Page 395 - Handbook Of Integral Equations

b
2
6.1-3. Equations of the Form y(x)+ K(x, t)y (t) dt = F (x)
a
b

λ 2
20. y(x)+ A x y (t) dt =0.
a
Solutions:
2λ +1 λ
y 1 (x)=0, y 2 (x)= – x .
A(b 2λ+1 – a 2λ+1 )
b
λ µ 2
21. y(x)+ A x t y (t) dt =0.
a
Solutions:
2λ + µ +1 λ
y 1 (x)=0, y 2 (x)= – x .
A(b 2λ+µ+1 – a 2λ+µ+1 )
b

y (t) dt =0.
22. y(x)+ A e –λx 2
a
Solutions:
2λ –λx
y 1 (x)=0, y 2 (x)= e .
A(e –2λb – e –2λa )

b
23. y(x)+ A e –λx–µt 2
y (t) dt =0.
a
Solutions:
2λ + µ –λx
y 1 (x)=0, y 2 (x)= e .
A[e –(2λ+µ)b – e –(2λ+µ)a ]
b
λ –µt 2
24. y(x)+ A x e y (t) dt =0.
a
λ
This is a special case of equation 6.2.20 with f(x)= Ax and g(t)= e –µt .

b
25. y(x)+ A e –µx λ 2
t y (t) dt =0.
a
λ
This is a special case of equation 6.2.20 with f(x)= Ae –µx and g(t)= t .
1
2
µ
26. y(x)+ A y (t) dt = Bx , µ > –1.
0
µ
This is a special case of equation 6.2.22 with g(t)= A, f(x)= Bx , a = 0, and b =1.
µ
A solution: y(x)= Bx + λ, where λ is determined by the quadratic equation

2
1 2AB B
2
λ + 1+ λ + =0.
A µ +1 2µ +1
b

β
µ
2
27. y(x)+ A t y (t) dt = Bx .
a
β
µ
This is a special case of equation 6.2.22 with g(t)= At and f(x)= Bx .
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 375