Page 387 - Handbook Of Integral Equations

x

λx –λx
14. y(x)+ sinh[λ(x – t)]f y(t) dt = Ae + Be + C.
a
1 . This is a special case of equation 5.8.23. The solution of this integral equation is
◦
determined by the solution of the second-order autonomous ordinary differential equation
2
2
y + λf(y) – λ y + λ C =0
xx
under the initial conditions

y(a)= Ae λa + Be –λa + C, y (a)= Aλe λa – Bλe –λa .
x
2 . Solution in an implicit form:
◦
y

2 2 2 2 2 –1/2
λ u – 2λ Cu – 2λF(u)+ λ (C – 4AB) du = ±(x – a),
y 0
u
F(u)= f(t) dt, y 0 = Ae λa + Be –λa + C.
y 0
x

15. y(x)+ sinh[λ(x – t)]f y(t) dt = A cosh(λx)+ B.
a
This is a special case of equation 5.8.14.
Solution in an implicit form:
y
2 2 2 2 2 2 –1/2
λ u – 2λ Bu – 2λF(u)+ λ (B – A ) du = ±(x – a),
y 0
u

F(u)= f(t) dt, y 0 = A cosh(λa)+ B.
y 0
x

16. y(x)+ sinh[λ(x – t)]f y(t) dt = A sinh(λx)+ B.
a
This is a special case of equation 5.8.23.
Solution in an implicit form:
y
2 2 2 2 2 2 –1/2
λ u – 2λ Bu – 2λF(u)+ λ (A + B ) du = ±(x – a),
y 0
u

F(u)= f(t) dt, y 0 = A sinh(λa)+ B.
y 0
x

17. y(x)+ sin[λ(x – t)]f y(t) dt = A sin(λx)+ B cos(λx)+ C.
a
1 . This is a special case of equation 5.8.25. The solution of this integral equation is
◦
determined by the solution of the second-order autonomous ordinary differential equation
2
2
y + λf(y)+ λ y – λ C =0
xx
under the initial conditions
y(a)= A sin(λa)+ B cos(λa)+ C, y (a)= Aλ cos(λa) – Bλ sin(λa).

x
◦
2 . Solution in an implicit form:
y
2 2 2 2 –1/2
λ D – λ u +2λ Cu – 2λF(u) du = ±(x – a),
y 0
u

2
2
2
y 0 = A sin(λa)+ B cos(λa)+ C, D = A + B – C , F(u)= f(t) dt.
y 0
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 367