Page 362 - Handbook Of Integral Equations

x

1
33. y(x)+ A x –λ–1 2 λ λ > – .
y (t) dt = Bx ,
2
0
λ
λ
Solutions: y 1 (x)= β 1 x and y 2 (x)= β 2 x , where β 1,2 are the roots of the quadratic equation
2
Aβ +(2λ +1)β – B(2λ + 1)=0.
x 2
y (t) dt
34. y(x)+ = A.
0 ax + bt
Solutions: y 1 (x)= λ 1 and y 2 (x)= λ 2 , where λ 1,2 are the roots of the quadratic equation
b 2

ln 1+ λ + bλ – Ab =0.
a
2
x y (t) dt
35. y(x)+ A 2 2 = Bx.
0 x + t
Solutions: y 1 (x)= λ 1 x and y 2 (x)= λ 2 x, where λ 1,2 are the roots of the quadratic equation
1 2
1 – π Aλ + λ – B =0.
4
2
x y (t) dt
36. y(x)+ √ = A.
2
0 ax + bt 2
Solutions: y 1 (x)= λ 1 and y 2 (x)= λ 2 , where λ 1,2 are the roots of the quadratic equation
1 dz
2
Iλ + λ – A =0, I = √ .
0 a + bz 2
x λ+1
n n – 2 λ

37. y(x)+ A ax + bt n y (t) dt = Bx .
0
λ
λ
Solutions: y 1 (x)= β 1 x and y 2 (x)= β 2 x , where β 1,2 are the roots of the quadratic equation
1 λ+1
2 2λ n –
AIβ + β – B =0, I = z a + bz n dz.
0
x

λt 2
38. y(x)+ A e y (t) dt = Be λx + C.
a
2
This is a special case of equation 5.8.11 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 + e λx – e λa =0, y 0 = Be λa + C.
2
Au – Bλ
y 0
x
39. y(x)+ A e λ(x–t) 2
y (t) dt = B.
a
This is a special case of equation 5.8.12. By differentiation, this integral equation can be
reduced to the separable ordinary differential equation
2
y + Ay – λy + λB =0, y(a)= B.

x
Solution in an implicit form:
du
y
+ x – a =0.
Au – λu + λB
2
B
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 342