I have already shown that the equation has a regular singular point at $x=0$ and started using the Frobenius method which is the method that I am supposed to use to answer this question. So far I have got it to the point
\begin{multline} \sum_{n=1}^{\infty}(n+\alpha+1)(n+\alpha+2)a_{n-1}x^{n+\alpha-1}-\sum_{n=0}^{\infty}(n+\alpha)(n+\alpha-1)a_nx^{n+\alpha-1}+\\\frac72\sum_{n=1}^{\infty}(n+\alpha-1)a_{n-1}x^{n+\alpha-1}-3/2\sum_{n=0}^{\infty}(n+\alpha)a_nx^{n+\alpha-1}+\frac32\sum_{n=1}^{\infty}a_{n-1}x^{n+\alpha-1}, \end{multline}
But I'm not sure how to get the first terms out and combine the rest of the terms.