## Confusing indicial equation

Hello everyone!

Now this is a pickle...

$2xy''-(1+2x^2)y'-xy=0$.

It is required to determine the indicial roots of this DE.
$(a) 1, 3/2.$
$(b) 0, 3/2.$
$(c) -1, 2/3.$
$(d) 0, -3/2.$
$(e) \text{None of the above.} \\$

After unifying the powers of the $x$s we get 2 pairs of sums, one pair starts with the index of $k=-2$ and the other with $k=0$.

So if we intend on not replacing any value for k, we gets $r=\frac{1-2k}{2} \text{and} r=-1-k$. But when we replace $k$ for $-1$ or $-2$ we get 2 different sets of values!!

What to do?!