Page 93 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes 3 2
or x x 3x 2 = c 0 c 1 c 2 c 3 c 1 2c 2 3c x
c 2 3 c x 2 c 3 x 3
2 2 3 6 ...(ii)
Equating coefficients of like powers of x on both sides of (ii), we get

c 0 c 1 c 2 c 3 = 2

c 1 2c 2 3c 3 = 3

1 3 c 3
c 2 c 3 = 1 and 1
2 2 6
Solving these, we get,
c 3 = 6,C 2 20, c 1 19, c 0 7 ...(iii)

Putting these values in (i) we get
x
L
x
x
L
x
L
x 3 x 2 3x 2 = 7L 0 ( ) 19 ( ) 20 ( ) 6 ( ).
1
3
2
Example 3: Prove that
x
L
x
x
xL ( ) (1 x ) ( ) nL ( ) = 0
n n n
and hence deduce that
L (0) = n
n
Solution: Since L n ( ) satisfies the Laguerre s equation
x
2
d y dy
x 2 (1 x ) ny 0, for n
dx dx
x
xL n ( ) (1 x ) ( ) nL n ( ) 0.
x
L
x
n
Putting x 0, we have
L n (0) nL n (0)
L
or L n (0) n since (0) 1
n
Self Assessment
x
x
L
15. Express L 4 ( ) in terms of L 3 ( ) and ( )
x
2
16. Show that
L n ( ) L n 1 ( ) L n ( )
x
x
x
17. Show that
L (1) nL (1) 0
n n

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