Chapter 6
To find functions
v
j
’s we need four determinants, the Wronskian
W
[
y
1
, y
2
, y
3
](
x
) and
W
1
(
x
),
W
2
(
x
), and
W
3
(
x
) given in (10) on page 340 of the text. Thus we compute
W e
−
x
, e
2
x
, xe
2
x
(
x
) =
e
−
x
e
2
x
xe
2
x
−
e
−
x
2
e
2
x
(1 + 2
x
)
e
2
x
e
−
x
4
e
2
x
(4 + 4
x
)
e
2
x
=
e
−
x
e
2
x
e
2
x
1
1
x
−
1
2
1 + 2
x
1
4
4 + 4
x
= 9
e
3
x
,
W
1
(
x
) = (
−
1)
3
−
1
W e
2
x
, xe
2
x
(
x
) =
e
2
x
xe
2
x
2
e
2
x
(1 + 2
x
)
e
2
x
=
e
4
x
,
W
2
(
x
) = (
−
1)
3
−
2
W e
−
x
, xe
2
x
(
x
) =
−
e
−
x
xe
2
x
−
e
−
x
(1 + 2
x
)
e
2
x
=
−
(1 + 3
x
)
e
x
,
W
3
(
x
) = (
−
1)
3
−
3
W e
−
x
, e
2
x
(
x
) =
e
−
x
e
2
x
−
e
−
x
2
e
2
x
= 3
e
x
.
Substituting these expressions into the formula (11) for determining
v
j
’s, we obtain
v
1
(
x
) =
g
(
x
)
W
1
(
x
)
W
[
e
−
x
, e
2
x
, xe
2
x
]
dx
=
e
2
x
e
4
x
9
e
3
x
dx
=
1
27
e
3
x
,
v
2
(
x
) =
g
(
x
)
W
2
(
x
)
W
[
e
−
x
, e
2
x
, xe
2
x
]
dx
=
−
e
2
x
(1 + 3
x
)
e
x
9
e
3
x
dx
=
−
1
9
(1 + 3
x
)
dx
=
−
x
9
−
x
2
6
,
v
3
(
x
) =
g
(
x
)
W
3
(
x
)
W
[
e
−
x
, e
2
x
, xe
2
x
]
dx
=
e
2
x
3
e
x
9
e
3
x
dx
=
x
3
,
where we have chosen zero integration constants. Then formula (12), page 340 of the text,
gives a particular solution
y
p
(
x
) =
1
27
e
3
x
e
−
x
−
x
9
+
x
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- Spring '10
- Hald,OH
- Math, Differential Equations, Linear Algebra, Algebra, Determinant, Equations, 1 W, Heterogeneity, Homogeneity, 3 W, e e2x 2e
-
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