3. If
λ >
0, then nontrivial solutions of (4) are possible. As mentioned earlier, the general
solution of (1) is
y
(
x
) =
c
1
cos
√
λx
+
c
2
sin
√
λx.
(5)
The condition
y
(0) = 0 implies that
c
1
= 0 so solutions will have the form
y
(
x
) =
c
2
sin
√
λx.
(6)
But
y
(
π
) = 0 implies that sin
√
λπ
= 0 which, in turn, implies that
√
λπ
=
nπ,
n
= 1
,
2
,
· · ·
(7)
or
λ
=
n
2
,
n
= 1
,
2
,
· · ·
.
(8)
Thus the solutions of the BVP (4) are (up to a constant)
y
n
(
x
) = sin
nx,
n
= 1
,
2
,
· · ·
.
(9)
•
The
discrete
λ
-values
λ
n
=
n
2
,
n
= 1
,
2
,
· · ·
, are called
eigenvalues
.
•
The corresponding functions
y
n
(
x
) = sin
nx
,
n
= 1
,
2
,
· · ·
, are called
eigenfunctions
. The
function
y
n
(
x
) has
n
−
1 nodes between 0 and
π
.
The reason for the “eigenvalue/eigenfunction” terminology is the following: Following stan-
dard practice, let us rewrite Eq. (1) as
Ly
+
λy
= 0
,
(10)
where
L
now represents the differential operator
d
2
dx
2
. We may rearrange this equation to have
the form
−
Ly
=
λy,
(11)
which may be viewed as an eigenvalue/eigenfunction equation: When operator
−
L
acts on the
function
y
, it produces a multiple of
y
. This is analogous to the matrix eigenvalue/eigenvector
problems that you have studied in linear algebra:
Av
=
λ
v
,
v
∈
R
n
.
(12)
2