I'm trying to solve $$y^{''}+\bigg(-\frac{1}{x}+\frac{1}{x^3}\bigg)y=0$$ using Frobenius method. What I did is following: $$y=\sum_{i=0}a_ix^{i+k}$$ $$y^{''}=\sum_{i=0}(i+k+2)(i+k+1)a_{i+2}x^{i+k}$$ and the original equation after some rearangment becomes $$\sum_{i=0}((i+k+2)(i+k+1)a_{i+2}-a_{i+1}+a_{i+3})x^{i+k}-\frac{a_0}{x}x^k+\frac{a_0}{x^3}x^k+\frac{a_1}{x^2}x^k+\frac{a_2}{x}x^k=0$$
Now comparing the coefficient in the above equation straightaway gives $$a_0=0$$ $$a_1=0$$ $$a_2=a_0$$
Now if I used the summation term I get $a_3=0$ from its first term and from the next term $a_4=0$, same goes for all the higher terms.
So ultimately I'm getting $y=0$ which is wrong. Where am I making the mistake?