= Solution
For the normal-form coefficient $p(x)=-1/(1-x)$, <Abel's identity> gives the <Wronskian>
$$
\boxed{W(x)=C\exp\left[-\int^xp(s)ds\right]=\frac{C}{1-x}.}
$$
Use <reduction of order>: if $y_2=y_1v$, then $W=y_1^2v'$. Absorbing a nonzero Wronskian constant into the second integration constant, the general solution is
$$
\boxed{y(x)=C_1y_1(x)+C_2y_1(x)\int^x\frac{ds}{(1-s)y_1(s)^2}.}
$$
This local representation applies on intervals where $y_1\ne0$, in particular sufficiently near $x=1$ because $y_1(1)=a_0\ne0$.
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