Page 93 - Handbook Of Integral Equations

x e µ(x–t)
45. y(t) dt = f(x), 0 < λ <1.
a [ln(x/t)] λ
Solution: x
sin(πλ) µx d f(t) dt
y(x)= e .
µt
π dx a te [ln(x/t)] 1–λ

1.7-3. Kernels Containing Exponential and Trigonometric Functions
x

46. e µ(x–t) cos[λ(x – t)]y(t) dt = f(x).
a
x

Solution: y(x)= f (x) – µf(x)+ λ 2 e µ(x–t) f(t) dt.
x
a
x

47. e µ(x–t) A 1 cos[λ 1 (x – t)] + A 2 cos[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 1.5.8:
x
–µx
A 1 cos[λ 1 (x – t)] + A 2 cos[λ 2 (x – t)] w(t) dt = e f(x).
a
x

2
48. e µ(x–t) cos [λ(x – t)]y(t) dt = f(x).
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 1.5.9.
Solution:
2λ 2 x µ(x–t) √

y(x)= ϕ(x)+ e sin[k(x – t)]ϕ(t) dt, k = λ 2, ϕ(x)= f (x) – µf(x).
x
k a
x

3
49. e µ(x–t) cos [λ(x – t)]y(t) dt = f(x).
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 1.5.15:
x
3
cos [λ(x – t)]w(t) dt = e –µx f(x).
a
x

4
50. e µ(x–t) cos [λ(x – t)]y(t) dt = f(x).
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 1.5.19:
x

4
cos [λ(x – t)]w(t) dt = e –µx f(x).
a
x n
51. e µ(x–t) cos(λx) – cos(λt) y(t) dt = f(x), n =1, 2, ...
a
The right-hand side of the equation is assumed to satisfy the conditions f(a)= f (a)= ··· =

x
f (n) (a)=0.
x
Solution:
(–1) n µx 1 d n+1 –µx
y(x)= e sin(λx) F µ (x), F µ (x)= e f(x).
n
λ n! sin(λx) dx

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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