Higher Engineering Mathematics

506 DIFFERENTIAL EQUATIONS

The complete solution of Bessel’s equation:

x^2

d^2 y

dx^2

+x

dy

dx

+

(

x^2 −v^2

)

y=0 is:

y=u+w=

Axv

{

1 −

x^2

22 (v+ 1 )

+

x^4

24 × 2 !(v+ 1 )(v+ 2 )

−

x^6

26 × 3 !(v+ 1 )(v+ 2 )(v+ 3 )

+···

}

+Bx−v

{

1 +

x^2

22 (v− 1 )

+

x^4

24 × 2 !(v− 1 )(v− 2 )

+

x^6

26 × 3 !(v− 1 )(v− 2 )(v− 3 )

+···

}

(39)

The gamma function

The solution of the Bessel equation of Problem 10
may be expressed in terms ofgamma functions.is
the upper case Greek letter gamma, and the gamma
function(x) is defined by the integral

(x)=

∫∞

0

tx−^1 e−tdt (40)

and is convergent forx> 0

From equation (40), (x+1)=

∫∞

0

txe−tdt

and by using integration by parts (see page 418):

(x+1)=

[

(

tx

)

(

e−t

− 1

)]∞

0

−

∫∞

0

(

e−t

− 1

)

xtx−^1 dx

=(0−0)+x

∫∞

0

e−ttx−^1 dt

=x(x) from equation (40)

This is an important recurrence relation for gamma
functions.

Thus, since (x+1)=x(x)

then similarly, (x+2)=(x+1)(x+1)

=(x+1)x(x) (41)

and (x+3)=(x+2)(x+2)

=(x+2)(x+1)x(x),

and so on.

These relationships involving gamma functions are

used with Bessel functions.

Bessel functions

The power series solution of the Bessel equation may

be written in terms of gamma functions as shown in

worked problem 11 below.

Problem 11. Show that the power series solu-

tion of the Bessel equation of worked problem 10

may be written in terms of the Bessel functions

Jv(x) andJ−v(x) as:

AJv(x)+BJ−v(x)

=

(x

2

)v{ 1

(v+1)

−

x^2

22 (1!)(v+2)

+

x^4

24 (2!)(v+4)

−···

}

+

(x

2

)−v{ 1

(1−v)

−

x^2

22 (1!)(2−v)

+

x^4

24 (2!)(3−v)

−···

}

From Problem 10 above,whenc=+v,

a 2 =

−a 0

22 (v+1)

If we leta 0 =

1

2 v(v+1)

then

a 2 =

− 1

22 (v+1) 2v(v+1)

=

− 1

2 v+^2 (v+1)(v+1)

=

− 1

2 v+^2 (v+2)

from equation (41)

Similarly, a 4 =

a 2

v^2 −(c+4)^2

from equation (37)

=

a 2

(v−c−4)(v+c+4)

=

a 2

−4(2v+4)

sincec=v