Thank you HallsofIvy for your help!
The polynomial equation for r that you get when you plug in y = x^{r} and try to solve the differential equation is:
r_{1}, r_{2} = \frac{-(\alpha-1) \pm \sqrt{(\alpha-1)^{2}-4\beta}}{2}
where r1 and r2 can be real and different, real and equal, or...