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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(
 1 ln(x) 0 1/x
) =

=    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)