MATH 353.06, Quiz 5 (Take-home)
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October 21, 2012
1. (6pts) Find the Laplace transform of the following functions:
(a) f (t) =
0,
t<
sin(t 4 ), t
4
4
(b) f (t) =
t
(t
0
)3 cos 2 d
Solution. (a) Observe that
f (t) = u (t) sin(t
4
th
MATH 353.06, Quiz 1
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SOLUTIONS
September 13, 2012
1. (5pts) Give an example of a scalar eld by specifying its three components as a function.
Solution. f (x, y ) = x2 + y 2 with domain = R2 and target set = R1
(answer not unique).
2.
EXAM 3
Math 353 ODE & PDE, Section 6
Instructor: Guangliang Chen
Duke University, Fall 2012
This is a take-home exam. Due date: next Monday in class.
You must show all necessary steps to receive full credit.
You may use textbook, lecture notes, and any ot
EXAM 2
Math 353 ODE & PDE, Section 6
Instructor: Guangliang Chen
Duke University, Fall 2012
You have 50 minutes.
No notes, no books, no calculators.
You must show ALL work and explain your reasoning CLEARLY
to receive full credit.
Please write your initia
MATH 353.06, Quiz 3
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SOLUTION
September 24, 2012
1. (5pts) Interpret the following second-order nonhomogeneous ODE for
y = y (t) in the setting of a mass-spring system:
Ay + By + Cy = f (t),
where A, B, C are constants, f (t) is a fun
MATH 353.06, Quiz 4 (Take-home)
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SOLUTIONS
October 6, 2012
1. (6pts) Solve the following ODEs:
(a). x2 y 3xy + 5y = 0,
x>0
Solution. This is an Euler equation with = 3, = 5. The characteristics equation is r2 4r + 5 = 0, which has two
MATH 353.06, Quiz 8
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November 30, 2012
1. (4pts) Write down the nite wave propagation problem, including the
PDE and boundary/initial conditions.
Answer.
utt = a2 uxx ,
0 < x < L,
u(0, t) = u(L, t) = 0,
t > 0;
u(x, 0) = f (x), ut (x,
MATH 353.06, Quiz 7
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November 22, 2012
1. (9pts) Consider
ut = 2 uxx + f (x, t),
ux (0, t) = ux (L, t) = 0,
u(x, 0) = h(x),
0 < x < L, t > 0;
t > 0;
0 x L.
(a) Interpret the above system.
Answer. The above system models how heat propo
MATH 353.06, Quiz 6 (Take-home)
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November 1, 2012
1. (8pts) Find the eigenvalues and eigenvectors of the following BVP
y + y = 0,
y (0) = 0,
y (L) = 0.
Solution. We consider the following three cases separately:
(1) = 0. In this case,
EXAM 1
Math 353 ODE & PDE, Section 6
Instructor: Guangliang Chen
Duke University, Fall 2012
You have 50 minutes.
No notes, no books, no calculators.
You must show ALL work and explain your reasoning CLEARLY
to receive full credit.
Please write your initia