(
f
g
)(
t
)
(b)
Using only the definition of the convolution as a definite integral,
not some fancy transform shenanigans, compute (
f
*
g
)(
t
) when
f
(
t
) = 3
t
2
and
g
(
t
) = 4
t
.
(
f
g
)(
t
)
(c)
Using the Laplace transform table, compute the Laplace transform of
f
*
g
when
f
(
t
) =
t
cos(
t
) and
g
(
t
) = exp(2
t
). [Do not attempt to simplify
the algebra after computing the transform.]
{(
f
g
)(
t
)}(
s
)
______________________________________________________________________
4. (10 pts.)
(a) Suppose that
f
(
t
) is defined for
t
> 0.
What is the definition of the Laplace transform of
f
,
{
f
(
t
)}, in terms of a definite integral??
{
f
(
t
)}(
s
)
(b)
Using only the definition, not the table, compute the Laplace transform of
f
(
t
)
0 ,
if
0 <
t
< 4
3 ,
if
4 <
t
.
{
f
(
t
)}(
s
)
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______________________________________________________________________
5.
(10 pts.)
The equation below has a regular singular point at
x
0
= 0.
x
2
y
xy
(
x
2
1)
y
0
(a) Obtain the indicial equation for the ODE at x
0
= 0 and its two roots. (b) Then use all the information available and
Theorem 6.3 to say what the two nontrivial linearly independent solutions given by theorem look like without attempting to
obtain the coefficients of the power series involved.
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 Fall '08
 STAFF
 Vector Space, #

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