Page 94 - Handbook Of Integral Equations

x
√
52. e µ(x–t) cos t – cos xy(t) dt = f(x).
a
Solution: x
2 µx 1 d
2 e –µt sin tf(t) dt
y(x)= e sin x √ .
π sin x dx cos t – cos x
a
x e µ(x–t) y(t) dt
53. √ = f(x).
a cos t – cos x
Solution:
1 µx d x e –µt sin tf(t) dt
y(x)= e √ .
π dx a cos t – cos x

x

λ
54. e µ(x–t) (cos t – cos x) y(t) dt = f(x), 0 < λ <1.
a
Solution:
x –µt

1 d
2 e sin tf(t) dt sin(πλ)
y(x)= ke µx sin x , k = .
sin x dx (cos t – cos x) λ πλ
a
x

λ
λ
55. e µ(x–t) (cos x – cos t)y(t) dt = f(x).
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 1.5.24:
x
λ
λ
(cos x – cos t)w(t) dt = e –µx f(x).
a
x

λ
λ

56. e µ(x–t) A cos x + B cos t y(t) dt = f(x).
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 1.5.25:
x

λ λ
–µx
A cos x + B cos t w(t) dt = e f(x).
a
x
e y(t) dt
µ(x–t)
57. = f(x), 0 < λ <1.
a (cos t – cos x) λ
The substitution w(x)= e –µx y(x) leads to an equation of the form 1.5.26:
x

w(t) dt –µx
= e f(x).
(cos t – cos x) λ
a
x

µ(x–t) ν
58. Ae + B cos (λx) y(t) dt = f(x).
a
µx
ν
This is a special case of equation 1.9.15 with g 1 (x)= Ae , h 1 (t)= e –µt , g 2 (x)= B cos (λx),
and h 2 (t)=1.
x

µ(x–t) ν
59. Ae + B cos (λt) y(t) dt = f(x).
a
µx
This is a special case of equation 1.9.15 with g 1 (x)= Ae , h 1 (t)= e –µt , g 2 (x)= B, and
ν
h 2 (t) = cos (λt).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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