Page 196 - Handbook Of Integral Equations

where Y = Y (x) is an arbitrary nontrivial solution of the second-order homogeneous differ-
ential equation

Y xx + g(x)Y + h(x)Y =0
x
satisfying the condition Y (a) ≠ 0.
x
13. y(x)+ (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 2.9.4:

x
u(x)+ (x – t)g(t)h(t)u(t) dt = f(x)/g(x).
a
x
n n–1
14. y(x) – g(x)+ λx + λ(x – t)x [n – xg(x)] y(t) dt = f(x).
a
n
This is a special case of equation 2.9.16 with h(x)= λx .
Solution:
x

y(x)= f(x)+ R(x, t)f(t) dt,
a
G(x) 2n n–1 H(x) x G(s)
n
R(x, t)=[g(x)+ λx ] + λ(λx + nx ) ds,
G(t) G(t) H(s)
t
x

λ
where G(x)=exp g(s) ds and H(x)=exp x n+1 .
a n +1
x

15. y(x) – g(x)+ λ +(x – t)[g (x) – λg(x)] y(t) dt = f(x).

x
a
This is a special case of equation 2.9.16.
Solution:
x

y(x)= f(x)+ R(x, t)f(t) dt,
a
x
e λ(s–t)
2

R(x, t)=[g(x)+ λ]e λ(x–t) + [g(x)] + g (x) G(x) ds,

x
t G(s)
x

where G(x)=exp g(s) ds .
a
x

16. y(x) – g(x)+ h(x)+(x – t)[h (x) – g(x)h(x)] y(t) dt = f(x).

x
a
Solution:
x

y(x)= f(x)+ R(x, t)f(t) dt,
a
x
G(x) 2 H(x) G(s)

R(x, t)=[g(x)+ h(x)] + {[h(x)] + h (x)} ds,
x
G(t) G(t) t H(s)
x x

where G(x)=exp g(s) ds and H(x)=exp h(s) ds .
a a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 175