Page 61 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Similarly putting r = 1, 2, ... in (iii) and simplifying stepwise, we have
P n ( ) = 0 P n iv ( ) 0 0... P n n ( ) 0 ...(vi)

But since

1.3...(2n 1)
n
P n ( ) . ! 0
x
n x = n ...(vii)
Therefore our assumption that P n ( ) 0 has a repeated root is not correct.
Hence all the roots of P n ( ) 0 are distinct.
x

x
Example: Find the roots of P 2 ( ) 0
1 2
x
As P 2 ( ) 0 = (3x 1)
2
1 2
P 2 ( ) = 0 (3x 1)
2
3 2 = 1

= 1/ 3

So the roots are

1
1 = 1/ 3,d 2
3
Self Assessment

x
6. Show that the roots of P 3 ( ) 0 are
3 3
, 0,
5 5

2.7 Summary

 Legendre’s Differential equation is obtained from Laplace equation in spherical polar
co-ordinates.

 Legendre’s Differential equation has x 1, as well as x as regular singular points.
 So Legendre’s Differential equation is solved as a power series.
x
n
 It is found that Legendre polynomial P n ( ) is a finite power series having x as the
highest power of x.

n
x
x
 The generating function for P n ( ) is found to be (1 2h h 2 ) 1/2 h P n ( )
n 0

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