Page 84 - Handbook Of Integral Equations

x

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

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

β γ
20. A arcsin (λx)+ B arcsin (µt)+ C y(t) dt = f(x).
a
γ
β
This is a special case of equation 1.9.6 with g(x)= A arcsin (λx) and h(t)= B arcsin (µt)+C.
1.6-3. Kernels Containing Arctangent

x

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

Solution: y(x)= (1 + λ x ) f (x) .
x
λ dx
x

22. A arctan(λx)+ B arctan(λt) y(t) dt = f(x).
a
For B =–A, see equation 1.6.21. This is a special case of equation 1.9.4 with g(x)=arctan(λx).
Solution:
sign x d – A x – B

y(x)= arctan(λx) A+B arctan(λt) A+B f (t) dt .
t
A + B dx a
x

23. A arctan(λx)+ B arctan(µt)+ C y(t) dt = f(x).
a
This is a special case of equation 1.9.6 with g(x)= A arctan(λx) and h(t)= B arctan(µt)+ C.
x
n
24. arctan(λx) – arctan(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:
n+1

1 2 2 d
y(x)= (1 + λ x ) f(x).
n
2 2
λ n!(1 + λ x ) dx
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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