Page 85 - Handbook Of Integral Equations

x
25. arctan(λx) – arctan(λt) y(t) dt = f(x).
a
Solution:
2 x

2 1 d ϕ(t)f(t) dt 1
y(x)= ϕ(x) √ , ϕ(x)= .
π ϕ(x) dx a arctan(λx) – arctan(λt) 1+ λ x
2 2
x y(t) dt
26. √ = f(x).
a arctan(λx) – arctan(λt)
Solution:
λ d x ϕ(t)f(t) dt 1
y(x)= √ , ϕ(x)= .
2 2
π dx a arctan(λx) – arctan(λt) 1+ λ x
x √ x – t
27. t arctan y(t) dt = f(x).
a t
The equation can be rewritten in terms of the Gaussian hypergeometric function in the form

x x
3
1
(x – t) γ–1 F α, β, γ;1 – y(t) dt = f(x), where α = , β =1, γ = .
2
2
a t
See 1.8.86 for the solution of this equation.
x

µ
28. arctan(λx) – arctan(λt) y(t) dt = f(x), 0 < µ <1.
a
Solution:
2
1 d ϕ(t)f(t) dt
x
y(x)= kϕ(x) ,
ϕ(x) dx [arctan(λx) – arctan(λt)] µ
a
1 sin(πµ)
ϕ(x)= , k = .
1+ λ x πµ
2 2
x

µ µ
29. arctan (λx) – arctan (λt) y(t) dt = f(x).
a
µ
This is a special case of equation 1.9.2 with g(x) = arctan (λx).
2 2

1 d (1 + λ x )f (x)
x
Solution: y(x)= µ–1 .
λµ dx arctan (λx)
x
y(t) dt

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

β γ
31. A arctan (λx)+ B arctan (µt)+ C y(t) dt = f(x).
a
γ
β
This is a special case of equation 1.9.6 with g(x)= A arctan (λx) and h(t)= B arctan (µt)+C.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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