I've tried but can't get the solution of this ode by Frobenius method.
$(x^2)y''-6y=0$
I tried with $y=\sum_{k=0}^{\infty}(a_k \cdot x^{(k+r)})$ where $a_k$ is coefficient. I can't find the recurrence relation. If any one finds the recurrence relation, that'll do it.
By Frobenius, the relation between coefficients is
$$x^2k(k-1)a_kx^{k-2}-6a_k=0,$$ i.e. $$a_k=0\lor k(k-1)=6,$$ hence the only nonzero terms are
$$a_3x^3+a_{-2}x^{-2}.$$