Page 335 - Handbook Of Integral Equations

◦
3 . Solution with λ = λ 2 ≠ λ 1 and f 1 = f 2 =0:

s 3
y(x)= f(x)+ Cy 2 (x), y 2 (x)= g(x) – h(x),
s 2
where C is an arbitrary constant and y 2 (x) is an eigenfunction of the equation corresponding
to the characteristic value λ 2 .
4 . The equation has no multiple characteristic values.
◦
b
16. y(x) – λ [g(x)h(t) – h(x)g(t)]y(t) dt = f(x).
a
The characteristic values of the equation:

1 1
, ,
λ 1 = λ 2 = –
2 2
s – s 2 s 3 s – s 2 s 3
1 1
where
b b b
2
2
s 1 = h(x)g(x) dx, s 2 = h (x) dx, s 3 = g (x) dx.
a a a
◦
1 . Solution with λ ≠ λ 1,2 :
y(x)= f(x)+ λ[A 1 g(x)+ A 2 h(x)],

where the constants A 1 and A 2 are given by

f 1 + λ(f 1 s 1 – f 2 s 2 ) –f 2 + λ(f 2 s 1 – f 1 s 3 )
A 1 = , A 2 = ,
2
2
2
2
(s 2 s 3 – s )λ +1 (s 2 s 3 – s )λ +1
1 1
b b

f 1 = f(x)h(x) dx, f 2 = f(x)g(x) dx.
a a
◦
2 . Solution with λ = λ 1 ≠ λ 2 and f 1 = f 2 =0:

2
s – s 2 s 3 – s 1
1
y(x)= f(x)+ Cy 1 (x), y 1 (x)= g(x)+ h(x),
s 2
where C is an arbitrary constant and y 1 (x) is an eigenfunction of the equation corresponding
to the characteristic value λ 1 .
3 . Solution with λ = λ 2 ≠ λ 1 and f 1 = f 2 =0:
◦

2
s – s 2 s 3 + s 1
1
y(x)= f(x)+ Cy 2 (x), y 2 (x)= g(x) – h(x),
s 2
where C is an arbitrary constant and y 2 (x) is an eigenfunction of the equation corresponding
to the characteristic value λ 2 .
◦
4 . The equation has no multiple characteristic values.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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