380_pdfsam_math 54 differential equation solutions odd -...

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