Ch10 Verified Test Bank + Partial Differential Equations And

Elementary Differential Equations, 12e (Boyce)

Chapter 10 Partial Differential Equations and Fourier Series

1) Which of the following represents all solutions of the boundary value problem

+ 36y = 0, y(0) = 1, y(π) = 1?

A) y = 0

B) y = 1

C) y = cos(6x) + sin(6x), where and are arbitrary real constants

D) y = cos(6x) + sin(6x), where is an arbitrary real constant

E) y = cos(6x) + sin(6x), where is an arbitrary real constant

Type: MC Var: 1

2) Consider the boundary value problem

+ 9y = 0, y(0) = 3, y(π) = 0

Which of these statements is true?

A) This boundary value problem has no solution.

B) y = 3 cos(3x) - 3 sin(3x) is the unique solution of this boundary value problem.

C) There are infinitely many solutions of this boundary value problem of the form , where C is an arbitrary real constant.

D) y = 0 is a solution of this boundary value problem.

Type: MC Var: 1

3) Consider the boundary value problem

+ 25y = 0, y(0) = -2, y = 0

Which of these statements is true?

A) This boundary value problem has no solution.

B) y = -2 cos(5x) + C sin(5x) is a solution of this boundary value problem, for any real constant C.

C) y = -2 cos(5x) is the unique solution of this boundary value problem.

D) y = 0 is the unique solution of this boundary value problem.

Type: MC Var: 1

4) Consider the boundary value problem

+ 49y = 0, y(0) = -5, y = 0

Which of these statements is true?

A) This boundary value problem has no solution.

B) y = -5 cos(7x) + C sin(7x) is a solution of this boundary value problem, for any real constant C.

C) y = -5 cos(7x) - 5 sin(7x) is the unique solution of this boundary value problem.

D) y = -5 cos(7x) + 5 sin(7x) is the unique solution of this boundary value problem.

Type: MC Var: 1

5) Consider the boundary value problem

+ 25y = 0, y(0) = 0, y = 0

Which of the following are eigenvalues for this boundary value problem? Select all that apply.

A)

B) 25

C)

D)

E) 225

Type: MC Var: 1

6) Assume that λ > 0. Consider the boundary value problem

+ λy = 0, y(0) = 0, (6) = 0

Which of the following are eigenvectors for this boundary value problem? Select all that apply.

A) y = 3 sin(x)

B) y = -0.2 cos(x)

C) y = 0.2 sin(x)

D) y = 3 sin(x)

E) y = -3 cos(x)

F) y = sin(x)

Type: MC Var: 1

7) What is the fundamental period of the periodic function f (x) = cos?

Type: SA Var: 1

8) Consider the following periodic function with period 6:

H(t) =

H(t + 12) = H(t)

The Fourier series representation for H(t) has the form

H(t) ~ + ,

where 12 is the period of H(t). What is the coefficient ? Express your answer as a simplified fraction.

Type: SA Var: 1

9) Consider the following periodic function with period 9:

f (t) =

f (t + 18) = f (t)

The Fourier series representation for f (t) has the form

f (t) ~ + ,

where 18 is the period of f (t). What is the coefficient ? Express your answer in exact form involving π. Do not approximate.

Type: SA Var: 1

10) Consider the following periodic function with period 4:

f (x) =

f (x + 4) = f (x)

Which of these is the Fourier representation for f (t)?

A) (1 + ) sin

B) (1 - ) cos

C) (1 + ) sin

D) (1 + ) cos

Type: MC Var: 1

11) Consider the following periodic function with period 8:

f (x) =

f (x + 8) = f (x)

Which of these is the Fourier representation for f (x)?

A)

B) - +

C)

D) - +

Type: MC Var: 1

12) Suppose f (x) is defined by f (x) = 4 on the interval [0, 9]. Consider the function , where = .

Compute F.

Type: SA Var: 1

13) Suppose f (x) is defined by f (x) = 8 on the interval [0, 9]. Consider the function , where = .

Compute F(-8).

Type: SA Var: 1

14) Consider the following periodic function with period 5:

f (t) =

f (t + 5) = f (t)

To what value does the Fourier series for f (t) converge for t = 4?

Type: SA Var: 1

15) Consider the following periodic function with period 4:

f (t) =

f (t + 4) = f (t)

To what value does the Fourier series for f (t) converge for t = 2?

Type: SA Var: 1

16) Consider the following periodic function with period 6:

f (t) =

f (t + 6) = f (t)

To what value does the Fourier series for f (t) converge for t = 2?

Type: SA Var: 1

17) Consider the following periodic function with period 7:

f (t) =

f (t + 7) = f (t)

To what value does the Fourier series for f (t) converge for t = 0?

Type: SA Var: 1

18) Consider the following periodic function with period 5:

f (t) =

f (t + 5) = f (t)

To what value does the Fourier series for f (t) converge for t = 5?

Type: SA Var: 1

19) Consider the following periodic function with period :

f (t) =

f = f (t)

The Fourier representation has the form

Compute f , where m is an odd integer.

Type: SA Var: 1

20) Consider the following periodic function with period :

f (t) =

f = f (t)

The Fourier representation has the form

What is the value of ?

Type: SA Var: 1

21) Consider the following periodic function with period :

f (t) =

f = f (t)

The Fourier representation has the form

Which of these are the coefficients ?

A) = , n = 1, 2, 3, ...

B) = - ∙ , n = 1, 2, 3, ...

C) = 0, = - , n = 1, 2, 3, ...

D) = - , = 0, n = 1, 2, 3, ...

E) = 0, n = 1, 2, 3, ...

Type: MC Var: 1

22) Consider the following periodic function with period :

f (t) =

f = f (t)

The Fourier representation has the form

Which of these are the coefficients ?

A) = , n = 1, 2, 3, ...

B) = - ∙ , n = 1, 2, 3, ...

C) = 0, = - , n = 1, 2, 3, ...

D) = - , = 0, n = 1, 2, 3, ...

E) = 0, n = 1, 2, 3, ...

Type: MC Var: 1

23) Consider the following periodic function with period 2π:

f (t) =

f (t + 2π) = f (t)

Which of these is the Fourier representation for f (t)?

A) +

B) +

C)

D)

Type: MC Var: 1

24) Consider the following periodic function with period 4π:

f (t) = 5t, -2π < t ≤ 2π

f (t + 4π) = f (t)

What is the Fourier series representation for f (t)?

Type: SA Var: 1

25) Consider the following periodic function with period 4π:

f (t) = 4t, -2π < t ≤ 2π

f (t + 4π) = f (t)

To what value does the Fourier series converge when t = -8π?

Type: SA Var: 1

26) Which of the following functions is even? Select all that apply.

A) y = -6

B) y = -6 - 7

C) y = sin(2t)

D) y = cos(3)

E) y = 3(t) + cos(2t)

F) y = 6

Type: MC Var: 1

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

A) If f (x) is an even function, then its graph is symmetric about the y-axis.

B) g(x) = + cos(7x) is an odd function.

C) h(x) = ∙ cos(7x) is an odd function.

D) If f (x) is an even function, then = 0.

E) If j(x) is an odd function with the Fourier series representation , then = 0, for all n.

Type: MC Var: 1

28) Consider the function f (x) = 7 + 3. Which of the following is the even periodic extension of f (x)?

A) g(x) =

g(x + 2) = g(x)

B) g(x) = 7 + 3, -1 < x < 1

C) g(x) =

g(x + 2) = g(x)

D) g(x) =

g(x + 2) = g(x)

Type: MC Var: 1

29) Consider the function f (x) = 5 + 3. Which of the following is the odd periodic extension of f (x)?

A) g(x) =

g(x + 6) = g(x)

B) g(x) = 5 + 3, -3 < x < 3

C) g(x) =

g(x + 6) = g(x)

D) g(x) =

g(x + 6) = g(x)

Type: MC Var: 1

30) Consider the following function:

f (x) =

What is the Fourier cosine series for f (x)?

A) + , where = 1

B) + , where = 1

C) , where = 1

D) + , where = 1

Type: MC Var: 1

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

A) The function f (x) defined by

f (x) = 2, 0 < x 4

f (x + 4) = f (x)

is even.

B) The odd periodic extension of f (x) = 5x, 0 < x < 1, is given by

g(x) =

C) The function f (x) defined by

f (x) =

f (x + 14) = f (x)

is even.

D) The even periodic extension of f (x) = 3 - x, 0 < x < 4, is given by

g(x) =

E) The Fourier series of the function

f (x) =

f (x + 6π) = f (x)

contains only sine terms.

Type: MC Var: 1

32) Find the Fourier series for f (x) = 4, 0 < x <

A) -

B)

C) -

D)

E)

Type: MC Var: 1

33) For which of these partial differential equations can the method of separation of variables be used to reduce it to a pair of ordinary differential equations? Select all that apply.

A) 3 + 6 = 0

B) 2 - 3 + 7 = 0

C) (3y + 8x) + = 0

D) f (x) + g(y) + 7 = 0, where f (x) and g(y) are continuous functions

E) 2 + 3x - u = 0

F) + + 4y( - ) = 0

Type: MC Var: 1

34) What is the solution of the following initial boundary value problem?

4 = , u(0, t) = 0, u(2, t) = 0, u(x, 0) = sin(4πx)

A) u(x, t) = sin

B) u(x, t) = sin(4πt)

C) u(x, t) =

D) u(x, t) =

Type: MC Var: 1

35) Consider the conduction of heat in a rod 30 cm in length whose ends are maintained at 0°C for all time t > 0. Find the expression for the temperature u(x, t) of position x in the rod at time t if the initial temperature distribution is given by

u(x, 0) =

Assume = 1 in the heat conduction partial differential equation.

Type: SA Var: 1

36) The ends of a rod 75 cm in length are connected to reservoirs that maintain the temperature at 11°C at x = 0 and 20°C at x = 75. The initial boundary value problem governing how heat conducts through the rod is as follows:

= 36, 0 < x < 75, t > 0

u(0, t) = 11, t > 0

u(75, t) = 20, t > 0

u(x, 0) = 5x + 4, 0 < x < 75

What is the steady-state temperature v(x) = u(x, t)?

Type: SA Var: 1

37) The ends of a rod 45 cm in length are connected to reservoirs that maintain the temperature at 13°C at x = 0 and 18°C at x = 45. The initial boundary value problem governing how heat conducts through the rod is as follows:

= 36, 0 < x < 45, t > 0

u(0, t) = 13, t > 0

u(45, t) = 18, t > 0

u(x, 0) = 7x + 5, 0 < x < 45

What is the transient temperature w(x, t) portion of the solution of the initial value boundary value problem?

A) w(x, t) = , where =

B) w(x, t) = , where =

C) w(x, t) = , where =

D) w(x, t) = , where =

Type: MC Var: 1

38) The ends of a rod 40 cm in length are connected to reservoirs that maintain the temperature at 12°C at x = 0 and 14°C at x = 40. The initial boundary value problem governing how heat conducts through the rod is as follows:

= 49, 0 < x < 40, t > 0

u(0, t) = 12, t > 0

u(40, t) = 14, t > 0

u(x, 0) = 5x + 8, 0 < x < 40

What is the solution u(x, t) of the initial boundary value problem?

A) u(x, t) = v(x) ∙ w(x, t)

B) u(x, t) = v(x) + w(x, t) + C, where C is an arbitrary real constant.

C) u(x, t) = v(x) + w(x, t)

D) u(x, t) = w(x, t) - v(x)

Type: MC Var: 1

39) What is the steady state solution for the heat conduction equation

= , 0 < x < 40, t > 0

equipped with the following boundary conditions:

u(0, t) - (0, t) = 5

u(40, t) + (40, t) = 4

A) v(x) = x + 201

B) v(x) = x +

C) v(x) = - x + 199

D) v(x) = - x +

Type: MC Var: 1

40) Suppose that both ends of a string of length 25 cm are attached to fixed points at height 0. Initially, the string is at rest and has shape 8 sin, where x is the horizontal coordinate along the string with zero at the left end. The speed of wave propagation along the string is 2 cm per sec. Formulate an initial boundary value problem that describes the shape of the string, u(x, t), over time.

u(0, t) = u(25, t) = 0, t > 0,

u(x, 0) = 8 sin, 0 < x < 25,

(x, 0) = 0

Type: ES Var: 1

41) Suppose the following initial boundary value problem governs the shape of a string 60 cm long, where t is measured in minutes:

= 70, 0 < x < 60, t > 0,

u(0, t) = u(60, t) = 0, t > 0,

u(x, 0) = (60 - x), 0 < x < 60,

(x, 0) =

What is the speed of wave propagation along the string?

Type: SA Var: 1

42) Suppose the following initial boundary value problem governs the shape of a string 140 cm long, where t is measured in minutes:

= 60, 0 < x < 140, t > 0,

u(0, t) = u(140, t) = 0, t > 0,

u(x, 0) = (140 - x), 0 < x < 140,

(x, 0) =

What is the initial displacement of the string at x = 110?

Type: SA Var: 1

43) Suppose the following initial boundary value problem governs the shape of a string 120 cm long, where t is measured in minutes:

= 20, 0 < x < 120, t > 0,

u(0, t) = u(120, t) = 0, t > 0,

u(x, 0) = (120 - x), 0 < x < 20,

(x, 0) =

What is the initial velocity of the string at the point x = 90?

Type: SA Var: 1

44) Determine the function u(x, y) satisfying Laplace's equation + = 0 in the rectangle , and satisfying the boundary conditions

u(x, 0) = 0, u(x, 4) = 0, 0 < x < 5

u(0, y) = 0, u(5, 0) =

A) u(x, y) = , where =

B) u(x, y) = , where =

C) u(x, y) = , where =

D) u(x, y) = , where =

Type: MC 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.