MATH 3705 * B
Test 2  Solutions.
February 2007
Questions 14 are multiple choice. Circle the correct answer. Only the answer will be marked.
1. [3 marks]
The general solution of 4
x
2
y
II
−
8
xy
I
+9
y
=0for
x
W
=0is
(
a
)
C
1

x

−
3
2
+
C
2

x

3
2
(
b
)

x

3
2
(
C
1
+
C
2
ln

x

)(
c
)

x

4+
√
7
(
C
1
+
C
2
ln

x

)
(
d
)

x

4
±
C
1
cos
p
√
7ln

x

Q
+
C
2
sin
p
√
7ln

x

Q=
(e) None of the above
Solution:
The indicial equation is 4
r
(
r
−
1)
−
8
r
+9=0, or 4
r
2
−
12
r
+ 9 = 0, with
r
1
=
r
2
=
12
8
=
3
2
,
⇒
Euler Equation, case (ii)
⇒
(
b
)
.
2. [3 marks]
The general solution of
x
2
y
II
+4
xy
I
−
10
y
=0for
x
W
=0is
(
a
)
C
1

x

2
+
C
2

x

−
5
(
b
)
C
1

x

−
2
+
C
2

x

5
(
c
)

x

−
2
(
C
1
+
C
2
ln

5
x

)
(
d
)

x

−
2
[
C
1
cos (5 ln

x

)+
C
2
sin (5 ln

x

)]
(e) None of the above
Solution:
The indicial equation is
r
(
r
−
1) + 4
r
−
10 = 0, or
r
2
+3
r
−
10 = 0, with
r
1
=2
,r
2
=
−
5
,
⇒
Euler equation, case (i)
⇒
(
a
)
.
3. [3 marks]
The equation 3
xy
II
+
x
2
y
I
+
1
5
y
=0has
(a)
one regular singular point
x
=5
.
(b)
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 Winter '08
 JaberAbdualrahman
 Math

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