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Differential Equations Quiz
#8 April
29, 2005
Name____________________ R. Hammack Score
______
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(1) Find the
general solution of the differential equation x y''+y'=x
First, let's find
x y''+y'=0
y''+x
y'=0
A.E.: =0
=
+
ln(x)
Next, apply variation of parameters to find .
Standard Form: y''+y'
= 1
W=Det(
|
|
) =![]() |
=∫
dx=-∫x
ln(x)dx = -
+
(by parts)
=∫
dx=∫x
dx =
Thus =-
+
+
ln(x)
=
SOLUTION: y=+
ln(x)
+
(2) Find the interval of
convergence of the power series
Ratio Test for Absolute Convergence:
=
=
2
|x|
= 2|x|
Thus, we get convergence if 2|x| <
1, or rather if -1/2 < x < 1/2.
What about the endpoint x = 1/2 ? Then the series becomes =
which
is the (convergent) alternating harmonic series.
What about the endpoint x = -1/2 ? Then the series becomes =
which
is the (divergent) harmonic series.
CONCLUSION: Interval of convergence
is (-,
]
Created by Mathematica (May 2, 2005) | ![]() |