Given the equation $$y''-2xy'+\mu y=0$$
$P(x) = -2x$ and $Q(x) = \mu$ so $x_{0}=0$ is an ordinary point.
I have the recurrence relation: $$a_{n+2}=\frac{a_{n}(\mu - 2n)}{(n+1)(n+2)}$$
With this I get the solutions $$y_{1}(x) = a_{0}(1 + \frac{\mu}{1 \cdot 2}x^{2} + \frac{\mu(\mu - 4)}{1 \cdot 2 \cdot 3 \cdot 4}x^{4} + ...)$$
and $$y_{2}(x) = a_{1}(\frac{(\mu - 2)}{1 \cdot 2 \cdot 3}x + \frac{(\mu - 2)(\mu - 6)}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5}x^{3} + \frac{(\mu - 2)(\mu - 6)(\mu - 10)}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7}x^{5} + ...)$$
Are these correct? I struggled with finding the general recurrence relation.
How do I find the radius of convergence when there are two solutions? How would I figure out for what values of $\mu$ the series would terminate?
Any hints and help appreciated :) thank you