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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