Finding radius of convergence without cauchy ratio test I am currently taking a differential equations course and we are learning about second order homogeneous equations and finding the general solution using series.
I noticed during lecture, that sometimes my professor is able to find the radius of convergence without using cauchy ratio test. 
For example, I noted down this equation:
$$ y''-2ty'+\lambda y=0$$
where $\lambda$ is a fixed constant. My professor said $2t$ and $\lambda$ are both polynomials, so the equation converges everywhere. Keep in mind that my professor speaks a bit fast, so I could have mistaken what he said above. 
Also, I would ask my professor but he does not like answering questions. 
There are times where my professor does use the ratio test but my question is:
How can we determine the radius of convergence without using the cauchy ratio test? I know there shall be times where you have to use the ratio test but I would like to know the cases in which you can determine the radius of convergence by just looking at the equation. 
 A: There is a theorem which answers this: (I summarize from the book Elementary Differential Equations and Boundary Value Problems)

Consider the second-order ode $y'' + p(t)y' + q(t) y = 0$. 
If $p(t)$ and $q(t)$ are analytic at a point $t_0$ (that is, the Taylor series expansions of $p$ and $q$ around $t_0$ converges to $p(t)$ and $q(t)$, respectively, in some interval around $t_0$) , then the general solution is given by
  \begin{equation}
y(t) = \sum_{n = 0}^{\infty}a_n(t - t_0)^n = a_0y_1(t) + a_1 y_2(t)
\end{equation}
  where $a_0$ and $a_1$ are arbitrary constants, and $y_1$, $y_2$ form a fundamental set of solutions. 
Furthermore, the radius of convergence for each of the series $y_1$ and $y_2$ is at least as large as the minimum of the radii of convergence of the series for $p$ and $q$.

In your example above, $p(t) = -2t$ and $q(t) = \lambda$ are polynomials, so analytic at every $t_0$ with infinite radius of convergence. Hence the solution $y(t)$ has infinite radius of convergence, i.e., converges everywhere.
