Solution
= Solution
Put $s=z-1$ and seek the leading <Frobenius series> term $y\sim s^\rho$. The coefficients of $y''$ and $y'$ become $-2s(1+s)$ and $2+s$, respectively. The lowest power, $s^{\rho-1}$, therefore has coefficient $-2\rho(\rho-1)+2\rho$. The <indicial equation> is
$$
\boxed{\rho(\rho-2)=0,\qquad\rho=0,2.}
$$
The term $-y$ contributes only at the next order and does not enter this equation.