Page 284 - Handbook Of Integral Equations

4.1-6. Kernels Containing Arbitrary Powers

b
µ
54. y(x) – λ (x – t)t y(t) dt = f(x).
a
µ
This is a special case of equation 4.9.8 with A =0, B = 1, and h(t)= t .
Solution:
y(x)= f(x)+ λ(A 1 + A 2 x),

where A 1 and A 2 are the constants determined by the formulas presented in 4.9.8.

b
ν
55. y(x) – λ (x – t)x y(t) dt = f(x).
a
ν
This is a special case of equation 4.9.10 with A =0, B = 1, and h(x)= x .
Solution:
ν
y(x)= f(x)+ λ(E 1 x ν+1 + E 2 x ),
where E 1 and E 2 are the constants determined by the formulas presented in 4.9.10.
b
µ
µ
56. y(x) – λ (x – t )y(t) dt = f(x).
a
µ
This is a special case of equation 4.9.3 with g(x)= x .
Solution:
µ
y(x)= f(x)+ λ(A 1 x + A 2 ),
where A 1 and A 2 are the constants determined by the formulas presented in 4.9.3.
b
µ
ν
ν
57. y(x) – λ (Ax + Bt )t y(t) dt = f(x).
a
ν
µ
This is a special case of equation 4.9.6 with g(x)= x and h(t)= t .
Solution:
ν
y(x)= f(x)+ λ(A 1 x + A 2 ),
where A 1 and A 2 are the constants determined by the formulas presented in 4.9.6.
b
ν
γ
µ
58. y(x) – λ (Dx + Et )x y(t) dt = f(x).
a
γ
This is a special case of equation 4.9.18 with g 1 (x)= x ν+γ , h 1 (t)= D, g 2 (x)= x , and
µ
h 2 (t)= Et .
Solution:
γ
y(x)= f(x)+ λ(A 1 x ν+γ + A 2 x ),
where A 1 and A 2 are the constants determined by the formulas presented in 4.9.18.
b

ν µ
γ δ
59. y(x) – λ (Ax t + Bx t )y(t) dt = f(x).
a
ν
γ
µ
This is a special case of equation 4.9.18 with g 1 (x)= x , h 1 (t)= At , g 2 (x)= x , and
δ
h 2 (t)= Bt .
Solution:
ν
γ
y(x)= f(x)+ λ(A 1 x + A 2 x ),
where A 1 and A 2 are the constants determined by the formulas presented in 4.9.18.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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