(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

consider the initial value problem (1-x)y^{,,}+xy^{,}-2y=0 find the series solution up to the term with x^{6}

2. Relevant equations

(1-x)y^{,,}+xy^{,}-2y=0

3. The attempt at a solution

assuming the answer has the form [itex]\Sigma[/itex]a_{n}x^{n}

that gives y^{,,}=[itex]\Sigma[/itex]na_{n}x^{n-1}and y^{,,}=[itex]\Sigma[/itex]n(n-1)a_{n}x^{n-2}then plugging these back in and getting rid of the x^{n-2}and x^{n-1}you get the equation [itex]\Sigma[/itex](n+2)(n+1)a_{n+2}x^{n}-[itex]\Sigma[/itex]n(n+1)a_{n+1}x^{n}+[itex]\Sigma[/itex]na_{n}x^{n}-2[itex]\Sigma[/itex]a_{n}x^{n}so what I'm wondering is how do you replace the a_{n+1}with an a_{n}so you can solve for a_{n+2}

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# Series solution up to a term, power series

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