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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