Page 361 - Handbook Of Integral Equations

x

28. y(t)y(ax – t) dt = Ae λx , a ≥ 1.
0
Solutions:
1
A exp(λx/a) dz
y(x)= ± √ , I = √ .
I x 0 z(a – z)
x

µ λx
29. y(t)y(ax – t) dt = Ax e , a ≥ 1.
0
Solutions:
1
A µ–1 µ–1 µ–1
y(x)= ± x 2 exp(λx/a), I = z 2 (a – z) 2 dz.
I 0

x
2
5.1-4. Equations of the Form y(x)+ K(x, t)y (t) dt = F (x)
a
x
2
30. y(x)+ A y (t) dt = Bx + C.
a
By differentiation, this integral equation can be reduced to a separable ordinary differential
equation.
◦
1 . Solution with AB >0:

(k + y a ) exp[2Ak(x – a)] + y a – k B
y(x)= k , k = , y a = aB + C.
(k + y a ) exp[2Ak(x – a)] – y a + k A
2 . Solution with AB <0:
◦

y a B
y(x)= k tan Ak(a – x) + arctan , k = – , y a = aB + C.
k A
3 . Solution with B =0:
◦
C
y(x)= .
AC(x – a)+1
x
2
2
31. y(x)+ k (x – t)y (t) dt = Ax + Bx + C.
a
2
This is a special case of equation 5.8.5 with f(y)= ky .
Solution in an implicit form:
y

2 –1/2
4Au – 2kF(u)+ B – 4AC du = ±(x – a),
y 0

3
3
2
F(u)= 1 u – y , y 0 = Aa + Ba + C.
3 0
x
λ 2
32. y(x)+ A t y (t) dt = Bx λ+1 + C.
a
2
This is a special case of equation 5.8.6 with f(y)= Ay . By differentiation, this integral
equation can be reduced to a separable ordinary differential equation.
Solution in an implicit form:
y
du
(λ +1) + x λ+1 – a λ+1 =0, y a = Ba λ+1 + C.
2
Au – B(λ +1)
y a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 341