Question 1

Which of the following represents all solutions of the boundary value problem  + 36y = 0, y (0) = 1, y ( $\pi$) = 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

Accepted Answer

D

Question 2

Consider the boundary value problem  + 9y = 0, y (0) = 3, y ( $\pi$) = 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.

Accepted Answer

A

Question 3

Consider the boundary value problem  + 25y = 0, y (0) = -2, y  = 0Which 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.

Accepted Answer

C

Question 4

Consider the boundary value problem  + 49y = 0, y (0) = -5, y  = 0Which 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.

Accepted Answer

The answer of Consider the boundary value problem  +...

Question 5

Consider the boundary value problem  + 25y = 0, y (0) = 0, y  = 0Which of the following are eigenvalues for this boundary value problem? Select all that apply.
A) 
B) 25
C) 
D) 
E) 225

Accepted Answer

The answer of Consider the boundary value problem  +...

Question 6

Assume that$\lambda$ > 0. Consider the boundary value problem  + ?y = 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)

Accepted Answer

The answer of Assume that$\lambda$ > 0. Consider the boundary...

Question 7

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

Accepted Answer

The answer of What is the fundamental period of the...

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

Accepted Answer

The answer of Consider the following periodic function with period...

Question 9

Consider the following periodic function with period 9:
  
where 18 is the period of f (t). What is the coefficient  ? Express your answer in exact form involving π. Do not approximate.

Accepted Answer

The answer of Consider the following periodic function with period...

Question 10

Consider the following periodic function with period 4:

Which of these is the Fourier representation for f (t)?
A)
B)
C)
D)

Accepted Answer

The answer of Consider the following periodic function with period...

Question 11

Consider the following periodic function with period 8:

Which of these is the Fourier representation for f (x)?
A)
B)
C)
D)

Accepted Answer

The answer of Consider the following periodic function with period...

Question 12

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

Accepted Answer

The answer of Suppose f (x) is defined by f...

Question 13

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

Accepted Answer

The answer of Suppose f (x) is defined by f...

Question 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?

Accepted Answer

The answer of Consider the following periodic function with period...

Question 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?

Accepted Answer

The answer of Consider the following periodic function with period...

Question 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?

Accepted Answer

The answer of Consider the following periodic function with period...

Question 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?

Accepted Answer

The answer of Consider the following periodic function with period...

Question 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?

Accepted Answer

The answer of Consider the following periodic function with period...

Question 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.

Accepted Answer

The answer of Consider the following periodic function with period...

Question 20

Consider the following periodic function with period  :
f (t) =  f  = f (t)
The Fourier representation has the form  
What is the value of  ?

Accepted Answer

The answer of Consider the following periodic function with period...

Quiz 10: Partial Differential Equations and Fourier Series

Consider the boundary value problem + 25y = 0, y (0) = -2, y = 0Which of these statements is true?

Consider the boundary value problem + 49y = 0, y (0) = -5, y = 0Which of these statements is true?

Consider the boundary value problem + 25y = 0, y (0) = 0, y = 0Which of the following are eigenvalues for this boundary value problem? Select all that apply.

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

Consider the following periodic function with period 9: where 18 is the period of f (t). What is the coefficient ? Express your answer in exact form involving π. Do not approximate.

Consider the following periodic function with period 4: Which of these is the Fourier representation for f (t) ?

Consider the following periodic function with period 8: Which of these is the Fourier representation for f (x) ?

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

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

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?

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?

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?

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?

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?

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.

Consider the following periodic function with period : f (t) = f = f (t) The Fourier representation has the form What is the value of ?

Quiz 10: Partial Differential Equations and Fourier Series

Which of the following represents all solutions of the boundary value problem + 36y = 0, y (0) = 1, y ( π\piπ ) = 1?

Consider the boundary value problem + 9y = 0, y (0) = 3, y ( π\piπ ) = 0 Which of these statements is true?

Consider the boundary value problem + 25y = 0, y (0) = -2, y = 0Which of these statements is true?

Consider the boundary value problem + 49y = 0, y (0) = -5, y = 0Which of these statements is true?

Consider the boundary value problem + 25y = 0, y (0) = 0, y = 0Which of the following are eigenvalues for this boundary value problem? Select all that apply.

Assume that λ\lambdaλ > 0. Consider the boundary value problem + ?y = 0, y (0) = 0, (6) = 0 Which of the following are eigenvectors for this boundary value problem? Select all that apply.

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

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

Consider the following periodic function with period 9: where 18 is the period of f (t). What is the coefficient ? Express your answer in exact form involving π. Do not approximate.

Consider the following periodic function with period 4: Which of these is the Fourier representation for f (t) ?

Consider the following periodic function with period 8: Which of these is the Fourier representation for f (x) ?

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

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

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?

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?

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?

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?

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?

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.

Consider the following periodic function with period : f (t) = f = f (t) The Fourier representation has the form What is the value of ?