Page 385 - Handbook Of Integral Equations

x

2. K(x, t)ϕ t, y(t) dt = f(x).
a

The substitution w(x)= ϕ x, y(x) leads to the linear equation
x

K(x, t)w(t) dt = f(x).
a
x

5.8-2. Equations of the Form y(x)+ K(x, t)G y(t) dt = F (x)
a
x

3. y(x)+ f y(t) dt = A.
a
Solution in an implicit form:
y
du

+ x – a =0.
f(u)
A
x

4. y(x)+ f y(t) dt = Ax + B.
a
Solution in an implicit form:
y
du
= x – a, y 0 = Aa + B.
A – f(u)
y 0
x

2
5. y(x)+ (x – t)f y(t) dt = Ax + Bx + C.
a
1 . This is a special case of equation 5.8.19. The solution of this integral equation is
◦
determined by the solution of the second-order autonomous ordinary differential equation
y + f(y) – 2A =0
xx
under the initial conditions
2

y(a)= Aa + Ba + C, y (a)=2Aa + B.
x
◦
2 . Solutions in an implicit form:
y
2 –1/2
4Au – 2F(u)+ B – 4AC du = ±(x – a),
y 0
u

2
F(u)= f(t) dt, y 0 = Aa + Ba + C.
y 0
x
λ

6. y(x)+ t f y(t) dt = Bx λ+1 + C.
a
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.
f(u) – B(λ +1)
y a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 365