
856 SECOND-ORDER ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES
and different boundary conditions in y. We assume that the coefficients of equation (1) and
the boundary conditions (2) meet the requirement
a(x), b(x), c(x) are continuous functions; a(x) > 0, |s
1
| + |k
1
| > 0, |s
2
| + |k
2
| > 0,
where x
1
≤ x ≤ x
2
.
In the general case, the Green’s function can be represented as
G(x, y, ξ, η) = ρ(ξ)
∞
X
n=1
u
n
(x)u
n
(ξ)
ku
n
k
2
Ψ
n
(y, η; λ
n
). (3)
Here,
ρ(x) =
1
a(x)
exp
Z
b(x)
a(x)
dx
, ku
n
k
2
=
Z
x
2
x
1
ρ(x)u
2
n
(x) dx, (4)
and the λ
n
and u
n
(x) are the eigenvalues and eigenfunctions of the homogeneous boundary
value problem for the ordinary differential equation
a(x)u
′′
xx
+ b(x)u
′
x
+ [λ +