Higher Engineering Mathematics

POWER SERIES METHODS OF SOLVING ORDINARY DIFFERENTIAL EQUATIONS 507

I

=

−a 2

23 (v+2)

=

− 1

23 (v+2)

− 1

2 v+^2 (v+2)

=

1

2 v+^4 (2!)(v+3)

since (v+2)(v+2)=(v+3)

and a 6 =

− 1

2 v+^6 (3!)(v+4)

and so on.

Therecurrence relationis:

ar=

(−1)r/^2

2 v+r

(r

2

!

)



(

v+

r

2

+ 1

)

And if we letr= 2 k, then

a 2 k=

(−1)k

2 v+^2 k(k!)(v+k+1)

(42)

fork=1, 2, 3,···

Hence, it is possible to write the new form for
equation (38) as:

y=Axv

{

1

2 v(v+1)

−

x^2

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

+

x^4

2 v+^4 (2!)(v+3)

−···

}

This is calledthe Bessel function of the first order
kind, of orderv,and is denoted byJv(x),

i.e. Jv(x)=

(x

2

)v{ 1

(v+1)

−

x^2

22 ( 1 !)(v+2)

+

x^4

24 ( 2 !)(v+ 3 )

−···

}

providedvis not a negative integer.

For the second solution,whenc=−v, replacingv
by−vin equation (42) above gives:

a 2 k=

(−1)k

22 k−v(k!)(k−v+1)

from which, when k=0,a 0 =

(− 1 )^0

2 −v(0!)(1−v)

=

1

2 −v(1−v)

since 0!=1 (see page 492)

whenk=1,a 2 =

(−1)^1

22 −v( 1 !)(1−v+1)

=

− 1

22 −v(1!)(2−v)

whenk=2,a 4 =

(−1)^2

24 −v(2!)(2−v+1)

=

1

24 −v(2!)(3−v)

whenk=3,a 6 =

(−1)^3

26 −v( 3 !)(3−v+1)

=

1

26 −v(3!)(4−v)

and so on.

Hence,y=Bx−v

{

1

2 −v(1−v)

−

x^2

22 −v(1!)(2−v)

+

x^4

24 −v(2!)(3−v)

−···

}

i.e. J−v(x)=

(x

2

)−v{ 1

( 1 −v)

−

x^2

22 ( 1 !)( 2 −v)

+

x^4

24 ( 2 !)( 3 −v)

−···

}

providedvis not a positive integer.

Jv(x) andJ−v(x) are two independent solutions of

the Bessel equation; the complete solution is:

y=AJv(x)+BJ−v(x) whereAandBare constants

i.e. y=AJv(x)+BJ−v(x)

=A

(x

2

)v{ 1

(v+ 1 )

−

x^2

22 ( 1 !)(v+ 2 )

+

x^4

24 ( 2 !)(v+ 4 )

−···

}

+B

(x

2

)−v{ 1

( 1 −v)

−

x^2

22 ( 1 !)( 2 −v)

+

x^4

24 ( 2 !)( 3 −v)

−···

}

In general terms:Jv(x)=

(x

2

)v∑∞

k= 0

(−1)kx^2 k

22 k(k!)(v+k+1)

and J−v(x)=

(x

2

)−v∑∞

k= 0

(−1)kx^2 k

22 k(k!)(k−v+1)