Page 368 - Handbook Of Integral Equations

x
λ–kλ–1
n n k λ
5. y(x)+ A ax + bt n y (t) dt = Bx .
a
λ
A solution: y = βx , where β is a root of the algebraic (or transcendental) equation
1
λ–kλ–1
k
AIβ + β – B =0, I = z kλ a + bz n n dz.
0
x
λt µ
6. y(x)+ A e y (t) dt = Be λx + C.
a
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.
µ
Au – Bλ
y 0
x

y (t) dt = Ae
7. y(x)+ k e λ(x–t) µ λx + B.
a
Solution in an implicit form:
y
dt λa

= x – a, y 0 = Ae + B.
µ
λt – kt – λB
y 0
x

µ
8. y(x)+ k sinh[λ(x – t)]y (t) dt = Ae λx + Be –λx + C.
a
µ
This is a special case of equation 5.8.14 with f(y)= ky .
Solution in an implicit form:
y
2 2 2 2 2 –1/2
λ u – 2λ Cu – 2kλF(u)+ λ (C – 4AB) du = ±(x – a),
y 0
1 µ+1 µ+1 λa –λa
F(u)= u – y 0 , y 0 = Ae + Be + C.
µ +1
x

µ
9. y(x)+ k sinh[λ(x – t)]y (t) dt = A cosh(λx)+ B.
a
µ
This is a special case of equation 5.8.15 with f(y)= ky .
Solution in an implicit form:
y
2 2 2 2 2 2 –1/2
λ u – 2λ Bu – 2kλF(u)+ λ (B – A ) du = ±(x – a),
y 0
1 µ+1 µ+1
F(u)= u – y 0 , y 0 = A cosh(λa)+ B.
µ +1
x

µ
10. y(x)+ k sinh[λ(x – t)]y (t) dt = A sinh(λx)+ B.
a
µ
This is a special case of equation 5.8.16 with f(y)= ky .
Solution in an implicit form:
y

2 2 2 2 2 2 –1/2
λ u – 2λ Bu – 2kλF(u)+ λ (A + B ) du = ±(x – a),
y 0
1 µ+1 µ+1
F(u)= u – y 0 , y 0 = A sinh(λa)+ B.
µ +1
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 348