Environmental Engineering Reference
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in which the self-conjugate relation (D.8) has been used. Since
ρ
(
x
)
>
0,
b
a
ρ
(
)
u
2
)
d
x
v
2
x
(
x
)+
(
x
>
0
¯
so that
λ
=
λ
. Thus
λ
must be real-valued.
Property 2.
Let
y
k
(
x
)
and
y
i
(
x
)
be the eigenfunctions corresponding to
λ
k
and
λ
i
respectively. If
λ
k
=
λ
i
,then
b
a
ρ
(
x
)
y
k
(
x
)
y
i
(
x
)
d
x
=
0
.
(D.12)
Proof.
By the definition, we have
Ly
k
(
x
)=
λ
k
ρ
(
x
)
y
k
(
x
)
,
Ly
i
(
x
)=
λ
i
ρ
(
x
)
y
i
(
x
)
.
By the self-conjugate relation (D.8)
b
a
ρ
(
b
a
[
(
λ
k
−
λ
i
)
x
)
y
k
(
x
)
y
i
(
x
)
d
x
=
y
i
(
x
)
Ly
k
(
x
)
−
y
k
(
x
)
Ly
i
(
x
)]
d
x
=
0
.
Since
b
a
ρ
(
λ
k
=
λ
,
we arrive at
x
)
y
k
(
x
)
y
i
(
x
)
d
x
=
0
.
i
Note that properties 1 and 2 are valid for both regular and periodic S-L problems.
Property 3.
Every eigenvalue of a regular S-L problem has a unique corresponding
eigenfunction up to a constant factor.
Proof.
Suppose that an eigenvalue has two linearly independent corresponding
eigenfunctions
y
1
(
x
)
and
y
2
(
x
)
. The Wronski determinant is thus
=
y
1
(
x
)
y
2
(
x
)
y
2
(
y
1
(
W
(
y
1
,
y
2
)=
y
1
(
x
)
x
)
−
y
2
(
x
)
x
)
.
y
1
(
y
2
(
x
)
x
)
By the theory of differential equations,
W
(
y
1
,
y
2
)
=
0 at every point in
[
a
,
b
]
.At
the end point
x
=
a
, however, we have, by the boundary condition (D.5) (assuming
α
2
=
0 without loss of the generality),
)
|
x
=
a
=
−
α
1
)+
α
1
(
,
(
)
(
(
)
(
)=
.
W
y
1
y
2
α
2
y
1
a
y
2
a
α
2
y
1
a
y
2
a
0
Therefore
y
1
(
must be linearly dependent.
Note that this property is not valid for periodic S-L problems. In the eigenvalue
problem (2.40) in Section 2.5, for example, one eigenvalue has two linearly inde-
pendent corresponding eigenfunctions.
x
)
and
y
2
(
x
)
Property 4.
The eigenvalues of regular S-L problems are positive semi-definite if
the coefficients in Eqs. (D.5) and (D.6) satisfy
−
α
1
0and
β
1
α
2
≥
β
2
≥
0.
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