I'm having trouble determining when to ignore the leading term when solving an O.D.E with a regular singular point, and when to solve for the indicial equation.
I think this pretty much comes down to determining which a_n (usually a_1 or a_2) are 0, however there doesn't seem to be any obvious way of figuring this out to me.
My mathematical typing skills are pretty weak, so I've attached an image with my work and with a brief explanation of exactly where I'm confused.
https://i.stack.imgur.com/ANK67.jpg
If any of this isn't allowed or if there's a question that is very similar to this one please let me know - but I promise i tried to research!!
You absolutely need $a_0\ne 0$ for the approach to make sense. If $a_0=0$ just shift the index, $\sum_{n=1}^\infty a_nx^{n+r}=\sum_{n=0}^{\infty}a_{n+1}x^{n+(r+1)}=\sum_{n=0}^\infty \tilde a_nx^{n+\tilde r}$, and you get the same argument, a vicious circle.
However nothing prevents $a_1=0$, which is what apparently happens in this case. The recursion connects $a_n$ to $a_{n-2}$, so the odd coefficients are zero and the series is in the even coefficients.