Page 90 - Handbook Of Integral Equations

x

λ
21. e µ(x–t) (sinh x – sinh t) y(t) dt = f(x), 0 < λ <1.
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 1.3.58:
x

λ
(sinh x – sinh t) w(t) dt = e –µx f(x).
a
x
λ
λ
22. e µ(x–t) (sinh x – sinh t)y(t) dt = f(x).
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 1.3.59:
x
λ
λ
(sinh x – sinh t)w(t) dt = e –µx f(x).
a
x

λ
λ

23. e µ(x–t) A sinh x + B sinh t y(t) dt = f(x).
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 1.3.60:
x

λ λ
–µx
A sinh x + B sinh t w(t) dt = e f(x).
a
x
µ(x–t) λ
24. Ae + B sinh x y(t) dt = f(x).
a
µx
λ
This is a special case of equation 1.9.15 with g 1 (x)= Ae , h 1 (t)= e –µt , g 2 (x)= B sinh x,
and h 2 (t)=1.
x

µ(x–t) λ
25. Ae + B sinh t y(t) dt = f(x).
a
µx
This is a special case of equation 1.9.15 with g 1 (x)= Ae , h 1 (t)= e –µt , g 2 (x)= B, and
λ
h 2 (t) = sinh t.
x e µ(x–t) y(t) dt
26. = f(x), 0 < λ <1.
a (sinh x – sinh t) λ
Solution:
sin(πλ) µx d x e –µt cosh tf(t) dt
y(x)= e .
π dx a (sinh x – sinh t) 1–λ
x

λ
λ

27. e µ(x–t) A tanh x + B tanh t y(t) dt = f(x).
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 1.3.77:
x

λ λ
–µx
A tanh x + B tanh t w(t) dt = e f(x).
a
x

β
λ
28. e µ(x–t) A tanh x + B tanh t + C y(t) dt = f(x).
a
λ
The substitution w(x)= e –µx y(x) leads to an equation of the form 1.9.6 with g(x)= A tanh x,
β
g(t)= B tanh t + C:
x
λ β
–µx
A tanh x + B tanh t + C w(t) dt = e f(x).
a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 68