Midterm 2 of Math 255.01 (Wi11)
February 25, 2011
The Ohio State University
1:30pm2:18pm
Including the cover sheer, this exam consists of ﬁve (5) pages and four (4) problems and is
worth a total of 100 points. The point value of each problem is indicated.
To obtain full
credit, you must have the correct answers along with relevant supporting work to jus
tify them.
Partial credit will be given based on the work that is shown. However,
answers
without sufﬁcient supporting work will receive no credit.
Also, you must complete the
exam in 48 minutes. Good luck!
Name:
Signature:
OSU Internet Username:
Problem #
Score
1
(30 pts)
2
(30 pts)
3
(15 pts)
4
(25 pts)
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View Full Document1. Suppose that
y
1
(
x
) = 1 +
x
2
and
y
2
(
x
)
are solutions to the equation
(1

x
2
)
d
2
y
dx
2
+ 2
x
dy
dx

2
y
= 0
,

1
< x <
1
,
such that
y
2
(0) = 0
and
y
0
2
(0) = 2
.
(a)
(12)
Use Abel’s theorem to calculate the Wronskian
W
(
x
) =
W
(
y
1
,y
2
)(
x
)
.
Solution:
By Abel’s theorem, there is a constant
c
1
such that
W
(
x
) =
c
1
exp
Z

2
x
1

x
2
dx
=
c
1
exp
Z
2
x
x
2

1
dx
=
c
1
exp ln
±
±
x
2

1
±
±
=
c
1

x
2

1

=
c
1
(1

x
2
)
.
On the other hand, by the deﬁnition of the Wronskian,
W
(2) =
y
1
(0)
y
0
2
(0)

y
2
(0)
y
0
1
(0) = 2
.
Hence
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 Spring '08
 Staff
 Math, characteristic equation, fundamental set, Abel

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