Page 131 - Handbook Of Integral Equations

x

2
12. y(x)+ A (xt – t )y(t) dt = f(x).
a
This is a special case of equation 2.9.4 with g(t)= At. Solution:
A x
y(x)= f(x)+ t y 1 (x)y 2 (t) – y 2 (x)y 1 (t) f(t) dt,
W a
where y 1 (x), y 2 (x) is a fundamental system of solutions of the second-order linear homo-

geneous ordinary differential equation y xx + Axy = 0; the functions y 1 (x) and y 2 (x) are
expressed in terms of Bessel functions or modified Bessel functions, depending on the sign
of the parameter A:
For A >0,
√ √ √ √
W =3/π, y 1 (x)= xJ 1/3 3 2 Ax 3/2 , y 2 (x)= xY 1/3 3 2 Ax 3/2 .
For A <0,
√ 3/2 √ 3/2
3
W = – , y 1 (x)= xI 1/3 3 2 |A| x , y 2 (x)= xK 1/3 3 2 |A| x .
2
x
2
13. y(x)+ A (x – xt)y(t) dt = f(x).
a
This is a special case of equation 2.9.3 with g(x)= Ax. Solution:
A x
y(x)= f(x)+ x y 1 (x)y 2 (t) – y 2 (x)y 1 (t) f(t) dt,
W a
where y 1 (x), y 2 (x) is a fundamental system of solutions of the second-order linear homo-
geneous ordinary differential equation y + Axy = 0; the functions y 1 (x) and y 2 (x) are
xx
expressed in terms of Bessel functions or modified Bessel functions, depending on the sign
of the parameter A:
For A >0,
√ √ 3/2 √ √ 3/2
2
2
W =3/π, y 1 (x)= xJ 1/3 3 Ax , y 2 (x)= xY 1/3 3 Ax .
For A <0,
√ 3/2 √ 3/2
2
3
2
W = – , y 1 (x)= xI 1/3 3 |A| x , y 2 (x)= xK 1/3 3 |A| x .
2
x

2
2
14. y(x)+ A (t – 3x )y(t) dt = f(x).
a
This is a special case of equation 2.1.55 with λ = 1 and µ =2.
x

2
15. y(x)+ A (2xt – 3x )y(t) dt = f(x).
a
This is a special case of equation 2.1.55 with λ = 2 and µ =1.
x

2
16. y(x) – (ABxt – ABx + Ax + B)y(t) dt = f(x).
a
This is a special case of equation 2.9.16 with g(x)= Ax and h(x)= B.
Solution:
x

y(x)= f(x)+ R(x, t)f(t) dt,
a
x
1 2 2 2 1 2 2
R(x, t)=(Ax + B)exp A(x – t ) + B exp A(s – t )+ B(x – s) ds.
2 2
t
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 110