Finding Solutions using the Frobenius Method

Question

So I am trying to find the terms of the ODE but I honestly can't seem to get it! I've been trying for over an hour and I'm pretty frustrated! If anyone could help me, I would really appreciate it!

Answer 1

The next step is to write down the recurrence equation for the coefficients.

EDIT: If $y = \sum_{n=0}^\infty a_n x^{n+r}$ is a solution, where $r$ is one of the indicial roots, then

$$\eqalign{y' &= \sum_{n=0}^\infty (n+r) a_n x^{n+r-1}\cr y'' &= \sum_{n=0}^\infty (n+r)(n+r-1) a_n x^{n+r-2} \cr} $$ Add the coefficients of $x^{n+r}$ in $2 x^2 y''$, $-2 x y''$, $3 x y'$, $-3 y'$, $-y$; these will involve $a_n$ and $a_{n+1}$, and the recurrence relation says that sum must be $0$. Note that the indicial equation ensures that you don't have a problem for the lowest $n$.