Series solution of $x{y}^{{}^{\u2033}}+2{y}^{{}^{\prime}}-xy=0$

I get$r(r+1)=0,(r+1)(r+2){c}_{1}=0$ and

$c}_{n+1}=\frac{{c}_{n-1}}{(n+1+r)(n+r+2)$

The first equation gives the indicial roots$r=-1$ and $r=0$ . The case for $r=0$ is fine.

For$r=-1$ , I don't see how ${c}_{1}=0$ is implied by the second equation. The way I see it, since $1+r=0$ when $r=-1,{c}_{1}$ is not necessarily zero here, but this leads to a different solution to what is in the text book. What am I missing? Why is $c}_{1$ forced to be zero?

I get

The first equation gives the indicial roots

For