What is the meaning of "perhaps" here? "[The solution] is valid in the common interval of convergence ... except perhaps for $x=x_0$ ..."

Question

I am unclear about why won't a Frobenius series solution be valid at $x=x_0$ or why is it uncertain at $x=x_0$? In the theorem it is stated as "except perhaps at $x=x_0$". I haven't studied real analysis yet, so I might be missing out important information.

Answer 1

A solution may go to $\infty$ at $_0$.

Example. $x_0 = 0$, consider the differential equation $$ x^2 y''(x)+3xy'(x)+y(x) = 0 . $$ Frobenius series solutions are: $$ y_1(x) = \frac{1}{x} + \dots\\ y_2(x) = \frac{\log x}{x} +\dots $$ so they go to $\infty$ at $x=0$. [In fact in this simple case, the $\dots$ are all zeros.]