Page 359 - Handbook Of Integral Equations

x

√
13. y(t)y(x – t) dt = A sin(λ x ).
0
√
Solutions: y = ± Aπ 1/4 –7/8 3/4 –1/8 J –1/4 λ 1 x , where J –1/4 is the Bessel function.
λ
x
2
2
x
2 µx
14. y(t)y(x – t) dt = A e cosh(λx).
0
A µx d x I 0 (λt) dt
Solutions: y(x)= ± √ e √ , where I 0 is the modified Bessel function.
π dx 0 x – t
x
15. y(t)y(x – t) dt = Ae µx sinh(λx).
0
√
µx
Solutions: y = ± Aλ e I 0 (λx), where I 0 is the modified Bessel function.
x

2 µx
16. y(t)y(x – t) dt = A e cos(λx).
0
A µx d x J 0 (λt) dt
Solutions: y(x)= ± √ e √ , where J 0 is the Bessel function.
π dx x – t
0
x

17. y(t)y(x – t) dt = Ae µx sin(λx).
0
√
µx
Solutions: y = ± Aλ e J 0 (λx), where J 0 is the Bessel function.
x
5.1-2. Equations of the Form K(x, t)y(t)y(x – t) dt = F (x)
0
x

λ
k
18. t y(t)y(x – t) dt = Ax , A >0.
0
Solutions:
1/2

AΓ(λ +1) λ–k–1
y(x)= ± λ+1+k λ+1–k x 2 ,
Γ Γ
2 2
where Γ(z) is the gamma function.
x

k
19. t y(t)y(x – t) dt = Ae λx .
0
Solutions:

1/2
A – k+1 λx
y(x)= ± k+1 1–k x 2 e ,
Γ Γ
2 2
where Γ(z) is the gamma function.
x
µ λx
k
20. t y(t)y(x – t) dt = Ax e .
0
Solutions:
1/2

µ–k–1
AΓ(µ +1) λx
y(x)= ± µ+k+1 µ–k+1 x 2 e ,
Γ Γ
2 2
where Γ(z) is the gamma function.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 339