Page 363 - Handbook Of Integral Equations

x

40. y(x)+ k e λ(x–t) 2 λx + B.
y (t) dt = Ae
a
Solution in an implicit form:
y
du λa
= x – a, y 0 = Ae + B.
2
λu – ku – λB
y 0
x
2
41. y(x)+ k sinh[λ(x – t)]y (t) dt = Ae λx + Be –λx + C.
a
2
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
3
3

F(u)= 1 u – y , y 0 = Ae λa + Be –λa + C.
3 0
x

2
42. y(x)+ k sinh[λ(x – t)]y (t) dt = A cosh(λx)+ B.
a
2
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
3
3

F(u)= 1 u – y , y 0 = A cosh(λa)+ B.
3 0
x
2
43. y(x)+ k sinh[λ(x – t)]y (t) dt = A sinh(λx)+ B.
a
2
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
3
3

F(u)= 1 u – y , y 0 = A sinh(λa)+ B.
3 0
x

2
44. y(x)+ k sin[λ(x – t)]y (t) dt = A sin(λx)+ B cos(λx)+ C.
a
2
This is a special case of equation 5.8.17 with f(y)= ky .
Solution in an implicit form:
y

2 2 2 2 –1/2
λ D – λ u +2λ Cu – 2kλF(u) du = ±(x – a),
y 0
2
2
3
2
3

y 0 = A sin(λa)+ B cos(λa)+ C, D = A + B – C , F(u)= 1 u – y .
3 0
x

5.1-5. Equations of the Form y(x)+ K(x, t)y(t)y(x – t) dt = F (x)
a
x

2
45. y(x)+ A y(t)y(x – t) dt = AB x + B.
0
A solution: y(x)= B.
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 343