Page 82 - Handbook Of Integral Equations

x

3. A arccos(λx)+ B arccos(µt)+ C y(t) dt = f(x).
a
This is a special case of equation 1.9.6 with g(x) = arccos(λx) and h(t)= B arccos(µt)+ C.
x

n
4. arccos(λx) – arccos(λt) y(t) dt = f(x), n =1, 2, ...
a
The right-hand side of the equation is assumed to satisfy the conditions f(a)= f (a)= ··· =

x
f (n) (a)=0.
x
Solution:
(–1) n √ d n+1
2 2
y(x)= √ 1 – λ x f(x).
2 2
n
λ n! 1 – λ x dx
x
5. arccos(λt) – arccos(λx) y(t) dt = f(x).
a
This is a special case of equation 1.9.38 with g(x)=1 – arccos(λx).
Solution:
2 x
2 1 d ϕ(t)f(t) dt 1

y(x)= ϕ(x) √ , ϕ(x)= √ .
π ϕ(x) dx a arccos(λt) – arccos(λx) 1 – λ x
2 2
x
y(t) dt
6. √ = f(x).
a arccos(λt) – arccos(λx)
Solution:
λ d x ϕ(t)f(t) dt 1
y(x)= √ , ϕ(x)= √ .
π dx a arccos(λt) – arccos(λx) 1 – λ x
2 2
x

µ
7. arccos(λt) – arccos(λx) y(t) dt = f(x), 0 < µ <1.
a
Solution:
2
x
1 d ϕ(t)f(t) dt
y(x)= kϕ(x) ,
ϕ(x) dx [arccos(λt) – arccos(λx)] µ
a
1 sin(πµ)
ϕ(x)= √ , k = .
2 2
1 – λ x πµ
x
µ µ
8. arccos (λx) – arccos (λt) y(t) dt = f(x).
a
µ
This is a special case of equation 1.9.2 with g(x) = arccos (λx).
√
1 d f (x) 1 – λ x
2 2

x
Solution: y(x)= – .
λµ dx arccos µ–1 (λx)
x
y(t) dt
9. µ = f(x), 0 < µ <1.
a arccos(λt) – arccos(λx)
Solution:
λ sin(πµ) d x ϕ(t)f(t) dt 1
y(x)= , ϕ(x)= √ .
π dx a [arccos(λt) – arccos(λx)] 1–µ 1 – λ x
2 2
x
β γ
10. A arccos (λx)+ B arccos (µt)+ C y(t) dt = f(x).
a
γ
β
This is a special case of equation 1.9.6 with g(x)=A arccos (λx) and h(t)=B arccos (µt)+C.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 60