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May 8th, 2014, 12:26 PM   #1
Joined: Oct 2013

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2nd order differential equation - variation of parameters

Hello, I have a short question regarding variation of parameters.

Suppose we have a differential equation $\displaystyle (x-1)y''-xy'+y=(x-1)^2 $, whose fundamental system is $\displaystyle \{x,e^{x}\}$

Now, my question is, if I am to search for the particular solution with the variation of parameters method, should I divide the whole equation by (x-1) first?

I.E. will the system of equations with the fundamental matrix look like this:

$\displaystyle \alpha(x) x+\beta(x) \mathrm{e}^{x}=0 \\
\alpha(x) +\beta(x) \mathrm{e}^{x} =(x-1)^2$

or this:

$\displaystyle \alpha(x) x+\beta(x)\mathrm{e}^{x}=0 \\
\alpha(x) +\beta(x) \mathrm{e}^{x} =(x-1)$

? Any help would be appreciated.
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