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December 2nd, 2010, 04:55 AM   #1
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laplace - separate variables

I'm supposed to find a nontrivial solution to tx'' + (t-2)x' + x = 0, x(0) = 0. You don't really need to know that but just in case.

I got to this point:
(s+1)X'(s) + 4X(s) = 0

Now I need to separate variables to find a solution but I've been working on this for two days and can't figure out how to get to the correct solution, which is:
X(s) = C/(s+1)^4

So my question is, how do I go from the first equation to the second using separation of variables?

Thanks!
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December 2nd, 2010, 09:03 AM   #2
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Re: laplace - separate variables

(s+1)X'(s) + 4X(s) = 0
(s+1)X'(s) = -4X(s)
X'(s)/X(s) = -4/(s + 1)
ln|X(s)| = -4ln|s + 1|+ lnC = ln|1/(s + 1)^4| + lnC = ln|C/(s + 1)^4|
X(s) = C/(s + 1)^4

This is the second technique (after plain-'ole integration!) in all of the diff EQ books that I have...
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