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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

| | | | ||

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