
84 Applied calculus of variations for engineers
Finally the remaining equation is of the Bessel kind:
r
2
d
2
u
1
dr
2
+ r
du
1
dr
+(k
2
r
2
− m
2
)u
1
=0.
The solution of such a differential equation when m is not an integer is of the
form
u
1
(r)=c
5
J
m
(kr)+c
6
J
−m
(kr),
where J are the Bessel functions of the first kind, defined by the formula
J
m
(x)=
∞
n=0
(−1)
n
n!(n + m)!
(
x
2
)
m+2n
,
This is a convergent series for any x = kr value. The J
−m
function in the
expression is simply defined by
J
−m
(x)=(−1)
m
J
m
(x).
However, in our case m is an integer and then the solution is
u
1
(r)=c
5
J
m
(kr)+c
6
Y
m
(kr).
The Bessel function of the second kind is defined as
Y
m
(x) = lim
p→m
cos(pπ)J
p
(x) − J
−p
(x)
sin(pπ)
.
The ...