Deck 10: Partial Differential Equations and Fourier Series

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

Answer 1

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

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.

Answer 2

This boundary value problem has no solution.

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.

Answer 3

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

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.

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

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)

Question 7

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

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.

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.

Question 10

Consider the following periodic function with period 4:

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

A)
B)
C)
D)

Question 11

Consider the following periodic function with period 8:

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

A)
B)
C)
D)

Question 12

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

Compute F .

Question 13

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

Compute F(-8).

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?

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?

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?

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?

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?

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.

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 ?

Question 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) $a_{n}=\frac{1}{(2 n-1) \pi}, n=1,2,3, \ldots$
B) $a_{n}=-\frac{2}{\pi} \cdot \frac{1}{2 n-1}, n=1,2,3, \ldots$
C) $a_{2 n}=0, a_{2 n-1}=-\frac{2}{\left(4 n^{2}-1\right) \pi}, n=1,2,3, \ldots$
D) $a_{2 n}=-\frac{2}{\left(4 n^{2}-1\right) \pi}, a_{2 n-1}=0, n=1,2,3, \ldots$
E) $a_{n}=0, n=1,2,3, \ldots$

Question 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) $b_{n}=\frac{1}{(2 n-1) \pi}, n=1,2,3, \ldots$
B) $b_{n}=-\frac{2}{\pi} \cdot \frac{1}{2 n-1}, n=1,2,3, \ldots$
C) $b_{2 n}=0, b_{2 n-1}=-\frac{2}{\left(4 n^{2}-1\right) \pi}, n=1,2,3, \ldots$
D) $b_{2 n}=-\frac{2}{\left(4 n^{2}-1\right) \pi}, b_{2 n-1}=0, n=1,2,3, \ldots$
E) $b_{n}=0, n=1,2,3, \ldots$

Question 23

Consider the following periodic function with period 2?:
F (t) =
F (t + 2 $\pi$ ) = f (t)
Which of these is the Fourier representation for f (t)?

A) $\frac{8}{9}+\frac{8}{\pi} \sum_{n=1}^{\infty} \frac{1}{n} \sin \frac{n \pi}{9} \sin (n t)$
B) $\frac{8}{9}+\frac{8}{\pi} \sum_{n=1}^{\infty} \frac{1}{n} \sin \frac{n \pi}{9} \cos (n t)$
C) $\frac{8}{\pi} \sum_{n=1}^{\infty} \frac{1}{n} \sin \frac{n \pi}{9} \sin (n t)$
D) $\frac{8}{\pi} \sum_{n=1}^{\infty} \frac{1}{n} \sin \frac{n \pi}{9} \cos (n t)$

Question 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)?

Question 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?

Question 26

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

A) $y=-6 t^{2}$
B) $y=-6 t^{12}-7 t^{8}$
C) $y=\sin (2 t)$
D) $y=\cos \left(3 t^{3}\right)$
E) $y=3 \sin ^{2}(t)+\cos (2 t)$
F) $y=6 t^{\frac{7}{9}}$

Question 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) $g(x)=x^{5}+\cos (7 x)$ is an odd function.
C) $h(x)=x^{5} \cdot \cos (7 x)$ is an odd function.
D) If $f(x)$ is an even function, then $\int_{-4}^{4} f(x) d x=0$ .
E) If $j(x)$ is an odd function with the Fourier series representation
$f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty} a_{n} \cos \frac{n \pi t}{5}+b_{n} \sin \frac{m \pi t}{5}$ , then $a_{n}=0$ , for all $n$ .

Question 28

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

A)

B)
C)
D)

Question 29

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

A)
B)
C)
D)

Question 30

Consider the following function:
F (x) =
What is the Fourier cosine series for f (x)?

A)
B)
C)
D)

Question 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
E) The Fourier series of the function

Question 32

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

A) $-\frac{8}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^{n}-1}{n} \cdot \sin \frac{2 n x}{3}$
B) $\frac{8}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^{n}+1}{n} \cdot \sin \frac{2 n x}{3}$
C) $-\frac{8}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^{n}-1}{n} \cdot \sin \frac{3 n \pi x}{2}$
D) $\frac{8}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^{n}-1}{n} \cdot \sin \frac{3 n \pi x}{2}$
E) $\frac{8}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^{n}+1}{n} \cdot \sin \frac{3 n \pi x}{2}$

Question 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) $3 u_{x x}+6 u_{y y}=0$
B) $2 u_{y}-3 u_{x x}+7 u_{x}=0$
C) $(3 y+8 x) u_{x}+u_{y}=0$
D) $f(x) u_{x x}+g(y) u_{y}+7=0$ , where $f(x)$ and $g(y)$ are continuous functions
E) $2 u_{y y}+3 x u_{y}-u=0$
F) $u_{x x}+u_{y y}+4 y\left(u_{x}-u_{y}\right)=0$

Question 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(4?x)

A) $u(x, t)=e^{-16 \pi^{2} t} \sin \left(\frac{\pi x}{4}\right)$
B) $u(x, t)=e^{-64 \pi^{2} t} \sin (4 \pi t)$
C) $u(x, t)=\sum_{n=1}^{\infty} e^{-n^{2} \pi^{2} t} \sin \frac{n \pi x}{4}$
D) $u(x, t)=\sum_{n=1}^{\infty} e^{-n^{2} \pi^{2} /} \sin (4 n \pi x)$

Question 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.

Question 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:

Question 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:

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

A)
B)
C)
D)

Question 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:

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)

Question 39

What is the steady state solution for the heat conduction equation

Equipped with the following boundary conditions:
U(0, t) - (0, t) = 5
U(40, t) + (40, t) = 4

A) $v(x)=\frac{1}{40} x+201$
B) $v(x)=\frac{1}{40} x+\frac{201}{40}$
C) $v(x)=-\frac{1}{40} x+199$
D) $v(x)=-\frac{1}{40} x+\frac{199}{40}$

Question 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.

Question 41

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

What is the speed of wave propagation along the string?

Question 42

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

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

Question 43

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

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

Question 44

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

A)
B)
C)
D)