Page 155 - Handbook Of Integral Equations

x

2 2
36. y(x)+ A exp λ(x – t ) y(t) dt = f(x).
a
Solution:
x

2 2
y(x)= f(x) – A exp λ(x – t ) – A(x – t) f(t) dt.
a
x

2 2
37. y(x)+ A exp λx + βt y(t) dt = f(x).
a
In the case β = –λ, see equation 2.2.36. This is a special case of equation 2.9.2 with
2 2
g(x)= –A exp λx ) and h(t)=exp βt .
∞ √

38. y(x)+ A exp –λ t – x y(t) dt = f(x).
x
√

This is a special case of equation 2.9.62 with K(x)= A exp –λ –x .
x

µ µ
39. y(x)+ A exp λ(x – t ) y(t) dt = f(x), µ >0.
a
µ µ
This is a special case of equation 2.9.2 with g(x)= –A exp λx and h(t)=exp –λt .
Solution:
x
µ µ
y(x)= f(x) – A exp λ(x – t ) – A(x – t) f(t) dt.
a
x 1
t
40. y(x)+ k exp –λ y(t) dt = g(x).
0 x x
This is a special case of equation 2.9.71 with f(z)= ke –λz .
n
For a polynomial right-hand side, g(x)= N A n x , a solution is given by
n=0
N n
A n n –λ
n! n! 1
y(x)= x , B n = n+1 – e n–k+1 .
1+ kB n λ k! λ
n=0 k=0
2.3. Equations Whose Kernels Contain Hyperbolic
Functions

2.3-1. Kernels Containing Hyperbolic Cosine

x
1. y(x) – A cosh(λx)y(t) dt = f(x).
a
This is a special case of equation 2.9.2 with g(x)= A cosh(λx) and h(t)=1.
Solution:
x
A
y(x)= f(x)+ A cosh(λx)exp sinh(λx) – sinh(λt) f(t) dt.
a λ

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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