
11.7. Higher-Order Linear Equations with Variable Coefficients 1051
Here G = G(x, y, t, τ) is the Green’s function; for t > τ ≥ 0, it satisfies the homogeneous
equation
∂
2
G
∂t
2
+ ψ(x, t)
∂G
∂t
−
n
X
k=0
a
k
(x, t)
∂
k
G
∂x
k
= 0 (5)
with the special semihomogeneous initial conditions
G = 0 at t = τ,
∂
t
G = δ(x − y) at t = τ
(6)
and the corresponding homogeneous boundary conditions
Γ
(1)
m
[G] = 0 at x = x
1
(m = 1, . . . , s),
Γ
(2)
m
[G] = 0 at x = x
2
(m = s + 1, . . . , n).
(7)
The quantities y and τ appear in problem (5)–(7) as free parameters (x
1
≤ y ≤ x
2
), and
δ(x) is the Dirac delta function.
One can verify formula (4) by a straightforward substitution into the equation and the ...