Solution
= Solution
For the normalized second-order equation $y''+P(x)y'+Q(x)y=0$, an <ordinary point> has analytic $P,Q$. A <regular singular point> is a <singular point> where $(x-x_0)P$ and $(x-x_0)^2Q$ 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 \b[both are regular singular points], with indicial roots $0,1/2$ and $0,2$, respectively.