v
à a
d
2 u
dx 2 þ b
du
dx
þ cu
&
'
À u
d
2 av
Ã
ð Þ
dx 2 À
d bv
Ã
ð Þ
dx
þ cv
Ã
&
'
¼
d
dx
av
à du
dx
À u
d av
Ã
ð Þ
dx
!
þ
d
dx
buv
Ã
½
Š:
ð10:57Þ
Hence, following the expressions of Sect. 10.2, we define L x
{ such that
L x
{ v
Â
à Ã
d
2 av
Ã
ð Þ
dx 2 À
d bv
Ã
ð Þ
dx
þ cv
Ã
:
ð10:58Þ
Taking a complex conjugate of both sides, we get
L x
{ v ¼
d
2 a
à v
ð Þ
dx 2 À
d b
à v
ð Þ
dx
þ c
à v:
ð10:59Þ
Considering the differential of a product function, we have as L x
{
L x
{
¼ a
à d
2
dx 2 þ 2
da
Ã
dx
À b
Ã
d
dx
þ
d
2 a
Ã
dx 2 À
db
Ã
dx
þ c
Ã
:
ð10:60Þ
Replacing (10.57) with (10.55) and (10.59), we have
v
à L x u
ð ÞÀ L x
{ v
Â
à à u ¼
d
dx
av
à du
dx
À u
d av
Ã
ð Þ
dx
þ buv
Ã
!
:
ð10:61Þ
Assuming that the relevant SOLDE is defined in [r, s] and integrating (10.61) within
that interval, we get
Z s
r
dx v
à L x u
ð ÞÀ L x
{ v
Â
à à u
h
i
¼ av
à du
dx
À u
d av
Ã
ð Þ
dx
þ buv
Ã
! s
r
:
ð10:62Þ
Using the definition of an inner product described in (1.128) and rewriting (10.62),
we have
vjL x u
h
iÀ L x
{ vju
¼ av
à du
dx
À u
d av
Ã
ð Þ
dx
þ buv
Ã
! s
r
:
Here if RHS of the above (i.e., the surface term of the above expression) vanishes,
we get
vjL x u
h
i¼ L x
{ vju
:
We find that this notation is consistent with (1.112).
390
10 Introductory Green’s Functions