Page 45 - Handbook Of Integral Equations

x

7. A cosh(λx)+ B cosh(µt)+ C y(t) dt = f(x).
a
This is a special case of equation 1.9.6 with g(x)= A cosh(λx) and h(t)= B cosh(µt)+ C.
x

8. A 1 cosh[λ 1 (x – t)] + A 2 cosh[λ 2 (x – t)] y(t) dt = f(x).
a
The equation is equivalent to the equation
x

B 1 sinh[λ 1 (x – t)] + B 2 sinh[λ 2 (x – t)] y(t) dt = F(x),
a
x

A 1 A 2
B 1 = , B 2 = , F(x)= f(t) dt,
λ 1 λ 2 a
of the form 1.3.41. (Differentiating this equation yields the original equation.)
x
2
9. cosh [λ(x – t)]y(t) dt = f(x).
a
Differentiation yields an equation of the form 2.3.16:
x

y(x)+ λ sinh[2λ(x – t)]y(t) dt = f (x).
x
a
Solution:
2λ 2 x √

y(x)= f (x) – sinh[k(x – t)]f (t) dt, where k = λ 2.
x t
k a
x
2 2

10. cosh (λx) – cosh (λt) y(t) dt = f(x), f(a)= f (a)=0.
x
a
1 d f (x)

x
Solution: y(x)= .
λ dx sinh(2λx)
x
2 2
11. A cosh (λx)+ B cosh (λt) y(t) dt = f(x).
a
2
For B =–A, see equation 1.3.10. This is a special case of equation 1.9.4 with g(x)=cosh (λx).
Solution:
2A x 2B
1 d – –

y(x)= cosh(λx) A+B cosh(λt) A+B f (t) dt .
t
A + B dx
a
x

2 2
12. A cosh (λx)+ B cosh (µt)+ C y(t) dt = f(x).
a
2
2
This is a special case of equation 1.9.6 with g(x)= A cosh (λx), and h(t)= B cosh (µt)+ C.
x

13. cosh[λ(x – t)] cosh[λ(x + t)]y(t) dt = f(x).
a
Using the formula
1
cosh(α – β) cosh(α + β)= [cos(2α) + cos(2β)], α = λx, β = λt,
2
we transform the original equation to an equation of the form 1.4.6 with A = B =1:
x
[cosh(2λx) + cosh(2λt)]y(t) dt =2f(x).
a
Solution: x
d 1 f (t) dt

t
y(x)= √ √ .
dx cosh(2λx) a cosh(2λt)
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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