
3.8. Equations Containing Arbitrary Functions 391
5
◦
. Suppose p(x) = s(x) = 1 and the function q = q(x) has a continuous derivative. Then
the following asymptotic relations hold for eigenvalues λ
n
and eigenfunctions y
n
(x) as
n → ∞:
p
λ
n
=
πn
x
2
− x
1
+
1
πn
Q(x
1
, x
2
) + O
1
n
2
,
y
n
(x) = sin
πn(x −x
1
)
x
2
− x
1
−
1
πn
(x
1
− x)Q(x, x
2
)
+ (x
2
− x)Q(x
1
, x)
cos
πn(x −x
1
)
x
2
− x
1
+ O
1
n
2
,
where
Q(u, v) =
1
2
Z
v
u
q(x) dx. (10)
◮ Second boundary value problem: the case of a
1
= a
2
= 1 and b
1
= b
2
= 0.
The solution of the second boundary value problem with the initial condition (1) and the
boundary conditions
∂
x
w = g
1
(t) at x = x
1
,
∂
x
w = g
2
(t) at x = x
2
is given by formulas (3)–(4) with
Λ
1
(x, t) =