MthSc 208, Fall 2011 (Differential Equations)
Dr. Macauley
HW 17
Due Friday April 14th, 2011
(1) Solve the following differential equations:
(a)
y
0
=
ky
(b)
y
0
=

ky
(c)
y
00
=
k
2
y
(d)
y
00
=

k
2
y
(e)
y
00
+ 3
y
0
+ 2
y
= 0
(f)
y
00
+ 2
y
0
+ 2
y
= 0
(g)
y
00
+ 2
y
0
+
y
= 0
(2) Suppose that
f
is a function defined on
R
(not necessarily periodic).
Show that there
is an odd function
f
odd
and an even function
f
even
such that
f
(
x
) =
f
odd
+
f
even
.
Hint
: As a guiding example, suppose
f
(
x
) =
e
ix
, and consider cos
x
=
1
2
(
e
ix
+
e

ix
) and
i
sin
x
=
1
2
(
e
ix

e

ix
).
(3) Express the
y
intercept of
f
(
x
) =
a
0
2
+
∞
X
n
=1
a
n
cos
nx
+
b
n
sin
nx
in terms of the
a
n
’s and
b
n
’s. (
Hint
: It’s
not
a
0
or
a
0
/
2!)
(4) Consider the 2
π
periodic function
f
(
x
) =
a
0
2
+
∞
X
n
=1
a
n
cos
nx
+
b
n
sin
nx
. Write the Fourier
series for the following functions:
(a) The reflection of
f
(
x
) across the
y
axis;
(b) The reflection of
f
(
x
) across the
x
axis;
(c) The reflection of
f
(
x
) across the origin.
(5)
(a) The Fourier series of an odd function consists only of sineterms. What additional
symmetry conditions on
f
will imply that the sine coefficients with even indices will
be zero (i.e., each
b
2
n
= 0)? Give an example of a nonzero function satisfying this
additional condition.
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 Spring '09
 Staufeneger
 Differential Equations, Equations, Fourier Series, Periodic function, Leonhard Euler, Dr. Macauley HW

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