- #1

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[tex](1 - x)\,y''\,+\,y\,=\,0[/tex]

I get:

[tex]2\,a_2\,+\,a_0\,+\,\sum_{n\,=\,1}^{\infty}\,\left[(n\,+\,2)\,(n\,+\,1)\,a_{n\,+\,2}\,-\,n\,(n\,+\,1)\,a_{n\,+\,1}\,+\,a_n\right]\,x_n\,=\,0[/tex]

Which leads to one equation:

[tex]2\,a_2\,+\,a_0\,=\,0[/tex]

[tex]a_2\,=\,-\frac{1}{2}\,a_0[/tex]

and the recursion formula:

[tex]a_{n\,+\,2}\,=\,\frac{n\,(n\,+\,1)\,a_{n\,+\,1}\,-\,a_n}{(n\,+\,2)\,(n\,+\,1)}[/tex]

Now I am totally lost, just like in this other thread, because I don't know how to move on from this step. Please help