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In-Depth Information
7.1 INTRODUCTION
The following nonhomogeneous equation,
px y
()
''
+
qx y
()
'
+
rx y
()
= Φ
()
x
(1)
can be established as follows:
2
d
d
^
^
Ly
,
L
=
p
+
q
+
r
(2)
2
dx
dx
^
xL is an exact operator:
It is known [1] that
(x)
d
(
)
'
ay+b =x
Φ
dx
(3)
if ξ satisfies the differential equation
(
)
(
)
''
'
'
''
'
p
î
+
2 p
q
î
+
p
q
r
î
=
0
(4)
and in this case,
αξ
=
p
,
βξ α
=
q
-
'
(5)
In several situations, the following property is true:
''
'
(6)
pqr
−+=
0
Therefore, it is evident that Eq. (4) is fulfilled for ξ = 1, which means
^
L is already exact, with
α =
p
β =
qp
'
that
and
. On the contrary, it is
straightforward to show the identity:
(
)
'
x
β
wp
q
w
'
q
=
-
,
w
=
exp
-
d
ξ
,
=
-
(7)
α
wp
p
w
p
such that Eq. (3) adopts the structure:
 
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