Page 418 - Handbook Of Integral Equations

6.6. Equations With Logarithmic Nonlinearity

6.6-1. Integrands With Nonlinearity of the Form ln[βy(t)]

b

1. y(x)+ A ln[βy(t)] dt = g(x).
a
This is a special case of equation 6.8.27 with f(t, y)= A ln(βy).
b

k
µ
2. y(x)+ A t ln [βy(t)] dt = g(x).
a
µ
k
This is a special case of equation 6.8.27 with f(t, y)= At ln (βy).
b

3. y(x)+ A ln(µt) ln[βy(t)] dt = g(x).
a
This is a special case of equation 6.8.27 with f(t, y)= A ln(µt) ln(βy).
b
4. y(x)+ A e λ(x–t) ln[βy(t)] dt = g(x).
a
This is a special case of equation 6.8.28 with f(t, y)= A ln(βy).
b
5. y(x)+ g(x) ln[βy(t)] dt = h(x).
a
This is a special case of equation 6.8.29 with f(t, y) = ln(βy).

b
6. y(x)+ A cosh(λx + µt) ln[βy(t)] dt = h(x).
a
This is a special case of equation 6.8.31 with f(t, y)= A ln(βy).

b
7. y(x)+ A sinh(λx + µt) ln[βy(t)] dt = h(x).
a
This is a special case of equation 6.8.32 with f(t, y)= A ln(βy).
b
8. y(x)+ A cos(λx + µt) ln[βy(t)] dt = h(x).
a
This is a special case of equation 6.8.33 with f(t, y)= A ln(βy).
b
9. y(x)+ A sin(λx + µt) ln[βy(t)] dt = h(x).
a
This is a special case of equation 6.8.34 with f(t, y)= A ln(βy).

6.6-2. Other Integrands
b

10. y(x)+ A ln[βy(x)] ln[γy(t)] dt = h(x).
a
This is a special case of equation 6.8.43 with g(x, y)= A ln(βy) and f(t, y) = ln(γy).
b

11. y(x)+ A y(xt) ln[βy(t)] dt =0.
a
This is a special case of equation 6.8.45 with f(t, y)= A ln(βy).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 398