So, I have this DE which is 2nd order, w/ variable coefficients, it goes;(adsbygoogle = window.adsbygoogle || []).push({});

xy''+(x-5)y'+(x^2-4)y=0 revolving around x_0=4.

I know there's a singular point at 0 and I assume to use a summation y(x)=[∞,Ʃ,n=0] a_n(x-x_0)^n

pardon me I don't know how to type the summation symbol, but that's supposed to be starting at n=0 going to infinity, I'm using x_0 for 'x sub 0', etc.

Well I take the 1st and 2nd derivatives of that sum and plug them into the ODE, having divided the y' and y coefficients by x so that y'' is alone. Then I'm left with the following;

http://files.royw.airpost.net/summm.jpg [Broken]

my problem is, I was told on other problems that any x-terms as coefficients outside (such as the two in blue, to the left of the summation symbols) should be re-written to include the form of the x-term inside the sums. So, I need them to involve (x-4), so I can factor them out later. But maybe my algebra is weak, because I don't know what to do here. I've already stumped two math majors with this situation so I'm thinking maybe I took a wrong turn to lead me into this.

Any help would be appreciated! thanks.

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# Stuck on Taylor series DE

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