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B
˙
ILKENT UNIVERSITY
Department of Mathematics
MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS
Homework set # 16
U.Mu˘gan
July 16, 2008
FUNDAMENTAL SET OF SOLUTIONS
1)
Find the Wronskian of the following given pair of functions:
a)
e
2
x
,
e

3
x/
2
.
b)
x,
xe
x
.
c)
e
x
sin
x,
e
x
cos
x
.
2)
Determine the largest interval in which the given I.V.P. is certain to have a unique twice
diﬀerentiable solution. Do not ﬁnd the solution.
a)
(
x

1)
y
00

3
xy
0
+ 4
y
= sin
x,
y
(

2) = 2
, y
0
(

2) = 1
.
b)
y
00
+ (cos
x
)
y
0
+ 3(ln

x

)
y
= 0
,
y
(2) = 3
, y
0
(2) = 1
.
c)
(
x

2)
y
00
+
y
0
+ (
x

2)(tan
x
)
y
= 0
,
y
(3) = 1
, y
0
(3) = 2
.
3)
Verify that
y
1
(
x
) = 1 and
y
2
(
x
) =
x
1
/
2
are two L.I. solutions of
yy
00
+ (
y
0
)
2
= 0
,
x >
0
.
Then show that
c
1
+
c
2
x
1
/
2
is not, in general a solution of the equation. Why not?
4)
If Wronskian
W
of
f
and
g
is
x
2
e
x
and
f
(
x
) =
x
, ﬁnd
g
(
x
).
5)
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