Page 181 - Handbook Of Integral Equations

x

2. y(x)+ A e µ(x–t) sinh[λ(x – t)]y(t) dt = f(x).
a
1 . Solution with λ(A – λ)>0:
◦
x
Aλ µ(x–t)

y(x)= f(x) – e sin[k(x – t)]f(t) dt, where k = λ(A – λ).
k a
2 . Solution with λ(A – λ)<0:
◦
Aλ x µ(x–t)
y(x)= f(x) – e sinh[k(x – t)]f(t) dt, where k = λ(λ – A).
k a

3 . Solution with A = λ:
◦
x
y(x)= f(x) – λ 2 (x – t)e µ(x–t) f(t) dt.
a

x

3. y(x)+ e µ(x–t) A 1 sinh[λ 1 (x – t)] + A 2 sinh[λ 2 (x – t)] y(t) dt = f(x).
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 2.3.18:
x

–µx
w(x)+ A 1 sinh[λ 1 (x – t)] + A 2 sinh[λ 2 (x – t)] w(t) dt = e f(x).
a
x
4. y(x)+ A te µ(x–t) sinh[λ(x – t)]y(t) dt = f(x).
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 2.3.23:

x

w(x)+ A t sinh[λ(x – t)]w(t) dt = e –µx f(x).
a

2.7-2. Kernels Containing Exponential and Logarithmic Functions

x

5. y(x) – A e µt ln(λx)y(t) dt = f(x).
a
µt
This is a special case of equation 2.9.2 with g(x)= A ln(λx) and h(t)= e .
x

6. y(x) – A e µx ln(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) = ln(λt).

x

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

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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