Page 323 - Handbook Of Integral Equations

b

27. y(x) – λ [AK µ (αx)+ BK ν (βt)]y(t) dt = f(x).
a
This is a special case of equation 4.9.5 with g(x)= AK µ (αx) and h(t)= BK ν (βt).

b
28. y(x) – λ [AK µ (x)K µ (t)+ BK ν (x)K ν (t)]y(t) dt = f(x).
a
This is a special case of equation 4.9.14 with g(x)= K µ (x) and h(t)= K ν (t).

b
29. y(x) – λ [AK µ (x)K ν (t)+ BK ν (x)K µ (t)]y(t) dt = f(x).
a
This is a special case of equation 4.9.17 with g(x)= K µ (x) and h(t)= K ν (t).

4.9. Equations Whose Kernels Contain Arbitrary

Functions

4.9-1. Equations With Degenerate Kernel: K(x, t)= g 1 (x)h 1 (t)+ ··· + g n (x)h n (t)

b
1. y(x) – λ g(x)h(t)y(t) dt = f(x).
a
b –1
1 . Assume that λ ≠ g(t)h(t) dt .
◦
a
Solution:
b –1 b

y(x)= f(x)+ λkg(x), where k = 1 – λ g(t)h(t) dt h(t)f(t) dt.
a a
b –1
2 . Assume that λ = g(t)h(t) dt .
◦
a
b

For h(t)f(t) dt = 0, the solution has the form
a
y = f(x)+ Cg(x),
where C is an arbitrary constant.
b
For h(t)f(t) dt ≠ 0, there is no solution.
a
The limits of integration may take the values a = –∞ and/or b = ∞, provided that the
corresponding improper integral converges.

b
2. 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 = √ ,
g 1 + (b – a)g 2 g 1 – (b – a)g 2

where

b b
2
g 1 = g(x) dx, g 2 = g (x) dx.
a a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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