I am trying to find two linearly independent power series solutions of $y''-xy'+y=0$. Upon solving for $C_{n+2}$ I got the following recursion: $$C_{n+2} = \frac{nC_n-C_n}{(n+2)(n+1)}$$ After that, I found a series of $C$s for $n=0,1,2,3,...$
Theses are some of them: $$C_2 = \frac{-1*C_0}{2!}$$ $$C_3 = 0$$ $$C_4 = \frac{-1*C_0}{4!}$$ $$C_5 = 0$$ $$C_6 = \frac{-1*3*C_0}{6!}$$ $$C_7 = 0$$ $$C_6 = \frac{-1*3*5*C_0}{8!}$$
Now I know that one of the power series solution is $n$ because the odd terms are just $0$. Now, I have difficulty finding the solution for the even terms because the pattern is not obvious. Please someone help me.