Page 332 - Handbook Of Integral Equations

3 . The solution with λ = λ 2 ≠ λ 1 and f 1 = f 2 = 0 is given by the formulas of item 2 in
◦
◦
which one must replace λ 1 and y 1 (x)by λ 2 and y 2 (x), respectively.
4 . Solution with λ = λ 1,2 = λ ∗ and f 1 = f 2 = 0, where the characteristic value λ ∗ =
◦
2
(A ≠ 0) is double:
A(b – a)+(B + C)h 1
y(x)= f(x)+ Cy ∗ (x),

where C is an arbitrary constant and y ∗ (x) is an eigenfunction of the equation corresponding

to λ ∗ . Two cases are possible.
(a) If A(b – a)+ Bh 1 ≠ Ch 1 , then

A(b – a)+(B – C)h 1
y ∗ (x)= A + Bxh(x) – h(x).
2h 0
2
2
(b) If A(b – a)+ Bh 1 = Ch 1 , then, in view of h 0 [A(b – a )+2Bh 2 ] = 0, we have

2
2
 A + Bxh(x) for h 0 ≠ 0 and A(b – a )= –2Bh 2 ,
2
2
y ∗ (x)= h(x) for h 0 = 0 and A(b – a ) ≠ –2Bh 2 ,
 2 2
C 1 [A + Bxh(x)] + C 2 h(x) for h 0 = 0 and A(b – a )= –2Bh 2 ,

where C 1 and C 2 are arbitrary constants.

b

12. y(x) – λ [g(x)g(t)+ h(x)h(t)]y(t) dt = f(x).
a
The characteristic values of the equation:

2
2
s 1 + s 3 + (s 1 – s 3 ) +4s 2 2 s 1 + s 3 – (s 1 – s 3 ) +4s 2 2
λ 1 = , λ 2 = ,
2
2
2(s 1 s 3 – s ) 2(s 1 s 3 – s )
2 2
where
b b b

2
2
s 1 = g (x) dx, s 2 = g(x)h(x) dx, s 3 = h (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 3 – f 2 s 2 ) f 2 – λ(f 2 s 1 – f 1 s 2 )
A 1 = 2 , A 2 = 2 ,
2
2
(s 1 s 3 – s )λ – (s 1 + s 3 )λ +1 (s 1 s 3 – s )λ – (s 1 + s 3 )λ +1
2
2
b b
f 1 = f(x)g(x) dx, f 2 = f(x)h(x) dx.
a a
◦
2 . Solution with λ = λ 1 ≠ λ 2 and f 1 = f 2 =0:
1 – λ 1 s 1
y(x)= f(x)+ Cy 1 (x), y 1 (x)= g(x)+ h(x),
λ 1 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 . The solution with λ = λ 2 ≠ λ 1 and f 1 = f 2 = 0 is given by the formulas of item 2 in
◦
◦
which one must replace λ 1 and y 1 (x)by λ 2 and y 2 (x), respectively.
◦
4 . Solution with λ = λ 1,2 = λ ∗ and f 1 = f 2 = 0, where the characteristic value λ ∗ =1/s 1 is
double:
y(x)= f(x)+ C 1 g(x)+ C 2 h(x),

where C 1 and C 2 are arbitrary constants.

© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 311