Page 202 - Handbook Of Integral Equations

x

32. y(x) – g(x)+ b cosh(λx)+ b(x – t)[λ sinh(λx) – cosh(λx)g(x)] y(t) dt = f(x).
a
This is a special case of equation 2.9.16 with h(x)= b cosh(λx).
Solution:
x
y(x)= f(x)+ R(x, t)f(t) dt,
a
x
G(x) 2 2 H(x) G(s)

R(x, t)=[g(x)+ b cosh(λx)] + b cosh (λx)+ bλ sinh(λx) ds,
G(t) G(t) t H(s)
x b
where G(x)=exp g(s) ds and H(x)=exp sinh(λx) .
a λ
x

33. y(x) – g(x)+ b sinh(λx)+ b(x – t)[λ cosh(λx) – sinh(λx)g(x)] y(t) dt = f(x).
a
This is a special case of equation 2.9.16 with h(x)= b sinh(λx).
Solution:

x
y(x)= f(x)+ R(x, t)f(t) dt,
a
G(x) 2 2 H(x) x G(s)
R(x, t)=[g(x)+ b sinh(λx)] + b sinh (λx)+ bλ cosh(λx) ds,
G(t) G(t) t H(s)
x b

where G(x)=exp g(s) ds and H(x)=exp cosh(λx) .
a λ
x

34. y(x)+ cos[λ(x – t)]g(t)y(t) dt = f(x).
a
Differentiating the equation with respect to x twice yields
x

y (x)+ g(x)y(x) – λ sin[λ(x – t)]g(t)y(t) dt = f (x), (1)

x x
a
x

2
y (x)+ g(x)y(x) – λ cos[λ(x – t)]g(t)y(t) dt = f (x). (2)

xx x xx
a
Eliminating the integral term from (2) with the aid of the original equation, we arrive at
the second-order linear ordinary differential equation
2 2

y xx + g(x)y x + λ y = f (x)+ λ f(x). (3)

xx
By setting x = a in the original equation and (1), we obtain the initial conditions for y = y(x):
y(a)= f(a), y (a)= f (a) – f(a)g(a). (4)

x
x
x
35. y(x)+ cos[λ(x – t)]g(x)h(t)y(t) dt = f(x).
a
The substitution y(x)= g(x)u(x) leads to an equation of the form 2.9.34:
x
u(x)+ cos[λ(x – t)]g(t)h(t)u(t) dt = f(x)/g(x).
a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 181