Higher Engineering Mathematics

508 DIFFERENTIAL EQUATIONS

Another Bessel function

It may be shown that another series forJn(x)is

given by:

Jn(x)=

(x

2

)n{ 1

n!

−

1

(n+1)!

(x

2

) 2

+

1

(2!)(n+2)!

(x

2

) 4

− ···

}

From this series two commonly used function are

derived,

i.e. J 0 (x)=

1

(0!)

−

1

(1!)^2

(x

2

) 2

+

1

(2!)^2

(x

2

) 4

−

1

(3!)^2

(x

2

) 6

+···

= 1 −

x^2

22 ( 1 !)^2

+

x^4

24 ( 2 !)^2

−

x^6

26 ( 3 !)^2

+···

and J 1 (x)=

x

2

{

1

(1!)

−

1

(1!)(2!)

(x

2

) 2

+

1

(2!)(3!)

(x

2

) 4

−···

}

=

x

2

−

x^3

23 (1!)(2!)

+

x^5

25 (2!)(3!)

−

x^7

27 (3!)(4!)

+···

2 4 6 8 10 12 14 x

−0.5

0

0.5

1

y

y = J 0 (x)

y = J 1 (x)

Figure 52.1

Tables of Bessel functions are available for a range

of values ofnandx, and in these,J 0 (x) andJ 1 (x) are

most commonly used.

Graphs ofJ 0 (x), which looks similar to a cosine,

andJ 1 (x), which looks similar to a sine, are shown

in Figure 52.1.

Now try the following exercise.

Exercise 198 Further problems on Bessel’s

equation and Bessel’s functions

1. Determine the power series solution of Bes-

sel’s equation:x^2

d^2 y

dx^2

+x

dy

dx

+(x^2 −v^2 )y= 0

whenv=2, up to and including the term

inx^6. [

y=Ax^2

{

1 −

x^2

12

+

x^4

384

− ···

}]

2. Find the power series solution of the Bessel
   function: x^2 y′′+xy′+

(

x^2 −v^2

)

y=0in

terms of the Bessel function J 3 (x) when

v=3. Give the answer up to and including

the term in⎡ x^7.

⎢

⎢

⎢

⎣

y=AJ 3 (x)=

(x

2

) 3 { 1

 4

−

x^2

22  5

+

x^4

25  6

−···

}

⎤

⎥

⎥

⎥

⎦