Higher Engineering Mathematics

POWER SERIES METHODS OF SOLVING ORDINARY DIFFERENTIAL EQUATIONS 511

I

Rodrigue’s formula

An alternative method of determining Legendre
polynomials is by usingRodrigue’s formula, which
states:

Pn(x)=

1

2 nn!

dn

(

x^2 − 1

)n

dxn

(48)

This is demonstrated in the following worked
problems.

Problem 15. Determine the Legendre polyno-

mialP 2 (x) using Rodrigue’s formula.

In Rodrigue’s formula, Pn(x)=

1

2 nn!

dn

(

x^2 − 1

)n

dxn
and whenn=2,

P 2 (x)=

1

222!

d^2 (x^2 −1)^2

dx^2

=

1

23

d^2 (x^4 − 2 x^2 +1)

dx^2

d

dx

(x^4 − 2 x^2 +1)

= 4 x^3 − 4 x

and

d^2

(

x^4 − 2 x^2 + 1

)

dx^2

=

d(4x^3 − 4 x)

dx

= 12 x^2 − 4

Hence, P 2 (x)=

1

23

d^2

(

x^4 − 2 x^2 + 1

)

dx^2

=

1

8

(

12 x^2 − 4

)

i.e. P 2 (x)=

1

2

(

3 x^2 − 1

)

the same as in Problem 13.

Problem 16. Determine the Legendre polyno-

mialP 3 (x) using Rodrigue’s formula.

In Rodrigue’s formula,Pn(x) =

1

2 nn!

dn

(

x^2 − 1

)n

dxn
and whenn=3,

P 3 (x)=

1

233!

d^3

(

x^2 − 1

) 3

dx^3

=

1

23 (6)

d^3

(

x^2 − 1

)(

x^4 − 2 x^2 + 1

)

dx^3

=

1

(8)(6)

d^3

(

x^6 − 3 x^4 + 3 x^2 − 1

)

dx^3

d

(

x^6 − 3 x^4 + 3 x^2 − 1

)

dx

= 6 x^5 − 12 x^3 + 6 x

d

(

6 x^5 − 12 x^3 + 6 x

)

dx

= 30 x^4 − 36 x^2 + 6

and

d

(

30 x^4 − 36 x^2 + 6

)

dx

= 120 x^3 − 72 x

Hence,P 3 (x)=

1

(8)(6)

d^3

(

x^6 − 3 x^4 + 3 x^2 − 1

)

dx^3

=

1

(8)(6)

(

120 x^3 − 72 x

)

=

1

8

(

20 x^3 − 12 x

)

i.e. P 3 (x)=

1

2

(

5 x^3 − 3 x

)

the same as in Prob-

lem 14.

Now try the following exercise.

Exercise 199 Legendre’s equation and

Legendre polynomials

1. Determine the power series solution of
   the Legendre equation:(
   1 −x^2

)

y′′− 2 xy′+k(k+1)y= 0 when

(a)k=0 (b)k=2, up to and including the

term inx^5.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

(a)y=a 0 +a 1

(

x+

x^3

3

+

x^5

5

+···

)

(b)y=a 0

{

1 − 3 x^2

}

+a 1

{

x−

2

3

x^3 −

1

5

x^5

}

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

2. Find the following Legendre polynomials:
   (a)P 1 (x) (b)P 4 (x) (c)P 5 (x)
   ⎡

⎢

⎣

(a)x (b)

1

8

(

35 x^4 − 30 x^2 + 3

)

(c)

1

8

(

63 x^5 − 70 x^3 + 15 x

)

⎤

⎥

⎦