Test 1
Introduction to PDE
MATH 336325820 (Fall 2009)
This exam has 4 questions, for a total of 20 points.
Please answer the questions in the spaces provided on the question sheets.
If you run out of room for an answer, continue on the back of the page.
Upon ﬁnishing PLEASE write and sign your pledge below:
On my honor I have neither given nor received any aid on this exam.
1
Rules
You may only use pencils, pens, erasers, and straight edges. No calculators, notes, books
or other aides are permitted. Scrap paper will be provided. Be sure to show a few key
intermediate steps when deriving results  answers only will not get full marks.
2
Given
You may assume the eigenvalues of the SturmLiouville problem
X
00
+
λX
= 0
,
0
< x <
1
X
0
(0) = 0
,
X
(1) = 0
.
are
λ
n
= (
n

1
2
)
2
π
2
and
X
n
(
x
) = cos
(
(
n

1
2
)
πx
)
, for
n
= 1
,
2
, . . .
, without derivation.
You may also assume the following orthogonality conditions for
m
,
n
positive integers:
Z
1
0
cos
(
(
m

1
2
)
πx
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 Spring '08
 Staff
 Math, Following, Partial differential equation, Sturm–Liouville theory

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