 Solution

For the normalized second-order equation $y^{\prime \prime }+P(x)y^{\prime }+Q(x)y=0$, an ordinary point has analytic $P,Q$. A regular singular point is a singular point where $(x-x_0)P$ and $(x-x_0{)}^{2}Q$ extend analytically through $x_0$. The two finite singular points here are $0$ and $1/2$. At zero, the respective limiting coefficients are $1/2,0$; at $1/2$, they are $-1,0$. Thus both are regular singular points, with indicial roots $0,1/2$ and $0,2$, respectively.