y) with subscript (4)

B) y(0.2) - (y) with subscript (4)

C) (y) with subscript (4) - 0.2

D) (y) with subscript (4) + 0.2

Type: MC Var: 1

39) Consider the system of initial value problems given by

x' = 2ty + 4x, x(0) = 0

y' = 5(y) with superscript (2) + 3x(t) with superscript (2), y(0) = 1

This problem can be expressed using matrix notation as

X' = F(t, X)

where

X = (table ( (x)(y) ))

F(t, X) = (table ( (f(t, x, y))(g(t, x, y)) )) = (table ( (2ty + 4x)(5(x) with superscript (2) + 3x(t) with superscript (2)) ))

X((t) with subscript (0)) = (table ( (0)(1) ))

When using the Euler method with h = 0.05, how many iterations n do you need in order to estimate the solution (y) with subscript (n) at t = 2 × 0.05?

n = ________

Type: SA Var: 1

40) Consider the system of initial value problems given by

x' = 2ty + 3x, x(0) = 0

y' = 5(y) with superscript (2) + 9x(t) with superscript (2), y(0) = 1

This problem can be expressed using matrix notation as

X' = F(t, X)

where

X = (table ( (x)(y) ))

F(t, X) = (table ( (f(t, x, y))(g(t, x, y)) )) = (table ( (2ty + 3x)(5(x) with superscript (2) + 9x(t) with superscript (2)) ))

X((t) with subscript (0)) = (table ( (0)(1) ))

When using the Euler method with h = 0.1, (t) with subscript (1) = ________

Type: SA Var: 1

41) Consider the system of initial value problems given by

x' = 2ty + 3x, x(0) = 0

y' = 6(y) with superscript (2) + 3x(t) with superscript (2), y(0) = 1

This problem can be expressed using matrix notation as

X' = F(t, X)

where

X = (table ( (x)(y) ))

F(t, X) = (table ( (f (t, x, y))(g(t, x, y)) )) = (table ( (2ty + 3x)(6(x) with superscript (2) + 3x(t) with superscript (2)) ))

X((t) with subscript (0)) = (table ( (0)(1) ))

When using the Euler method with h = 0.05, what are the values of (x) with subscript (1) and (y) with subscript (1) when using the Euler method?

A) (x) with subscript (1) = 0, (y) with subscript (1) = 0.3

B) (x) with subscript (1) = 0, (y) with subscript (1) = 1 + 0.3

C) (x) with subscript (1) = 1, (y) with subscript (1) = 0.3

D) (x) with subscript (1) = 1, (y) with subscript (1) = 1 + 0.3

Type: MC Var: 1

42) Consider the system of initial value problems given by

x' = 2ty + 3x, x(0) = 0

y' = 5(y) with superscript (2) + 3x(t) with superscript (2), y(0) = 1

This problem can be expressed using matrix notation as

X' = F(t, X)

where

X = (table ( (x)(y) ))

F(t, X) = (table ( (f (t, x, y))(g(t, x, y)) )) = (table ( (2ty + 3x)(5(x) with superscript (2) + 3x(t) with superscript (2)) ))

X((t) with subscript (0)) = (table ( (0)(1) ))

When using the Euler method with h = 0.1, (t) with subscript (2) = ________

Type: SA Var: 1

43) Consider the system of initial value problems given by

x' = 3ty + 4x, x(0) = 0

y' = 6(y) with superscript (2) + 3x(t) with superscript (2), y(0) = 1

This problem can be expressed using matrix notation as

X' = F(t, X)

where

X = (table ( (x)(y) ))

F(t, X) = (table ( (f (t, x, y))(g(t, x, y)) )) = (table ( (3ty + 4x)(6(x) with superscript (2) + 3x(t) with superscript (2)) ))

X((t) with subscript (0)) = (table ( (0)(1) ))

When using the Euler method with h = 0.05, what are the values of (x) with subscript (2) and (y) with subscript (2) when using the Euler method?

A) (x) with subscript (2) = 3 × (0.05) with superscript (2)(1 + 6 × 0.05), (y) with subscript (2) = 1 + 2 × 6 × 0.05 + 2 × (6) with superscript (2) × (0.05) with superscript (2) + (6) with superscript (3) × (0.05) with superscript (3)

B) (x) with subscript (2) = 3 × 6 × (0.05) with superscript (3), (y) with subscript (2) = 2 × 6 × 0.05 + 2 × (6) with superscript (2) × (0.05) with superscript (2) + (6) with superscript (3) × (0.05) with superscript (3)

C) (x) with subscript (2) = 3 × 0.05(1 + 6 × 0.05), (y) with subscript (2) = 1 + 2 × 6 × 0.05 + 2 × (6) with superscript (2) × (0.05) with superscript (2)

D) (x) with subscript (2) = 3×(0.05) with superscript (2)(1 + 6 × 0.05), (y) with subscript (2) = 1 + 2 × 6 × 0.05 + 2 × (6) with superscript (2) × (0.05) with superscript (2)

Type: MC Var: 1

44) Consider the system of initial value problems given by

x' = 3ty + 3x, x(0) = 0

y' = 6(y) with superscript (2) + 3x(t) with superscript (2), y(0) = 1

This problem can be expressed using matrix notation as

X' = F(t, X)

where

X = (table ( (x)(y) ))

F(t, X) = (table ( (f (t, x, y))(g(t, x, y)) )) = (table ( (3ty + 3x)(6(x) with superscript (2) + 3x(t) with superscript (2)) ))

X((t) with subscript (0)) = (table ( (0)(1) ))

When using the improved Euler method with h = 0.05, how many iterations n do you need in order to estimate the solution (y) with subscript (n) at t = 0.10?

n = ________

Type: SA Var: 1

45) Consider the system of initial value problems given by

x' = 3ty + 3x, x(0) = 0

y' = 6(y) with superscript (2) + 3x(t) with superscript (2), y(0) = 1

This problem can be expressed using matrix notation as

X' = F(t, X)

where

X = (table ( (x)(y) ))

F(t, X) = (table ( (f (t, x, y))(g(t, x, y)) )) = (table ( (3ty + 3x)(6(x) with superscript (2) + 3x(t) with superscript (2)) ))

X((t) with subscript (0)) = (table ( (0)(1) ))

When using the improved Euler method with h = 0.05, (t) with subscript (1) = ________

Type: SA Var: 1

46) Consider the system of initial value problems given by

x' = 2ty + 3x, x(0) = 0

y' = 5(y) with superscript (2) + 3x(t) with superscript (2), y(0) = 1

This problem can be expressed using matrix notation as

X' = F(t, X)

where

X = (table ( (x)(y) ))

F(t, X) = (table ( (f (t, x, y))(g(t, x, y)) )) = (table ( (2ty + 3x)(5(x) with superscript (2) + 3x(t) with superscript (2)) ))

X((t) with subscript (0)) = (table ( (0)(1) ))

What are the values of (x) with subscript (1) and (y) with subscript (1) when using the improved Euler method with h = 0.05?

A) (x) with subscript (1) = (2 × (0.05) with superscript (2)/2), (y) with subscript (1) = 1 + (0.05/2)(5 + (5) with superscript (3) × (0.05) with superscript (2))

B) (x) with subscript (1) = 2 × (0.05) with superscript (2)(1 + 5 × 0.05), (y) with subscript (1) = 1 + (0.05/2)(5 + 5 × ((1 + 5 × 0.05)) with superscript (2) + 5 × 3 × (0.05) with superscript (3))

C) (x) with subscript (1) = 2 × 0.05(1 + 5 × 0.05), (y) with subscript (1) = 1 + (0.05/2)(5 + 5 × ((1 + 5 × 0.05)) with superscript (2) + 5 × 3 × (0.05) with superscript (3))

D) (x) with subscript (1) = 2 × (0.05) with superscript (2)(1 + 5 × 0.05), (y) with subscript (1) = (0.05/2)(5 + 5 × ((1 + 5 × 0.05)) with superscript (2) + 5 × 3 × (0.05) with superscript (3))

Type: MC Var: 1

47) Consider the initial value problem

y' - y = (e) with superscript (2t)(y) with superscript (2), y(0) = 1

(Note: The exact solution is y = - (3(e) with superscript (t)/(e) with superscript (3t) - 4))

To apply the improved Euler method, which of the following expressions would you use for f (t, y)?

A) y - (e) with superscript (2t)(y) with superscript (2)

B) (e) with superscript (2t)(y) with superscript (2) - y

C) (e) with superscript (2t)(y) with superscript (2)

D) y + (e) with superscript (2t)(y) with superscript (2)

Type: MC Var: 1

48) Consider the initial value problem

y' - y = (e) with superscript (2t)(y) with superscript (2), y(0) = 1

(Note: The exact solution is y = - (3(e) with superscript (t)/(e) with superscript (3t) - 4))

Using a step size of h = 0.25, compute the following approximations. Give your answers accurate to 6 decimal places.

(i) (y) with subscript (1) = ________

(ii) (y) with subscript (2) = ________

(iii) (y) with subscript (3) = ________

(iv) (y) with subscript (4) = ________

(ii) 8.446435

(iii) 1993.872136

(iv) 18345839793141.130763

Type: ES Var: 1

49) Consider the initial value problem

y' - y = (e) with superscript (2t)(y) with superscript (2), y(0) = 1

(Note: The exact solution is y = - (3(e) with superscript (t)/(e) with superscript (3t) - 4))

Which of the following expressions represents the error in the estimation for the improved Euler method?

(e) with subscript (4) = ________

A) |(y) with subscript (k) - (3(e) with superscript (0.25)/(e) with superscript (0.75) - 4)|

B) |(3(e) with superscript (0.25)/(e) with superscript (0.75) - 4) - (y) with subscript (k)|

C) |- (3e/(e) with superscript (3) - 4) - (y) with subscript (k)|

D) |(3e/(e) with superscript (3) - 4) - (y) with subscript (k)|

Type: MC Var: 1

50) Consider the initial value problem

y' - y = (e) with superscript (2t)(y) with superscript (2), y(0) = 1

(Note: The exact solution is y = - (3(e) with superscript (t)/(e) with superscript (3t) - 4))

The size of the error (e) with subscript (k) is large because

A) there is a vertical asymptote between [0, 1].

B) the step size is far from the initial time (t) with subscript (0).

C) the estimation y(1) is estimated far from the initial time (t) with subscript (0).

D) the function f (t) contains an exponential function (e) with superscript (2t).

Type: MC Var: 1

51) Consider the initial value problem

y' = 2 - 3t(y) with superscript (3), y(1) = 2

This question is related to using the Runge-Kutta method for approximating the solution y at t = 1.1 with a step size of h = 0.05.

How many approximations (y) with subscript (n) are needed to estimate a solution at y(1.1) if h = 0.05?

n = ________

Type: SA Var: 1

52) Consider the initial value problem

y' = 2 - 3t(y) with superscript (3), y(1) = 2

This question is related to using the Runge-Kutta method for approximating the solution y at t = 1.1 with a step size of h = 0.05.

To find (y) with subscript (1), first calculate the following in order to apply the Runge-Kutta method. Express your answers accurate to seven decimal places.

(i) (k) with subscript (1) = ________

(ii) (k) with subscript (2) = ________

(iii) (k) with subscript (3) = ________

(iv) (k) with subscript (4) = ________

(v) So, using the Runge-Kutta method, (y) with subscript (1) = ________

(ii) -7.3745219

(iii) -16.4048448

(iv) -3.1723640

(v) 1.3939075

Type: ES Var: 1

53) Consider the initial value problem

y' = 2 - 3t(y) with superscript (3), y(1) = 2

This question is related to using the Runge-Kutta method for approximating the solution y at t = 1.1 with a step size of h = 0.05.

To find (y) with subscript (2), first calculate the following in order to apply the Runge-Kutta method. Express your answers accurate to seven decimal places.

(i) (k) with subscript (1) = ________

(ii) (k) with subscript (2) = ________

(iii) (k) with subscript (3) = ________

(iv) (k) with subscript (4) = ________

(v) So, using the Runge-Kutta method, (y) with subscript (2) = ________

(ii) -4.0104693

(iii) -4.9819359

(iv) -2.9512518

(v) 1.1650133

Type: ES Var: 1

54) Consider the initial value problem

y' = 3y + 4t + 3, y(0) = 0

Use the backward Euler method with step size h = 0.01 to find the following approximation. Express your answer to 5 decimal places.

(y) with subscript (1) = ________

Type: SA Var: 1

55) Consider the initial value problem

y' = 3y + 4t + 3, y(0) = 0

Use the backward Euler method with step size h = 0.01 to find the following approximation. Express your answer to 5 decimal places.

(y) with subscript (2) = ________

Type: SA Var: 1

56) Consider the initial value problem

y' = 2y + 4t + 3, y(0) = 0

Use the backward Euler method with step size h = 0.01 to find the following approximation. Express your answer to 5 decimal places.

(y) with subscript (3) = ________

Type: SA Var: 1

57) Consider the following initial value problem on the interval [0, 1]

y' = 2y - cos(t), y(0) = 2

Approximate the solution y(t) at t = 2.5 using the fourth-order backward differentiation formula with step size h = 0.05. Using the Runge-Kutta method you are provided with some initial (y) with subscript (n) values to start the fourth-order backward differentiation formula.

(y) with subscript (1) = 2.157777

(y) with subscript (2) = 2.3322791

(y) with subscript (3) = 2.5253944

Use the fourth-order backward Euler differentiation formula to compute the following approximation. Express your answer accurate to 6 decimal places.

(y) with subscript (4) = ________

Type: SA Var: 1

58) Consider the following initial value problem on the interval [0, 1]

y' = 2y - cos(t), y(0) = 2

Approximate the solution y(t) at t = 2.5 using the fourth-order backward differentiation formula with step size h = 0.05. Using the Runge-Kutta method you are provided with some initial (y) with subscript (n) values to start the fourth-order backward differentiation formula.

(y) with subscript (1) = 2.157777

(y) with subscript (2) = 2.3322791

(y) with subscript (3) = 2.5253944

Use the fourth-order backward Euler differentiation formula to compute the following approximation. Express your answer accurate to 6 decimal places.

(y) with subscript (5) = ________

Type: SA Var: 1

59) Consider the initial value problem

y' - y = (e) with superscript (t)(y) with superscript (2), y(0) = 1

The following table provides the estimation for the solution at different t values using the Euler and Improved Euler methods with a step size of 0.25.

t

Euler

Improved Euler

0

1

1

0.25

1.5

1.7986321

0.5

2.5972643

5.1799111

0.75

6.0270589

94.9082602

1.0

26.7590621

8114447.0397517

Based on the table above, which of the following is true regarding the given initial value problem?

A) There is no numerical solution at y(0) = 1.

B) A numerical solution can never be close to the exact solution.

C) has a vertical asymptote for y(t) between [0.25, 1].

D) There is a solution which contains a horizontal asymptote.

E) There is a solution which contains a vertical asymptote.

Type: MC Var: 1

60) For the following differential equation, use the fourth-order Adams-Moulton formula to estimate the solution at t = 0.4 using a step size of h = 0.1. To start the process, you are given some (y) with subscript (i) values.

y' = y + 2t + 1, y(0) = 0

(y) with subscript (1)

(y) with subscript (2)

(y) with subscript (3)

0.1155125

0.264207713

0.449575491

The fourth-order Adams-Moulton formula is

(y) with subscript (n + 1) = (y) with subscript (n) + (h/24)(9(f) with subscript (n + 1) + 19(f) with subscript (n) - 5(f) with subscript (n - 1) + (f) with subscript (n - 2))

Calculate the following approximations. Express your answers accurate to 7 decimal places.

(i) (f) with subscript (0) = ________

(ii) (f) with subscript (1) = ________

(iii) (f) with subscript (2) = ________

(iv) (f) with subscript (3) = ________

(ii) 1.3155125

(iii) 1.6642077

(iv) 2.0495755

Type: ES Var: 1

61) For the following differential equation, use the fourth-order Adams-Moulton formula to estimate the solution at t = 0.4 using a step size of h = 0.1. To start the process, you are given some (y) with subscript (i) values.

y' = y + 2t + 1, y(0) = 0

(y) with subscript (1)

(y) with subscript (2)

(y) with subscript (3)

0.1155125

0.264207713

0.449575491

The fourth-order Adams-Moulton formula is

(y) with subscript (n + 1) = (y) with subscript (n) + (h/24)(9(f) with subscript (n + 1) + 19(f) with subscript (n) - 5(f) with subscript (n - 1) + (f) with subscript (n - 2))

Using the fourth-order Adams-Moulton formula, approximate (y) with subscript (4). Express your answer accurate to 7 decimal places.

(y) with subscript (4) = ________

Type: SA Var: 1

62) For the following differential equation, use the fourth-order Adams-Moulton formula to estimate the solution at t = 0.4 using a step size of h = 0.1. To start the process, you are given some (y) with subscript (i) values.

y' = y + 2t + 1, y(0) = 0

(y) with subscript (1)

(y) with subscript (2)

(y) with subscript (3)

0.1155125

0.264207713

0.449575491

The fourth-order Adams-Moulton formula is

(y) with subscript (n + 1) = (y) with subscript (n) + (h/24)(9(f) with subscript (n + 1) + 19(f) with subscript (n) - 5(f) with subscript (n - 1) + (f) with subscript (n - 2))

Calculate (f) with subscript (4) = ________

Type: SA Var: 1

63) For the following differential equation, use the fourth-order Adams-Moulton formula to estimate the solution at t = 0.4 using a step size of h = 0.1. To start the process, you are given some (y) with subscript (i) values.

y' = y + 2t + 1, y(0) = 0

(y) with subscript (1)

(y) with subscript (2)

(y) with subscript (3)

0.1155125

0.264207713

0.449575491

The fourth-order Adams-Moulton formula is

(y) with subscript (n + 1) = (y) with subscript (n) + (h/24)(9(f) with subscript (n + 1) + 19(f) with subscript (n) - 5(f) with subscript (n - 1) + (f) with subscript (n - 2))

Using the fourth-order Adams-Moulton formula, approximate (y) with subscript (5). Express your answer accurate to 7 decimal places.

(y) with subscript (5) = ________

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.

Document Information

Document Type:
DOCX
Chapter Number:
8
Created Date:
Aug 21, 2025
Chapter Name:
Chapter 8 Numerical Methods
Author:
William E. Boyce

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