Page 324 - Handbook Of Integral Equations

1 . Solution with λ ≠ λ 1,2 :
◦
y(x)= f(x)+ λ[A 1 g(x)+ A 2 ],
where the constants A 1 and A 2 are given by
f 1 – λ[f 1 g 1 – (b – a)f 2 ] f 2 – λ(f 2 g 1 – f 1 g 2 )
A 1 = 2 , A 2 = 2 ,
2
2
[g – (b – a)g 2 ]λ – 2g 1 λ +1 [g – (b – a)g 2 ]λ – 2g 1 λ +1
1 1
b b
f 1 = f(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:
◦

g 2
y(x)= f(x)+ Cy 1 (x), y 1 (x)= g(x)+ ,
b – a
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:

g 2
y(x)= f(x)+ Cy 2 (x), y 2 (x)= g(x) – ,
b – a
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

3. y(x) – λ [g(x) – g(t)]y(t) dt = f(x).
a
The characteristic values of the equation:
1 1
, ,
λ 1 = λ 2 = –
2 2
g – (b – a)g 2 g – (b – a)g 2
1
1
where
b b
2
g 1 = g(x) dx, g 2 = g (x) dx.
a a
1 . Solution with λ ≠ λ 1,2 :
◦
y(x)= f(x)+ λ[A 1 g(x)+ A 2 ],
where the constants A 1 and A 2 are given by
f 1 + λ[f 1 g 1 – (b – a)f 2 ] –f 2 + λ(f 2 g 1 – f 1 g 2 )
A 1 = , A 2 = ,
2
2
2
2
[(b – a)g 2 – g ]λ +1 [(b – a)g 2 – g ]λ +1
1 1
b b

f 1 = f(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:
◦
1 – λ 1 g 1
y(x)= f(x)+ Cy 1 (x), y 1 (x)= g(x)+ ,
λ 1 (b – a)
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 . The equation has no multiple characteristic values.
◦
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 303