Page 40 - Handbook Of Integral Equations

x

λx µx µt βx γx γt
23. Ae (e – e )+ Be (e – e ) y(t) dt = f(x).
a
βx
µt
λx
This is a special case of equation 1.9.45 with g 1 (x)= Ae , h 1 (t)= –e , g 2 (x)= Be , and
γt
h 1 (t)= –e .
x

24. A exp(λx + µt)+ B exp[(λ + β)x +(µ – β)t]
a

– (A + B) exp[(λ + γ)x +(µ – γ)t] y(t) dt = f(x).
λx
This is a special case of equation 1.9.47 with g 1 (x)= e .
x

λx λt n
25. e – e y(t) dt = f(x), n =1, 2, ...
a
Solution:
1 λx 1 d
n+1
y(x)= e f(x).
n
λ n! e λx dx
x √

26. e λx – e λt y(t) dt = f(x), λ >0.
a
Solution: x
λt
2 λx –λx d
2 e f(t) dt
y(x)= e e √ .
π dx λx λt
a e – e
x
y(t) dt

27. √ = f(x), λ >0.
a e λx – e λt
Solution:
λt
λ d x e f(t) dt
y(x)= √ .
π dx λx λt
a e – e
x

λt µ
28. (e λx – e ) y(t) dt = f(x), λ >0, 0< µ <1.
a
Solution:
d e f(t) dt sin(πµ)

2 x λt
λx
–λx
y(x)= ke e , k = .
λt µ
dx a (e λx – e ) πµ
x y(t) dt
29. = f(x), λ >0, 0< µ <1.
λt µ
a (e λx – e )
Solution:
λt
λ sin(πµ) d x e f(t) dt
y(x)= .
λt 1–µ
π dx a (e λx – e )
1.2-2. Kernels Containing Power-Law and Exponential Functions
x

λ(x–t)
30. A(x – t)+ Be y(t) dt = f(x).
a
Differentiating with respect to x, we arrive at an equation of the form 2.2.4:
x
λ(x–t)
By(x)+ A + Bλe y(t) dt = f (x).
x
a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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