I was given the equation $$y''-2xy'+\mu y = 0 $$ where $\mu$ is a parameter $\geq{0}$.
I got the relation of sums: $$ \sum_{n=0}^{\infty} a_{n+2}(n+1)(n+2)x^{n} - 2\sum_{n=0}^{\infty} a_{n}nx^{n} + \mu\sum_{n=0}^{\infty} a_{n}x^{n} = 0 $$
I then got the recurrence relation: $$a_{n+2}=\frac{a_{n}(2n - \mu)}{(n+1)(n+2)}$$
so I got the following even terms: $$a_{2}=\frac{-a_{0}\mu}{1 \cdot 2}$$
$$a_{4}=\frac{a_{2}(4 - \mu)}{3 \cdot 4}=\frac{-a_{0}\mu(4 - \mu)}{1 \cdot 2 \cdot 3 \cdot 4}$$
and the following odd terms:
$$a_{3}=\frac{a_{1}(2 - \mu)}{2 \cdot 3}$$
$$a_{5}=\frac{a_{3}(6 - \mu)}{4 \cdot 5}=\frac{a_{1}(2 - \mu_)(6 - \mu)}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5}$$
However, I am not sure if the term $a_{2}$ is $0$ since the coefficient of $x^{0}$ is $a_{2}(1)(2)$ which must equal $0$ so then $a_{2}=0$ and all of the even terms disappear? I am not sure if this is correct I am getting confused by it. Are all of the even terms zero?
I also have to find the radius of convergence which I used the ratio test for: $$\lim_{n\to\infty} \mid \frac{(n+1)(n+2)}{(2n - \mu)} \mid = \infty$$ so the radius of convergence is infinite? Would this be for both the odd and even series?
I don't have access to the answer to the question, and I am just trying to make sure I understand the concept of the Frobenius method and ensure I am doing it correctly.
And do the series terminate whenever $\mu = 2n $?
Thank you for the help.
