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Differential Equations                                        Quiz #7                                                      April 13, 2005

Name____________________                   R.  Hammack                                                 Score ______
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(1)    Find the general solution of the differential equation  y' '- y = x + 4

Looking at the associated homogeneous equation   y''-y = 0,
you can see that its auxiliary equation is -1 = 0, with roots 1 and -1.
It follows that the complementary function is = +.

Now, the D.E. that we want to solve is (-1)y = x + 4,
so we apply the annihilator operator to get rid of the function  x + 4:

the roots are -1, 1, 1, 1, 0 so the general solution is
y = ++ A x + B + C,
with = + and =A x +B +C.

Note ' = A x + A + B + 2B x
And   '' = A x + A + A + B + 2B x + 2B x + 2B
Or  '' = (2A + 2B)+ (A x+4B) x + B

Plugging into  y''-y = x + 4 gives
(2A + 2B)+(A x + 4B) x + B -(A x + B +C) = x + 4
(2A + 2B)+ 4B x -C = x + 4

From this we see C = -4, B = 1/4, and A = -1/4.

SOLUTION: y = +- x + -4