Page 149 - Handbook Of Integral Equations

x

λ(x+t) 2λt λt
11. y(x)+ Ae – Ae + Be y(t) dt = f(x).
a
λx
λt
The transformation z = e , τ = e , Y (z)= y(x) leads to an equation of the form 2.1.5:
z

Y (z)+ B 1 (z – τ)+ A 1 Y (τ) dτ = F(z), F(z)= f(x),
b
λa
where A 1 = B/λ, B 1 = A/λ, b = e .
x

λ(x+t) 2λt λt
12. y(x)+ Ae + Be + Ce y(t) dt = f(x).
a
λx
λt
The transformation z = e , τ = e , Y (z)= y(x) leads to an equation of the form 2.1.6:
z

Y (z) – (A 1 z + B 1 τ + C 1 )Y (τ) dτ = F(z), F(z)= f(x),
b
λa
where A 1 = –A/λ, B 1 = –B/λ, C 1 = –C/λ, b = e .
x

λ(x–t) µx+λt λx+µt
13. y(x)+ λe + A µe – λe y(t) dt = f(x).
a
This is a special case of equation 2.9.23 with h(t)= A.
Solution:
x
1 d F(t) e 2λt
y(x)= Φ(x) dt ,
e λx dx a e λt t Φ(t)
λ – µ (λ+µ)x
x
Φ(x)=exp A e , F(x)= f(t) dt.
λ + µ a
x

–λ(x–t) λx+µt µx+λt
14. y(x) – λe + A µe – λe y(t) dt = f(x).
a
This is a special case of equation 2.9.24 with h(x)= A.
Assume that f(a) = 0. Solution:
x
x
d e 2λx f(t)
y(x)= w(t) dt, w(x)= e –λx Φ(t) dt ,
dx Φ(x) e λt
a a t
λ – µ (λ+µ)x

Φ(x)=exp A e .
λ + µ
x
λ(x–t) βt µx+λt λx+µt
15. y(x)+ λe + Ae µe – λe y(t) dt = f(x).
a
βt
This is a special case of equation 2.9.23 with h(t)= Ae .
Solution:
x
d F(t) e (2λ+β)t
–(λ+β)x
y(x)= e Φ(x) dt ,
dx a e λt t Φ(t)
x
λ – µ (λ+µ+β)x
Φ(x)=exp A e , F(x)= f(t) dt.
λ + µ + β a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 128