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.
Best Answer
As you say, one solution is $y_1(x)=c_1 x$. The other solution has no nice expression, as far as I can tell. According to Wolfram Alpha, it is $$ y_2(x)=c_2 \sqrt{-x^2}\left( 2i \sqrt π\left( \text{Erfi}\left(\sqrt{\frac{x^2}2}\right) + 1\right) - \frac{2i \sqrt2 \,e^{x^2/2}}{\sqrt{x^2}} - 2 \sqrt π\right) $$