I am currently solving the Frobenius Method for the question $xy'' +y = 0$ given the ICs $y(0) = 0, y'(0)=1$
I have done some work into solving that the first series solution for $y_1 = \sum_{n=0}^\infty {(-1)^n x^{n+1} \over (n+1)!(n)!}$
I am just completely stuck now at working out the second series solution for $y_2$
I've figured out that the we use the smaller root of the indicial equation $r(r-1)=0 \rightarrow{r=0}$
and that by plugging this into the recurrence relation, we get $a_0 =0$ (which can't be the case)
But from here I am just stuck, could someone help me out please?
Many thanks!