Environmental Engineering Reference
In-Depth Information
Thus we define
J
γ
(
)
γπ
−
J
−
γ
(
)
x
cos
x
Y
n
(
x
)=
lim
γ
→
n
Y
γ
(
x
)=
lim
γ
→
.
sin
γπ
n
This is called the
Bessel function of integer-order of the second kind
and can be
found by using the L'Hôpital's rule. It can also be proven that
Y
n
(
x
)
and
J
n
(
x
)
are
linearly independent. Thus the general solution of Eq. (A.10) is for all
γ
(
γ
≥
0
without loss of generality),
y
=
C
1
J
γ
(
x
)+
C
2
Y
γ
(
x
)
.
Remark 1.
J
γ
(
, the symbols of Bessel functions, can be used like
trigonometric and logarithmic functions. We also have tables of Bessel functions.
By using these symbols we may concisely express general solutions of Bessel equa-
tions. For example, the general solution of the Bessel equation
x
)
and
Y
γ
(
x
)
x
2
y
9
25
x
2
y
+
xy
+
−
=
0
is
y
=
C
1
J
5
(
x
)+
C
2
Y
5
(
x
)
or
y
=
C
1
J
5
(
x
)+
C
2
J
5
(
x
)
,
−
where
C
1
and
C
2
are arbitrary constants. The general solution of
1
x
2
y
1
x
y
+
16
y
+
−
=
0
is
y
,where
C
1
and
C
2
are arbitrary constants.
Remark 2.
By a variable transformation of
t
=
C
1
J
4
(
x
)+
C
2
Y
4
(
x
)
=
mx
, the equation
xy
+
m
2
x
2
n
2
y
x
2
y
+
−
=
0
is transformed into a Bessel equation of
n
-th order
t
2
y
+
ty
+
t
2
n
2
y
−
=
0
.
9
x
2
y
1
x
y
+
4
Example 1.
Find the general solution of
y
+
−
=
0.
Solution.
Consider a variable transformation
t
=
3
x
. The equation is transformed
into a Bessel equation of second order
t
2
y
+
ty
+(
t
2
−
4
)
y
=
0
.
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