Page 370 - Handbook Of Integral Equations

x
t
2
16. y(x)+ f y(t) dt = Ax .
x
0
2 2
Solutions: y k (x)= B x , where B k (k = 1, 2) are the roots of the quadratic equations
k
1
2
B ± IB – A =0, I = zf(z) dz.
0
x
t
k
a
17. y(x) – t f y (t) dt =0, k ≠ 1.
0 x
A solution:
1

1+a a+k
y(x)= Ax 1–k , A 1–k = z 1–k f(z) dz.
0
∞

k
18. y(x) – e λt+βx f(x – t)y (t) dt =0, k ≠ 1.
x
A solution:
λ + β 1–k λ + βk
∞
y(x)= A exp x , A = exp z f(–z) dz.
1 – k 0 1 – k
x
k
19. y(x) – e λt+βx f(x – t)y (t) dt =0, k ≠ 1.
–∞
A solution:

λ + β 1–k λ + βk
∞
y(x)= A exp x , A = exp – z f(z) dz.
1 – k 1 – k
0
5.4. Equations With Exponential Nonlinearity

5.4-1. Equations Containing Arbitrary Parameters

x
1. y(x)+ A exp[λy(t)] dt = B.
a
Solution:
1 –Bλ
y(x)= – ln Aλ(x – a)+ e .
λ

x

2. y(x)+ A exp[λy(t)] dt = Bx + C.
a
For B = 0, see equation 5.4.1.
Solution with B ≠ 0:

1 A A λB(a–x)
y(x)= – ln + e –λy 0 – e , y 0 = aB + C.
λ B B

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 350