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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.
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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