Page 192 - Handbook Of Integral Equations

2.9. Equations Whose Kernels Contain Arbitrary
Functions

2.9-1. Equations With Degenerate Kernel: K(x, t)= g 1 (x)h 1 (t)+ ··· + g n (x)h n (t)
x
g(x)
1. y(x) – λ y(t) dt = f(x).
a g(t)
Solution:
x
g(x)
y(x)= f(x)+ λ e λ(x–t) f(t) dt.
a g(t)
x

2. y(x) – g(x)h(t)y(t) dt = f(x).
a
Solution:
x x

y(x)= f(x)+ R(x, t)f(t) dt, where R(x, t)= g(x)h(t)exp g(s)h(s) ds .
a t
x
3. y(x)+ (x – t)g(x)y(t) dt = f(x).
a
This is a special case of equation 2.9.11.
1 . Solution:
◦
x

1
y(x)= f(x)+ Y 1 (x)Y 2 (t) – Y 2 (x)Y 1 (t) g(x)f(t) dt, (1)
W
a
where Y 1 = Y 1 (x) and Y 2 = Y 2 (x) are two linearly independent solutions (Y 1 /Y 2 /≡ const) of
the second-order linear homogeneous differential equation Y xx + g(x)Y = 0. In this case, the

Wronskian is a constant: W = Y 1 (Y 2 ) – Y 2 (Y 1 ) ≡ const.

x
x
2 . Given only one nontrivial solution Y 1 = Y 1 (x) of the linear homogeneous differential
◦
equation Y + g(x)Y = 0, one can obtain the solution of the integral equation by formula (1)
xx
with
x
dξ
W =1, Y 2 (x)= Y 1 (x) ,
2
Y (ξ)
b 1
where b is an arbitrary number.
x

4. y(x)+ (x – t)g(t)y(t) dt = f(x).
a
This is a special case of equation 2.9.12.
◦
1 . Solution:
1 x
y(x)= f(x)+ Y 1 (x)Y 2 (t) – Y 2 (x)Y 1 (t) g(t)f(t) dt, (1)
W a
where Y 1 = Y 1 (x) and Y 2 = Y 2 (x) are two linearly independent solutions (Y 1 /Y 2 /≡ const) of
the second-order linear homogeneous differential equation Y xx + g(x)Y = 0. In this case, the

Wronskian is a constant: W = Y 1 (Y 2 ) – Y 2 (Y 1 ) ≡ const.

x
x
◦
2 . Given only one nontrivial solution Y 1 = Y 1 (x) of the linear homogeneous differential

equation Y xx + g(x)Y = 0, one can obtain the solution of the integral equation by formula (1)
with
x dξ
W =1, Y 2 (x)= Y 1 (x) 2 ,
b Y (ξ)
1
where b is an arbitrary number.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 171