Page 409 - Handbook Of Integral Equations

b
38. y(x)+ f(t)y(x – t)y(t) dt = Ae λx .
a
1 . Solutions:
◦
λx
λx
y 1 (x)= k 1 e , y 2 (x)= k 2 e ,
where k 1 and k 2 are the roots of the quadratic equation
b
2
Ik + k – A =0, I = f(t) dt.
a
βx
2 . The substitution y(x)= e w(x) leads to an equation of the same form,
◦
b

w(x)+ f(t)w(t)w(x – t) dt = Ae (λ–β)x .
a
b
39. y(x)+ f(t)y(t)y(x – t) dt = A sinh λx.
a
A solution:
y(x)= p sinh λx + q cosh λx. (1)
Here p and q are roots of the algebraic system
2
2
2
2
p + I 0 pq + I cs (p – q )= A, q + I cc q – I ss p = 0, (2)
where
b b

I 0 = f(t) dt, I cs = f(t) cosh(λt) sinh(λt) dt,
a a
b b

2 2
I cc = f(t) cosh (λt) dt, I ss = f(t) sinh (λt) dt.
a a
Different solutions of system (2) generate different solutions (1) of the integral equation.
b

40. y(x)+ f(t)y(t)y(x – t) dt = A cosh λx.
a
A solution:
y(x)= p sinh λx + q cosh 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 6.2.39. Different solutions of system (2) generate
different solutions (1) of the integral equation.
b
41. y(x)+ f(t)y(t)y(x – t) dt = A sin 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 )= A, q + I cc q – I ss p = 0, (2)
where
b b

I 0 = f(t) dt, I cs = f(t) cos(λt) sin(λt) dt,
a a
b b

2
2
I cc = f(t) cos (λt) dt, I ss = f(t) sin (λt) dt.
a a
Different solutions of system (2) generate different solutions (1) of the integral equation.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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