Page 264 - Handbook Of Integral Equations

b
40. f(t)y(x + βt) dt = Ae λx .
a
Solution:
A λx b
y(x)= e , B = f(t) exp(λβt) dt.
B a

b
41. f(t)y(x + βt) dt = A sin λx + B cos λx.
a
Solution:
y(x)= p sin λx + q cos λx,

where

AI c + BI s BI c – AI s
p = , q = ,
2
2
I + I 2 I + I 2
c s c s
b b
I c = f(t) cos(λβt) dt, I s = f(t) sin(λβt) dt.
a a
1

42. y(ξ) dt = f(x), ξ = g(x)t.
0
Assume that g(0) = 0, g(1) = 1, and g ≥ 0.

x
1
1 . The substitution z = g(x) leads to an equation of the form 3.1.41: y(zt) dt = F(z),
◦
0
where the function F(z) is obtained from z = g(x) and F = f(x) by eliminating x.
2 . Solution y = y(z) in the parametric form:
◦
g(x)
y(z)= f (x)+ f(x), z = g(x).

x
g (x)

x
1

λ
43. t y(ξ) dt = f(x), ξ = g(x)t.
0

Assume that g(0) = 0, g(1) = 1, and g ≥ 0.
x
1
λ
1 . The substitution z = g(x) leads to an equation of the form 3.1.42: t y(zt) dt = F(z),
◦
0
where the function F(z) is obtained from z = g(x) and F = f(x) by eliminating x.
2 . Solution y = y(z) in the parametric form:
◦
g(x)
y(z)= f (x)+(λ +1)f(x), z = g(x).

x
g (x)

x
b

β
44. f(t)y(ξ) dt = Ax , ξ = xϕ(t).
a
Solution:
A β b β
y(x)= x , B = f(t) ϕ(t) dt. (1)
B
a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 243