Page 43 - Handbook Of Integral Equations

x

2 2
49. A exp(λx )+ B exp(λt )+ C y(t) dt = f(x).
a
2
This is a special case of equation 1.9.5 with g(x) = exp(λx ).
x
2 2
50. A exp(λx )+ B exp(µt ) y(t) dt = f(x).
a
2
2
This is a special case of equation 1.9.6 with g(x)= A exp(λx ) and h(t)= B exp(µt ).
x √
2
2
51. x – t exp[λ(x – t )]y(t) dt = f(x).
a
Solution:
2
2 2 d 2 x exp(–λt )
y(x)= exp(λx ) √ f(t) dt.
π dx 2 x – t
a
2
2
x exp[λ(x – t )]
52. √ y(t) dt = f(x).
a x – t
Solution:
2
1 2 d x exp(–λt )
y(x)= exp(λx ) √ f(t) dt.
π dx a x – t
x

2
λ
2
53. (x – t) exp[µ(x – t )]y(t) dt = f(x), 0 < λ <1.
a
Solution:
2
d 2 x exp(–µt ) sin(πλ)
2
y(x)= k exp(µx ) f(t) dt, k = .
dx 2 a (x – t) λ πλ
x

β β
54. exp[λ(x – t )]y(t) dt = f(x).
a

Solution: y(x)= f (x) – λβx β–1 f(x).
x
1.3. Equations Whose Kernels Contain Hyperbolic
Functions
1.3-1. Kernels Containing Hyperbolic Cosine

x
1. cosh[λ(x – t)]y(t) dt = f(x).
a
x

Solution: y(x)= f (x) – λ 2 f(x) dx.
x
a
x

2. cosh[λ(x – t)] – 1 y(t) dt = f(x), f(a)= f (a)= f (x)=0.
x xx
a
1
Solution: y(x)= f xxx (x) – f (x).

x
λ 2
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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