Page 220 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 13: Orthogonality of Solutions

–x
are orthogonal w.r.t. the weight function p(x) = e on the interval 0 x Notes
x
x
L
x
L m ( ). ( ) e dx
n
0
d n n x
x
= L m ( ) n (x e )dx
dx
0
d n 1 d n 1
n
n
x
x
= L m ( ) (x e x ) L m ( ) (x e x )dx
dx n 1 dx n 1
0 0
d n 1 n x
x
= L m ( ) n 1 (x e )dx
dx
0
proceeding similarly
d n m n x
m
m
= ( 1) L ( ) (x e ) dx n m
x
m n m
dx
0
n
m
= ( 1) m ( 1) m ! d n m (x e x ) dx n m
n m
dx
0
d n m 1 n x
= m ! n m 1 (x e ) 0
dx
0
Now,
x
x
L 2 n ( ). e dx
0
d n
n
= L n ( ) (x e x ) dx
x
dx n
0
n
= ( 1) n L n ! n ( )(x e x ) dx
x
0

x
n
n
n
n
= ( 1) ( 1) n ! x e dx ( !) 2
0
Thus the functions of the orthogonal system are
e x /2 L n ( )
x
(x) = (n = 0, 1, 2,...)
v ! n

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