Midterm Exam, Math 112 B
9 February, 2007
Name:
ID:
Instructions: Put away any electronic devices, papers, notes, and books. Use only
a pencil and eraser (or a pen if you are daring). Write neatly and only on the paper
provided. Use the back of the page i
Homework, Section 30, Math 112 B
26 February, 2007
1. Using the Greens function for the corresponding Poisson problem, solve
2u 2u
+
= 0, 0 < x < , 0 < y < A,
x2 y 2
u(x, 0) = sin(3x), u(x, A) = 0, 0 x ,
u(0, y ) = 0, u(, y ) = 0, 0 y A.
You should be abl
Practice Exam I, Math 112 B
2 February, 2007
Name:
ID:
Instructions: The actual midterm exam will have four to ve problems, that look very
much like a subset of the following nine. Complete these problems with the aid of the
book. The use of group work an
Practice Final, Math 112 B
12 March, 2007
Name:
ID:
Instructions: The actual nal exam will have between six and eight problems that look
very much like a subset of the following 17. Complete these problems with the aid of the
book. The use of group work a
Quiz I, Math 112 B, 18 January, 2007
Name:
ID:
1 (10 points): Find the general solution to the following equation using separation of
variables.
Hint: Let u(x, t) = X (x)T (t). You can assume the ODE for X has only nonnegative
eigenvalues.
u
2u
k 2 = 0,
Quiz II, Math 112 B, 26 January, 2007
Name:
ID:
1 (10 points): Find the general solution to the following equation using separation of
variables.
Hint: Let u(x, y ) = X (x)Y (y ). You can assume the ODE for X has only positive
eigenvalues.
2u 2u
+
= 0, fo
Quiz III, Math 112 B, 1 February, 2007
Name:
ID:
1 (10 points): Solve by separation of variables
2u
u
2u
+ 2a
c2 2 = 0, for 0 < x < , t > 0,
t2
t
x
u(0, t) = u(, t) = u(x, 0) = 0,
u
(x, 0) = g (x).
t
(1)
(2)
(3)
where a > nc.
Hint: Let u(t, x) = T (t)X (
Quiz IV, Math 112 B, 22 February, 2007
Name:
ID:
1 (20 points): Solve
2
u = sin x, for 0 < x < , 0 < y < 1,
u(x, 0) = u(x, 1) = u(0, y ) = u(, y ) = 0.
(1)
(2)
Assume the ODE for X has eigenfunctions sin nx for n = 1, 2, .
Suppose u(x, y ) =
1
bn (y ) sin
Quiz V, Math 112 B, 1 March, 2007
Name:
ID:
1 (20 points): Using the Greens function to solve:
2
u = sin 2x sin y,
for 0 < x < , 0 < y < ,
u(x, 0) = u(x, ) = u(0, y ) = u(, y ) = 0.
G(x, y, , ) =
1
1
2 sinh n( y ) sinh n sin nx sin n
n sinh n
for y
Soluti
Quiz VI, Math 112 B, 8 March, 2007
Name:
ID:
1 (20 points): Solve the problem
2
u = 0, for 0 < x < , 0 < y < , 0 < z < ,
u(0, y, z ) = u(, y, z ) = u(x, y, ) = 0,
u
u
(x, 0, z ) =
(x, , z ) = 0,
y
y
u(x, y, 0) = sin 3x.
(1)
(2)
(3)
(4)
Solution: Letting u
Final Exam, Math 112 B
19 March, 2007
Name:
Solutions
ID:
Instructions: Put away any electronic devices, papers, notes, and books. Use only a
pencil and an eraser, or a pen. Write neatly and only on the paper provided. Use the
back of the page if necessar
Midterm Exam, Math 112 B
9 February, 2007
Name:
ID:
Instructions: Put away any electronic devices, papers, notes, and books. Use only
a pencil and eraser (or a pen if you are daring). Write neatly and only on the paper
provided. Use the back of the page i
Quiz VI, Math 112 B, 8 March, 2007
Name:
ID:
1 (20 points): Solve the problem
2
u = 0, for 0 < x < , 0 < y < , 0 < z < ,
u(0, y, z ) = u(, y, z ) = u(x, y, ) = 0,
u
u
(x, 0, z ) =
(x, , z ) = 0,
y
y
u(x, y, 0) = sin 3x.
(1)
(2)
(3)
(4)
Solution: Letting u
Final Exam, Math 112 B
19 March, 2007
Name:
Solutions
ID:
Instructions: Put away any electronic devices, papers, notes, and books. Use only a
pencil and an eraser, or a pen. Write neatly and only on the paper provided. Use the
back of the page if necessar
Homework, Section 30, Math 112 B
26 February, 2007
1. Using the Greens function for the corresponding Poisson problem, solve
2u 2u
+
= 0, 0 < x < , 0 < y < A,
x2 y 2
u(x, 0) = sin(3x), u(x, A) = 0, 0 x ,
u(0, y ) = 0, u(, y ) = 0, 0 y A.
You should be abl
Practice Exam I, Math 112 B
2 February, 2007
Name:
ID:
Instructions: The actual midterm exam will have four to ve problems, that look very
much like a subset of the following nine. Complete these problems with the aid of the
book. The use of group work an
Practice Final, Math 112 B
12 March, 2007
Name:
ID:
Instructions: The actual nal exam will have between six and eight problems that look
very much like a subset of the following 17. Complete these problems with the aid of the
book. The use of group work a
Quiz I, Math 112 B, 18 January, 2007
Name:
ID:
1 (10 points): Find the general solution to the following equation using separation of
variables.
Hint: Let u(x, t) = X (x)T (t). You can assume the ODE for X has only nonnegative
eigenvalues.
u
2u
k 2 = 0,
Quiz II, Math 112 B, 26 January, 2007
Name:
ID:
1 (10 points): Find the general solution to the following equation using separation of
variables.
Hint: Let u(x, y ) = X (x)Y (y ). You can assume the ODE for X has only positive
eigenvalues.
2u 2u
+
= 0, fo
Quiz III, Math 112 B, 1 February, 2007
Name:
ID:
1 (10 points): Solve by separation of variables
2u
u
2u
+ 2a
c2 2 = 0, for 0 < x < , t > 0,
t2
t
x
u(0, t) = u(, t) = u(x, 0) = 0,
u
(x, 0) = g (x).
t
(1)
(2)
(3)
where a > nc.
Hint: Let u(t, x) = T (t)X (
Quiz IV, Math 112 B, 22 February, 2007
Name:
ID:
1 (20 points): Solve
2
u = sin x, for 0 < x < , 0 < y < 1,
u(x, 0) = u(x, 1) = u(0, y ) = u(, y ) = 0.
(1)
(2)
Assume the ODE for X has eigenfunctions sin nx for n = 1, 2, .
Suppose u(x, y ) =
1
bn (y ) sin
Quiz V, Math 112 B, 1 March, 2007
Name:
ID:
1 (20 points): Using the Greens function to solve:
2
u = sin 2x sin y,
for 0 < x < , 0 < y < ,
u(x, 0) = u(x, ) = u(0, y ) = u(, y ) = 0.
G(x, y, , ) =
1
1
2 sinh n( y ) sinh n sin nx sin n
n sinh n
for y
Soluti
24.962 Advanced phonology
20 April 2005
Some other types of opacity
(1) A case of counterfeeding in the environment: Japanese rendaku
Second element becomes voiced in certain types of compounds
From ren sequential + daku(on) voiced; examples from It