Page 373 - Handbook Of Integral Equations

1 x t
16. y(x)+ f exp[λy(t)] dt = A.
x 0 x
A solution: y(x)= β, where β is a root of the transcendental equation

1
β + Ie λβ – A =0, I = f(z) dz.
0

5.5. Equations With Hyperbolic Nonlinearity

5.5-1. Integrands With Nonlinearity of the Form cosh[βy(t)]

x

1. y(x)+ k cosh[βy(t)] dt = A.
a
This is a special case of equation 5.8.3 with f(y)= k cosh(βy).
x
2. y(x)+ k cosh[βy(t)] dt = Ax + B.
a
This is a special case of equation 5.8.4 with f(y)= k cosh(βy).

x

2
3. y(x)+ k (x – t) cosh[βy(t)] dt = Ax + Bx + C.
a
This is a special case of equation 5.8.5 with f(y)= k cosh(βy).
x

λ
4. y(x)+ k t cosh[βy(t)] dt = Bx λ+1 + C.
a
This is a special case of equation 5.8.6 with f(y)= k cosh(βy).
x

5. y(x)+ g(t) cosh[βy(t)] dt = A.
a
This is a special case of equation 5.8.7 with f(y) = cosh(βy).
x
cosh[βy(t)]
6. y(x)+ dt = A.
ax + bt
0
This is a special case of equation 5.8.8 with f(y) = cosh(βy).
x
cosh[βy(t)]
7. y(x)+ √ dt = A.
2
0 ax + bt 2
This is a special case of equation 5.8.9 with f(y) = cosh(βy).
x

8. y(x)+ k e λt cosh[βy(t)] dt = Be λx + C.
a
This is a special case of equation 5.8.11 with f(y)= k cosh(βy).

x
9. y(x)+ k e λ(x–t) cosh[βy(t)] dt = A.
a
This is a special case of equation 5.8.12 with f(y)= k cosh(βy).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 353