singular or ordinary point of a differential equation

Question

Is $x=0$ singular or regular point of the following differential equation

$p_2(x)y''+p_1(x)y'+p_0(x)y=0$

We know that $s_1=x$ and $s_2=x^2$ are two solutions of the equation

I am having trouble figuring out how I can use the solutions to find out if it is a singular or ordinary point. Any help would help alot :)

Answer 1

Consider the general homogeneous second order linear differential equation $$u''+P(x)u'+Q(x)u=0$$ where $z \in D \subseteq \mathbb{C}$.

The point $x_0 \in D$ is said to be an ordinary point of the above the given differential equation if $P(x)$ and $Q(x)$ are analytic at $x_0$.

If either $P(x)$ or $Q(x)$ fails to be analytic at $x_0$, the point $x_0$ is called a singular point of the given differential equation.

A singular point $x_0$ of the given differential equation is said to be regular singular point if the function $(x-x_0)P(x)$ and $(x-x_0)^2 Q(x)$ are analytic at $x_0$ and irregular otherwise.

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