This is what I've got so far:
$$2ty''+y'+2ty=0$$ $$y''+\frac{1}{2t} y + y = 0$$ $$\sum_{n=0}^{\infty}(n+r-1)(n+r)c_n t^{n+r-2} + \frac{1}{2t}\sum_{n=0}^{\infty} (n+r)c_n t^{n+r-1} - \sum_{n=0}^{\infty}c_n t^{n+r}$$ $$\sum_{n=0}^{\infty}(n+r-1)(n+r)c_n t^{n+r-2} + \sum_{n=0}^{\infty} 2(n+r)c_n t^{n+r-2} - \sum_{n=0}^{\infty}c_n t^{n+r}$$ $$\sum_{n=0}^{\infty}(n+r-1)(n+r) +2(n+r)c_n t^{n+r-2} - \sum_{n=0}^{\infty}c_n t^{n+r}$$ $$(r(r-1)+2r) c_{0}t^{r-2}+\sum_{n=1}^{\infty}(n+r-1)(n+r) +2(n+r)c_n t^{n+r-2} - \sum_{n=0}^{\infty}c_n t^{n+r}$$ $$(r^2-r)c_{0}t^{r-2}+\sum_{n=1}^{\infty}(n+r-1)(n+r) +2(n+r)c_n t^{n+r-2} - \sum_{n=0}^{\infty}c_n t^{n+r}$$ $$(r^2-r)c_{0}t^{r-2}+\sum_{n=0}^{\infty}(n+r-1)(n+r) +2(n+r)c_n t^{n+r-1} - \sum_{n=0}^{\infty}c_n t^{n+r}$$ $$(r^2-r)c_{0}t^{r-2}+\sum_{n=0}^{\infty}(n+r-1)(n+r) +2(n+r)c_n t^{n+r-1} -c_n t^{n+r}$$
In the notes my professor provided, he solves for his $r$ values first, but I cannot follow his steps, and other examples I've found on the internet seem to start off this way. Where do I want to go from here? Should I try to pull out my $c_0$ and $c_1$ terms? Or? Do I need to find my potential $r$ values first? If so, can someone elaborate on how I go about this?