Page 358 - Handbook Of Integral Equations

x

5. y(t)y(x – t) dt =(Ax + B)e λx , A, B >0.
0
Solutions:

√ 1 A A A
y(x)= ± Be λx √ exp – x + erf x ,
πx B B B
2 z 2
where erf z = √ exp –t dt is the error function.
π
0
x
µ λx
2
6. y(t)y(x – t) dt = A x e .
0
Solutions: √
A Γ(µ +1) µ–1 λx
y(x)= ± µ+1 x 2 e .
Γ
2
x

µ–1 µ λx
7. y(t)y(x – t) dt = Ax + Bx e .
0
Solutions:
√
AΓ(µ) µ–2
B µ +1 µ B
y(x)= ± x 2 exp λ – µ x Φ , ; µ x ,
Γ(µ/2) A 2 2 A
where Φ(a, c; x) is the degenerate hypergeometric function (Kummer’s function).

x
2
8. y(t)y(x – t) dt = A cosh(λx).
0
A d x I 0 (λt) dt
Solutions: y(x)= ± √ √ , where I 0 is the modified Bessel function.
π dx 0 x – t
x

9. y(t)y(x – t) dt = A sinh(λx).
0
√
Solutions: y = ± Aλ I 0 (λx), where I 0 is the modified Bessel function.
x

√
10. y(t)y(x – t) dt = A sinh(λ x ).
0
√
x
2
Solutions: y = ± Aπ 1/4 –7/8 3/4 –1/8 I –1/4 λ 1 x , where I –1/4 is the modified Bessel
λ
2
function.
x

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

12. y(t)y(x – t) dt = A sin(λx).
0
√
Solutions: y = ± Aλ J 0 (λx), where J 0 is the Bessel function.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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