)
2
−
2
λ
2
n
i
a
0
y
(
x
)
2
x dx
= 0
You will need to use integration by parts on the second term.
c) If
y
=
J
0
(
λ
n
x
), justify the equality
y
(
a
) = 0 and get
[
y
′
(
a
)]
2
=
2
λ
2
n
a
2
i
a
0
x
[
y
(
x
)]
2
dx
d) Use part c) and the fact that
J
′
0
(
x
) =
−
J
1
(
x
) to show that
b
J
0
(
λ
n
x
)
b
2
=
a
2
J
1
(
α
n
)
2
/
2
(keep in mind that the norm is worked out on the interval [0
, a
] and that the weight function
is
w
(
x
) =
x
). (Note: this is a modi²ed version of problem 36 in section 4.8, if you want to
read the original.)
4) (10 points) As on the transcendental ODE handout, let
E
(
x
) be the unique solution of
y
′
=
y
,
y
(0) = 1, and let
L
(
x
) be the unique solution of
xy
′
= 1,
L
(1) = 0. As with the
handout, you must do this problem using the ODE information ONLY.
a) Show that for all
x
,
L
(
E
(
x
)) =
x
.
b) Show EITHER
E
(
x
1
+
x
2
) =
E
(
x
1
)
E
(
x
2
) OR
L
(
x
1
x
2
) =
L
(
x
1
) +
L
(
x
2
). You don’t need
to do both of these, and I will only grade one (at random) if you choose to submit them
both.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document5) (10 points)
a) Use the eigenfunction method to solve the following PDE on the square 0
≤
x
≤
1,
0
≤
y
≤
1, with boundary conditions
u
(
x,
0) =
u
(
x,
1) =
u
(0
, y
) =
u
(1
, y
) = 0. (Hint: Your
answer will NOT be an inFnite sum.)
4
∂
2
u
∂x
2
+ 5
∂
2
u
∂y
2
= sin(3
πx
) sin(2
πy
)
b) Same set up, di±erent
f
(
x, y
). Use the eigenfunction method to solve the following PDE
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 NormanKatz
 Eigenvalue, eigenvector and eigenspace, wave equation, Eigenfunction, infinite sum, Vibrations of a circular drum

Click to edit the document details