MATH 220: PRACTICE MIDTERM
This is a closed book, closed notes, no calculators exam.
There are 5 problems. Solve all of them. Total score: 100 points.
Problem 1.
(i)
(13 points)
Find the general
C
1
solution of the PDE
3
y
2
u
x
+
u
y
= 0
.
(ii)
(6 points)
Solve the initial value problem with initial condition
u
(
x,
0) =
x
2
.
Problem 2.
(i)
(10 points)
Find the general
C
2
solution of the PDE
u
xx
+ 3
u
xt
+ 2
u
tt
= 0
.
(ii)
(10 points)
Solve the initial value problem with initial condition
u
(
x,

x
) =
φ
(
x
)
, u
t
(
x,

x
) =
ψ
(
x
)
,
with
φ,ψ
given.
Problem 3.
(20 points)
Find the
bounded
solution of Laplace’s equation on the
infinite strip
R
x
×
(0
,
1)
y
:
u
xx
+
u
yy
= 0
,
x
∈
R
, y
∈
(0
,
1)
,
u
(
x,
0) =
φ
(
x
)
,
x
∈
R
,
u
(
x,
1) = 0
,
x
∈
R
,
where
φ
(
x
) =
xe

x
for
x
≥
0,
φ
(
x
) = 0 for
x<
0. You may leave your solution as
the inverse Fourier transform of a function that you have evaluated explicitly.
Problem 4.
(17 points)
Consider the damped wave equation on
R
x
×
[0
,
∞
)
t
:
u
tt
+
a
(
x
)
u
t
= (
c
(
x
)
2
u
x
)
x
,
where
a
≥
0,
c>
0 depending on
x
only, and there are constants
c
1
,c
2
>
0 such
that
c
1
≤
c
(
x
)
≤
c
2
for all
x
. Suppose that
u
(
x,
0) and
u
t
(
x,
0) vanish for

x
 ≥
R
.
Let
E
(
t
) =
1
2
integraldisplay
R
(
u
t
(
x,t
)
2
+
c
(
x
)
2
u
x
(
x,t
)
2
)
dx.
Show that
E
is a decreasing (i.e. nonincreasing) function of
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 '09
 Math, Differential Equations, Applied Mathematics, Equations, Partial Differential Equations, Constant of integration, Boundary value problem, Partial differential equation, damped wave equation, PDE uxx

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