values of the coefficients of the series involved, and then (b) do obtain
the the numerical values of the coefficients
c
1
, ... ,
c
5
to the first
solution,
y
1
(
x
), given by Theorem 6.3 when
c
0
= 1. [
Keep the parts separate.
]
______________________________________________________________________
7.
(10 pts.) Solve the following second order initialvalue problem using
only the Laplace transform machine.
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______________________________________________________________________
8.
(10 pts.) (a)
Given
f
(
x
) =
x
is a nonzero solution to
,
obtain a second, linearly independent solution by reduction of order.
(b) Use the Wronskian to prove the two solutions are linearly independent.
______________________________________________________________________
9.
(10 pts.)
An 8lb weight is attached to the lower end of a coil spring suspended from the ceiling and comes to
rest in its equilibrium position, thereby stretching the spring 0.4 ft.
The weight is then pulled down 6 inches below its
equilibrium position and released at t = 0.
The resistance of the medium in pounds is numerically equal to 2x
′
, where x
′
is
the instantaneous velocity in feet per second.
(a)
Set up the differential equation for the motion and list the initial conditions.
(b)
Solve the initialvalue problem set up in part (a) to determine the displacement of the weight as a function of
time.
___________________________________________________________________________
Bonkers 10 Point Bonus:
Obtain a condition that implies that
will
be an integrating factor of the differential equation
and show how to compute
μ
when that sufficient condition is true.
Say
where your work is for it won’t fit here.
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 Fall '08
 STAFF
 3 pts, 4 pts, 0.4 ft

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