For those that are interested, the problem has been taken from the Elementary Differential Equations and Boundary Value Problems, 8th Edition, by Boyce. It is problem 1 from Section 5.4.

The problem states to find all singular points of the given equation and determine whether each one is regular or irregular.

$\displaystyle

x y'' + (1 - x) y' + x y = 0

$

$\displaystyle

P(x) = x = 0

$

The point x = 0 is a singular point.

So all other points are ordinary points.

$\displaystyle

\lim_{x \to 0} (x - 0) \frac {(1 - x)} {(x)} = 1

$

$\displaystyle

\lim_{x \to 0} (x - 0)^2 \frac {(x)} {(x)} = 0

$

So as my conclusion the two limits are finite numbers and we can say that the point x = 0 is a regular singular point