I have been given the equation $$y'' -2xy' +(\mu)y = 0 $$ where $\mu$ is a fixed parameter $\geq0$. I solve it with $x_{0} = 0$.
After plugging in $y = \sum_{n=0}^{\infty} a_{n}x^{n} $ into the equation above, I equate the coefficients to $0$:
$$a_{n+2}(n+1)(n+2) + a_{n}(\mu - 2n) = 0$$
I am hoping this is correct.
From this I get the recurrence relation (which I also hope is correct):
$$a_{n+2}=\frac{a_{n}(\mu -2n)}{(n+1)(n+2)}.$$
I expand and get some terms:
$$a_{2}=\frac{a_{0}(\mu)}{(1)(2)}$$
$$a_{3}=\frac{a_{1}(\mu - 2)}{(1)(2)(3)}$$
$$a_{4}=\frac{a_{0}(\mu)(\mu - 4)}{(1)(2)(3)(4)}$$
and so on.
How do I find the first three non-zero terms of each series?
Any help would be appreciated. Thank you.