an SL type equation with
p
(
x
) =
e
x/
2
,
q
(
x
) =
1
2
xe
x/
2
,
σ
(
x
) =
1
2
e
x/
2
.
6. The problem is
x
2
y
′′
+
xy
′
+ (2
λ

1)
y
= 0
,
0
< x <
1
,
y
(
x
)
, y
′
(
x
) bounded as
x
→
0+
,
y
(1)

y
′
(1)
.
Solution.
Once again, the only question here is whether this is an SL problem; if it is, it is
singular
because the boundary condition at 0 is not of the regular type and, more importantly, while
p
(
x
) is not
x
2
it
will still turn out to be 0 at 0. If one proceeds as in the previous problem, one finds that the function
μ
by
which one should multiply the ODE is
μ
(
x
) = 1
/x
, after which the ODE can be rewritten in the form
(
xy
′
)
′

1
x
y
+ 2
λy
= 0
,
an SL type equation with
p
(
x
) =
x
,
q
(
x
) =

1
x
,
σ
(
x
) = 2.
7. This was the Quiz 2 problem:
Compute the eigenvalues and eigenfunctions of the following regular SL problem:
y
′′
+
λy
= 0
,
0
< x < π
y
(0) = 0
, y
′
(
π
) = 0
.
Solution.
One sees first there are no nonnegative eigenvalues.
Once this is established, we search for
eigenvalues of the form
l
=
μ
2
,
μ >
0. As done already many times in this course, the general solution of the
ODE is
y
(
x
) =
C
1
cos
μx
+
C
2
sin
μx.
The boundary condition
y
(0) = 0 implies
C
1
= 0 so that
y
(
x
) =
C
2
sin
μx
.
Then
y
′
(
x
) =
C
2
μ
cos
μx
and
y
′
(
π
) = 0 implies
C
2
μ
cos
μπ
= 0.
There will be a non zero solution for
μ
such that cos
μπ
= 0, thus
μπ
= (2
n

1)
π/
2,
n
= 1
,
2
, . . .
. The final answer is:
The eigenvalues are the numbers
λ
n
=
(
2
n

1
2
)
2
,
for
n
= 1
,
2
,
3
,
4
, . . .
;
the corresponding eigenfunctions are the functions
y
n
(
x
) =
C
sin
(2
n

1)
x
2
,
n
= 1
,
2
,
3
, . . .
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3
8. Compute the eigenvalues and eigenfunctions of the following regular SL problem:
y
′′
+
λy
= 0
,
0
< x <
1
y
′
(0) = 0
, y
(1) = 0
.
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 Spring '13
 Schonbek
 Boundary value problem, Sturm–Liouville theory, SL problem, qy + λσy

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