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Differential Equations Quiz
#9 May
6, 2005
Name____________________ R. Hammack Score
______
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(1) Find find
two linearly independent power series solutions about the ordinary point x=0
of the differential equation y''-2x y'+ y=0.
Try solution y=.
Then y'=
n
,
and y''=
n(n-1)
Plugging this in:
y''+y'+y=0
n(n-1)
-2x
n
+
=
0
n(n-1)
-
n
+
=
0
(k+2)(k+1)
-
2
k
+
=
0
(
(k+2)(k+1)+(1-2k)
)
=
0
Thus: (k+2)(k+1)+(1-2k)
=
0
So =
=
=-
=
=-
=
=-
=
=-
=-
Thus =
(1-
-
-
-
... )
=
=
=
=
=
=
=
=
=
Thus =
(x+
+
+
+
... )