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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