I was thinking of trying to solve for the solution of Bessel's differential equation from series solution using direct method (i.e. changing all the summation indices to one) but I couldn't solve it. So I found a new method for solving this using frobenius method which I am unfamiliar with. But still I was watching the proof in the youtube where I got confused in a part. According to the video, $\sum_{n=0}^\infty (n+r)(n+r+1)a_{n}x^{n+r+1} + \frac{1}{x} \sum_{n=0}^\infty (n+r)a_{n}x^{n+r-1}+\sum_{n=0}^\infty a_{n}x^{n+r}-p^2\sum_{n=0}^\infty a_{n}x^{n+r-2}=0$
where for third part of summation, it is supposed that n=m-2 and solved while again the indices is changed back to n for simplicity which makes the third part
$\sum_{n=2}^\infty a_{n-2}x^{n+r-2}$
but what i am confused about it if we were solving it by direct method then changing the index for n=0 to n=2 would make
$\sum_{n=2}^\infty a_{n+2}x^{n+r+2}$,isn't it?
So is it wrong about the direct method procedure?
How do we know if we have to solve it through Frobenius method? How do we check for the regular singular point ?? I understood it by the frobenius method too but what I am confused about is that it has different value than what we'd have got from direct method. Am i wrong to think about it this way?