Confusing indicial equation

Jan 2010
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?!