Page 87 - Handbook Of Integral Equations

x

µ µ
39. arccot (λx) – arccot (λt) y(t) dt = f(x).
a
µ
This is a special case of equation 1.9.2 with g(x) = arccot (λx).
2 2
1 d (1 + λ x )f (x)

x
Solution: y(x)= – µ–1 .
λµ dx arccot (λx)
x
y(t) dt
40. µ = f(x), 0 < µ <1.
a arccot(λt) – arccot(λx)
Solution:
λ sin(πµ) d x ϕ(t)f(t) dt 1
y(x)= , ϕ(x)= .
2 2
π dx a [arccot(λt) – arccot(λx)] 1–µ 1+ λ x
x

β γ
41. A arccot (λx)+ B arccot (µt)+ C y(t) dt = f(x).
a
γ
β
This is a special case of equation 1.9.6 with g(x)= A arccot (λx) and h(t)= B arccot (µt)+C.
1.7. Equations Whose Kernels Contain Combinations of
Elementary Functions
1.7-1. Kernels Containing Exponential and Hyperbolic Functions

x

1. e µ(x–t) A 1 cosh[λ 1 (x – t)] + A 2 cosh[λ 2 (x – t)] y(t) dt = f(x).
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 1.3.8:
x
–µx
A 1 cosh[λ 1 (x – t)] + A 2 cosh[λ 2 (x – t)] w(t) dt = e f(x).
a
x
2
2. e µ(x–t) cosh [λ(x – t)]y(t) dt = f(x).
a
Solution:
2λ 2 x µ(x–t) √

y(x)= ϕ(x) – e sinh[k(x – t)]ϕ(x) dt, k = λ 2, ϕ(x)= f (x) – µf(x).
x
k a
x

3
3. e µ(x–t) cosh [λ(x – t)]y(t) dt = f(x).
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 1.3.15:
x
3
cosh [λ(x – t)]w(t) dt = e –µx f(x).
a
x
4
4. e µ(x–t) cosh [λ(x – t)]y(t) dt = f(x).
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 1.3.19:
x
4
cosh [λ(x – t)]w(t) dt = e –µx f(x).
a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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