Page 167 - Handbook Of Integral Equations

x

11. y(x) – A ln(kx)+ B – AB(x – t) ln(kx) y(t) dt = f(x).
a
This is a special case of equation 2.9.7 with λ = B and g(x)= A ln(kx).
x

12. y(x)+ A ln(kt)+ B + AB(x – t) ln(kt) y(t) dt = f(x).
a
This is a special case of equation 2.9.8 with λ = B and g(t)= A ln(kt).

∞

n
13. y(x)+ a (t – x) ln(t – x)y(t) dt = f(x), n =1, 2, ...
x
For f(x)= m A k exp(–λ k x), where λ k > 0, a solution of the equation has the form
k=1
m
A k an!
y(x)= exp(–λ k x), B k =1 + n+1 1+ 1 2 + 1 3 + ··· + n 1 – ln λ k – C ,
B k λ
k=1 k
where C = 0.5772 ... is the Euler constant.
∞ ln(t – x)

14. y(x)+ a √ y(t) dt = f(x).
x t – x
This is a special case of equation 2.9.62 with K(–x)= ax –1/2 ln x.
For f(x)= m A k exp(–λ k x), where λ k > 0, a solution of the equation has the form
k=1
m
A k π

y(x)= exp(–λ k x), B k =1 – a ln(4λ k )+ C ,
B k λ k
k=1
where C = 0.5772 ... is the Euler constant.

2.5. Equations Whose Kernels Contain Trigonometric
Functions

2.5-1. Kernels Containing Cosine

x

1. y(x) – A cos(λx)y(t) dt = f(x).
a
This is a special case of equation 2.9.2 with g(x)= A cos(λx) and h(t)=1.
Solution:
x

y(x)= f(x)+ A cos(λx)exp A sin(λx) – sin(λt) f(t) dt.
a λ
x

2. y(x) – A cos(λt)y(t) dt = f(x).
a
This is a special case of equation 2.9.2 with g(x)= A and h(t) = cos(λt).
Solution:
x
A
y(x)= f(x)+ A cos(λt)exp sin(λx) – sin(λt) f(t) dt.
a λ

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 146