Page 411 - Handbook Of Integral Equations

b
y (t) dt = g(x).
6. y(x)+ A e λ(x–t) β
a
β
This is a special case of equation 6.8.28 with f(t, y)= Ay .
b

β
7. y(x) – g(x)y (t) dt =0.
a
A solution:
1
b 1–β

β
y(x)= λg(x), λ = g (t) dt .
a
For β > 0, the equation also has the trivial solution y(x) ≡ 0.
b

β
8. y(x) – g(x)y (t) dt = h(x).
a
β
This is a special case of equation 6.8.29 with f(t, y)= –y .
b
β
9. y(x)+ A cosh(λx + µt)y (t) dt = h(x).
a
β
This is a special case of equation 6.8.31 with f(t, y)= Ay .
b

β
10. y(x)+ A sinh(λx + µt)y (t) dt = h(x).
a
β
This is a special case of equation 6.8.32 with f(t, y)= Ay .
b

β
11. y(x)+ A cos(λx + µt)y (t) dt = h(x).
a
β
This is a special case of equation 6.8.33 with f(t, y)= Ay .
b
β
12. y(x)+ A sin(λx + µt)y (t) dt = h(x).
a
β
This is a special case of equation 6.8.34 with f(t, y)= Ay .
∞ t

2
13. y(x)+ f y(t) dt = Ax .
0 x
2 2
Solutions: y k (x)= β x , where β k (k = 1, 2) are the roots of the quadratic equations
k
∞

2
β ± Iβ – A =0, I = zf(z) dz.
0
∞ t β

λ
14. y(x) – t f y(t) dt =0, β ≠ 1.
0 x
A solution:
1+λ ∞ λ+β
y(x)= Ax 1–β , A 1–β = z 1–β f(z) dz.
0
∞ β

λt

15. y(x) – e f(ax + bt) y(t) dt =0, b ≠ 0, aβ ≠ –b.
–∞
A solution:
aλ 1–β λb
∞
y(x)= A exp – x , A = exp z f(bz) dz.
aβ + b aβ + b
–∞
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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