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PHY2061
R. D. Field
Department of Physics
chp36_11.doc
University of Florida
Another Differential Equation
Consider the
2
nd
order
differential equation
dxt
dt
D
dx t
Cx t
2
2
0
()
++
=
,
where
C
and
D
are constants.
We solve this equation by turning it into an
algebraic equation
by looking for a solution of the form
xt
Ae
at
=
.
Substituting this into the differential equation yields,
aD
a
C
2
0
=
or
a
DD
C
=−
±
−
22
2
.
Case I (C > (D/2)
2
, damped oscillations):
For
C > (D/2)
2
,
i
C
D
D
i
±
−
±
′
/(
/
)
/
2
2
ω
, where
′
CD
(/
)
2
2
, and the most general solution has the form:
e
Be
e
A
t
B
t
t
t
D
t
it
Dt
co
s
(
)
s
in
(
)
s
(
)
s
(
)
/
/
/
/
=+
=
′
+
′
=
′ +
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 Spring '08
 FRY
 Physics

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