Page 148 - Handbook Of Integral Equations

x

λ 1 (x–t) λ 2 (x–t)
10. y(x)+ A 1 e + A 2 e y(t) dt = f(x).
a
◦
1 . Introduce the notation
x x
I 1 = e λ 1 (x–t) y(t) dt, I 2 = e λ 2 (x–t) y(t) dt.
a a
Differentiating the integral equation twice yields (the first line is the original equation)
y + A 1 I 1 + A 2 I 2 = f, f = f(x), (1)

y +(A 1 + A 2 )y + A 1 λ 1 I 1 + A 2 λ 2 I 2 = f , (2)
x
x
2
2

y xx +(A 1 + A 2 )y +(A 1 λ 1 + A 2 λ 2 )y + A 1 λ I 1 + A 2 λ I 2 = f . (3)
x
xx
1
2
Eliminating I 1 and I 2 , we arrive at the second-order linear ordinary differential equation with
constant coefficients

y xx +(A 1 + A 2 – λ 1 – λ 2 )y +(λ 1 λ 2 – A 1 λ 2 – A 2 λ 1 )y = f xx – (λ 1 + λ 2 )f + λ 1 λ 2 f. (4)

x
x
Substituting x = a into (1) and (2) yields the initial conditions

y(a)= f(a), y (a)= f (a) – (A 1 + A 2 )f(a). (5)
x
x
Solving the differential equation (4) under conditions (5), we can find the solution of the
integral equation.
2 . Consider the characteristic equation
◦
2
µ +(A 1 + A 2 – λ 1 – λ 2 )µ + λ 1 λ 2 – A 1 λ 2 – A 2 λ 1 = 0 (6)
which corresponds to the homogeneous differential equation (4) (with f(x) ≡ 0). The structure
of the solution of the integral equation depends on the sign of the discriminant
2
D ≡ (A 1 – A 2 – λ 1 + λ 2 ) +4A 1 A 2
of the quadratic equation (6).
If D > 0, the quadratic equation (6) has the real different roots
√ √
1
1
µ 1 = (λ 1 + λ 2 – A 1 – A 2 )+ 1 D, µ 2 = (λ 1 + λ 2 – A 1 – A 2 ) – 1 D.
2 2 2 2
In this case, the solution of the original integral equation has the form
x

µ 1 (x–t) µ 2 (x–t)
y(x)= f(x)+ B 1 e + B 2 e f(t) dt,
a
where
µ 1 – λ 2 µ 1 – λ 1 µ 2 – λ 2 µ 2 – λ 1
B 1 = A 1 + A 2 , B 2 = A 1 + A 2 .
µ 2 – µ 1 µ 2 – µ 1 µ 1 – µ 2 µ 1 – µ 2
If D < 0, the quadratic equation (6) has the complex conjugate roots
√
1
µ 1 = σ + iβ, µ 2 = σ – iβ, σ = (λ 1 + λ 2 – A 1 – A 2 ), β = 1 –D.
2 2
In this case, the solution of the original integral equation has the form
x
σ(x–t) σ(x–t)
y(x)= f(x)+ B 1 e cos[β(x – t)] + B 2 e sin[β(x – t)] f(t) dt.
a
where
1
B 1 = –A 1 – A 2 , B 2 = A 1 (λ 2 – σ)+ A 2 (λ 1 – σ) .
β
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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