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Differential Equations Quiz
#5 April
1, 2005
Name____________________ R. Hammack Score
______
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(1) Verify that
the set of functions ,
forms a fundamental set of solutions to y''-y'-12y=0.
First, let's confirm that
=
is a solution to the D.E.
'
= -3
and
''
= 9
.
Plugging this in,
y''-y'-12y = 9-(-3
)-12
= 0, so it is a solution.
Next, let's confirm that =
is a solution to the D.E.
'
= 4
and
''
= 16
.
Plugging this in,
y''-y'-12y = 16-4
-12
= 0, so it is a solution.
Also, these functions are linearly independent, for
=
|
|
4
-(-3
)
=4
+3
=7
≠0.
Thus, as two linearly independent solutions to a second order homogeneous D.E.,
these functions form a fundamental set of solutions.
(2) Given that =
is a solution to
y''-7x
y'+16y=0, find another solution
for which the set {
,
}
is a fundamental set of solutions.
First, we divide through by
to put this equation into a standard form.
y''-
y'+
y
= 0