= Solution
Set $t=y-1$. Then $y^2-1=t(t+2)$, and a <Frobenius solution> has the form
$$
x=t^\sigma\sum_{j=0}^{\infty}a_jt^j,\qquad a_0\ne0.
$$
In the <differential equation>, the lowest power is $t^{\sigma-1}$, whose coefficient is
$$
[2\sigma(\sigma-1)+\sigma]a_0=\sigma(2\sigma-1)a_0.
$$
The <indicial equation> and its <indicial roots> are consequently
$$
\boxed{\sigma(2\sigma-1)=0,\qquad \sigma_1=0,\quad\sigma_2=\frac12.}
$$
Their difference is not an integer, so the <Frobenius method> produces two independent series without a logarithmic term:
$$
\boxed{x_1(y)=\sum_{j=0}^{\infty}a_j(y-1)^j,\qquad x_2(y)=(y-1)^{1/2}\sum_{j=0}^{\infty}b_j(y-1)^j,\quad a_0b_0\ne0.}
$$
Choose a branch of $(y-1)^{1/2}$, or take $y>1$ for real solutions. A general local solution is $C_1x_1+C_2x_2$ by <linear independence> and the <superposition principle>. The analytic factors converge at least for $|y-1|<2$, up to the next singular point at $y=-1$.
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