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.
(a)
Set
y
=
vx
. Substituting y into the ODE above and simplifying the
algebra results in
Then, setting
w
=
v
′
, and a little additional algebra provides us with the
linear, 1st order, homogeneous equation
An integrating factor for this ODE is
μ
=
x
2
/(
x
2
 1). Solving the ODE, an
additional integration, and a suitable choice of constants allows us to
obtain a second solution
Hence,
f
and
g
are linearly independent functions.
______________________________________________________________________
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.
(a)
Let
x
(
t
) denote the displacement, in feet, from the equilibrium position at time
t
, in seconds. Since the
mass,
m
= 8/32 = 1/4 slug, and the spring constant is
k
= 20 pounds per foot, an initialvalue problem describing the motion
of the mass is
(b)
The ODE is equivalent to x
″
+ 8x
′
+ 80x = 0 with
as the general solution. Using the initial conditions to build a linear system and solving the system provides us with the
solution to the IVP,
.
___________________________________________________________________________
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
 Ode

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