Page 410 - Handbook Of Integral Equations

b
42. y(x)+ f(t)y(t)y(x – t) dt = A cos λx.
a
A solution:
y(x)= p sin λx + q cos λx. (1)

Here p and q are roots of the algebraic system
2
2
2
2
p + I 0 pq + I cs (p + q )=0, q + I cc q – I ss p = A, (2)
where we use the notation introduced in 6.2.41. Different solutions of system (2) generate
different solutions (1) of the integral equation.

6.3. Equations With Power-Law Nonlinearity

b

6.3-1. Equations of the Form G(···) dt = F (x)
a
b
β
λ µ
1. t y (x)y (t) dt = f(x).
a
A solution: 1
1 b β – µ+β
µ λ µ
y(x)= A f(x) , A = t f(t) dt .
a
b

λt µ β
2. e y (x)y (t) dt = f(x).
a
A solution: 1
1 b β – µ+β
µ λt µ
y(x)= A f(x) , A = e f(t) dt .
a
∞ s

k
b
a
c

3. f(x t)t y x t y(t) dt = Ax .
0
A solution:
1
A s+1 λ a + c + ab

y(x)= x , λ = ,
I k – a – as
∞ a + c + as + bk + cs

β
I = f(t)t dt, β = .
k – a – as
0
b
β
6.3-2. Equations of the Form y(x)+ K(x, t)y (t) dt = F (x)
a
b
λ β
4. y(x)+ A t y (t) dt = g(x).
a
λ β
This is a special case of equation 6.8.27 with f(t, y)= At y .
b

µt β
5. y(x)+ A e y (t) dt = g(x).
a
µt β
This is a special case of equation 6.8.27 with f(t, y)= Ae y .
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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