Page 182 - Handbook Of Integral Equations

x

8. y(x) – A e µ(x–t) ln(λt)y(t) dt = f(x).
a
Solution:
x
(λx) Ax
y(x)= f(x)+ A e (µ–A)(x–t) ln(λt) f(t) dt.
(λt) At
a
x

9. y(x)+ A e µ(x–t) (ln x – ln t)y(t) dt = f(x).
a
Solution: x
1 µ(x–t)

y(x)= f(x)+ e u (x)u (t) – u (x)u (t) f(t) dt,
2
1
2
1
W a
where the primes stand for the differentiation with respect to the argument specified in the
parentheses, and u 1 (x), u 2 (x) is a fundamental system of solutions of the second-order linear
–1
homogeneous ordinary differential equation u +Ax u = 0, with u 1 (x) and u 2 (x) expressed
xx
in terms of Bessel functions or modified Bessel functions, depending on the sign of A:
√ √ √ √
1
W = , u 1 (x)= xJ 1 2 Ax , u 2 (x)= xY 1 2 Ax for A >0,
π
√ √ √ √
1
W = – , u 1 (x)= xI 1 2 –Ax , u 2 (x)= xK 1 2 –Ax for A <0.
2
∞

10. y(x)+ a e λ(x–t) ln(t – x)y(t) dt = f(x).
x
This is a special case of equation 2.9.62 with K(x)= ae λx ln(–x).
2.7-3. Kernels Containing Exponential and Trigonometric Functions
x
11. y(x) – A e µt cos(λx)y(t) dt = f(x).
a
µt
This is a special case of equation 2.9.2 with g(x)= A cos(λx) and h(t)= e .
x
12. y(x) – A e µx cos(λt)y(t) dt = f(x).
a
This is a special case of equation 2.9.2 with g(x)= Ae µx and h(t) = cos(λt).
x
13. y(x)+ A e µ(x–t) cos[λ(x – t)]y(t) dt = f(x).
a
◦
1 . Solution with |A| >2|λ|:
x
y(x)= f(x)+ R(x – t)f(t) dt,
a

A 2
1 1 2 2
R(x)=exp (µ – A)x sinh(kx) – A cosh(kx) , k = A – λ .
2 4
2k
2 . Solution with |A| <2|λ|:
◦
x

y(x)= f(x)+ R(x – t)f(t) dt,
a
A 2

1 2 1 2
R(x)=exp (µ – A)x sin(kx) – A cos(kx) , k = λ – A .
2 2k 4
1
3 . Solution with λ = ± A:
◦
2
x
1 2 1
y(x)= f(x)+ R(x – t)f(t) dt, R(x)= A x – A exp µ – A x .
2 2
a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 161