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Math 293 Practice Prelim #3
Spring 2000
(The actual exam will not be as long as this one.)
Integration formulas which may be useful:
Z
x
sin
ax dx
=

x
cos
ax
a
+
sin
ax
a
2
Z
x
cos
ax dx
=
x
sin
ax
a
+
cos
ax
a
2
Z
x
2
sin
ax dx
=

x
2
cos
ax
a
+2
x
sin
ax
a
2
cos
ax
a
3
Z
x
2
cos
ax dx
=
x
2
sin
ax
a
x
cos
ax
a
2

2
sin
ax
a
3
1
. Let
f
(
x
)=
x
2
when

1
≤
x
≤
1, and let
f
(
x
) be periodic with period 2.
(a) Graph
f
on the interval

3
≤
x
≤
3.
(b) Is
f
even, odd, or neither? Is
f
continuous?
(c) For what values of
x
does the Fourier series of
f
actually converge to
f
(
x
)?
(d) Compute the Fourier series of
f
.
2
. Solve the following initialboundary value problem for the heat equation. It is not
necessary to give any derivation or justiﬁcation, for this problem.
u
t
=3
u
xx
,u
(0
,t
)=0
(
π,t
(
x,
0) = 3 sin
x
+ sin 2
x

1
4
sin 5
x.
3
. Consider the second order equation
y
00

2
y
0
+
λy
= 0 with the boundary conditions
y
(0)=0
,
y
(
π
) = 0. Find all the eigenvalues
λ
for which the boundary value problem has
nontrivial solutions, and ﬁnd the nontrivial solutions. For full credit you should consider
each of the three cases
λ>
1,
λ
= 1, and
λ<
1.
4
. Consider the boundary value problem, which is
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This homework help was uploaded on 01/21/2008 for the course MATH 2930 taught by Professor Terrell,r during the Spring '07 term at Cornell University (Engineering School).
 Spring '07
 TERRELL,R
 Math, Differential Equations, Equations, Formulas

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