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UNIVERSITY OF CALIFORNIA
Santa Barbara
Department of Electrical and Computer Engineering
Problem Set No. 8
Issued:
November 26, 2008
Fall 2008
Due:
December 3, 2008
ECE 201A
Final:
Final exam will be held on December 9, 2008 between 8:00 and 11:00 a.m.
All the material
covered will be included in the final.
Exam will be closed book.
Formulas and equations too hard to
remember or derive will be provided.
Please bring a blue book to the exam.
1.
Consider the following differential equation which is an eigenvalue equation.
d
2
dx
2
y
(
x
)
+
k
2
y
(
x
)
=
0
for 0 <
x
< 1
with the boundary condition
y
(0) =
y
(1) = 0.
(a)
Find an analytical solution for this differential equation.
(b)
Discretize the region 0 <
x
< 1 into N+1 equal segments choosing N points such that
x
i
=
i
×Δ
x
for
i
=
1.......
N
where
Δ
x
=
1
N
+
1
This allows us to discretize
y(x)
such that
y
i
=
y
(
x
i
) .
(c)
Discretize the differential equation using the finite difference approximation to the
second derivative with respect
x
, which is
2
11
1
1
22
2
()
2
(
)()
2
ii
i
i
i
i
y
xy
x
y
x
y
y
y
d
yx
dx
x
x
+
−+
−
−
+
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