Page 127 - Handbook Of Integral Equations

2. Let the solution of the integral equation (1) have the form

x
d d

y(x)= L 1 x, f(x)+ L 2 x, R(x, t)f(t) dt, (5)
dx dx
a
where L 1 and L 2 are some linear differential operators.
The solution of the more complicated integral equation
x

K ϕ(x), ϕ(t) y(t) dt = f(x), (6)
a

where ϕ(x) is an arbitrary monotone function (differentiable sufficiently many times, ϕ > 0), is
x
determined by the formula

1 d

y(x)= ϕ (x)L 1 ϕ(x), f(x)
x
ϕ (x) dx

x
x (7)
1 d

+ ϕ (x)L 2 ϕ(x), R ϕ(x), ϕ(t) ϕ (t)f(t) dt.

x t
ϕ (x) dx

x a
Below are formulas for the solutions of integral equations of the form (6) for some specific
functions ϕ(x). In all cases, it is assumed that the solution of equation (1) is known and is
determined by formula (5).
λ
(a) For ϕ(x)= x ,
1 d 2 λ–1 λ 1 d λ λ
λ–1
x
λ
λ–1
y(x)= λx L 1 x , f(x)+ λ x L 2 x , R x , t t f(t) dt.
λx λ–1 dx λx λ–1 dx a
λx
(b) For ϕ(x)= e ,
1 d 2 λx λx 1 d λx λt
λt
x
λx
λx
y(x)= λe L 1 e , f(x)+ λ e L 2 e , R e , e e f(t) dt.
λe λx dx λe λx dx a
(c) For ϕ(x) = ln(λx),
1 d 1 d x 1
y(x)= L 1 ln(λx), x f(x)+ L 2 ln(λx), x R ln(λx), ln(λt) f(t) dt.
x dx x dx a t
(d) For ϕ(x) = cos(λx),

–1 d
y(x)= –λ sin(λx)L 1 cos(λx), f(x)
λ sin(λx) dx
–1 d
x
2
+ λ sin(λx)L 2 cos(λx), R cos(λx), cos(λt) sin(λt)f(t) dt.
λ sin(λx) dx a
(e) For ϕ(x) = sin(λx),

1 d
y(x)= λ cos(λx)L 1 sin(λx), f(x)
λ cos(λx) dx
x
1 d
2
+ λ cos(λx)L 2 sin(λx), R sin(λx), sin(λt) cos(λt)f(t) dt.
λ cos(λx) dx a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 105