Confusing indicial equation

Hello everyone!

Now this is a pickle...

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

It is required to determine the indicial roots of this DE.

$\displaystyle (a) 1, 3/2. $

$\displaystyle (b) 0, 3/2. $

$\displaystyle (c) -1, 2/3. $

$\displaystyle (d) 0, -3/2. $

$\displaystyle (e) \text{None of the above.} \\$

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

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

What to do?!