Ordinary and Singular Points

Definition

Let $y^{\prime \prime }+p(x)+y^{\prime }+q(x)y=0$ be a 2nd-order ODE

Ordinary point

$p(x),q(x)$ are analytic at $x=x_0$ $\iff p,q$ is rational and no $(x-x_0)$ in the denominators

Singular point

Regular singular point

$(x-x_0)p(x)$ and $(x-x_0{)}^{2}q(x)$ are analytic at $x=x_0$ $\iff p,q$ is rational and no $(x-x_0)$ in the denominators of $(x-x_0)p(x),(x-x_0{)}^{2}q(x)$ $\iff \underset{x\to x_0}{\lim }(x-x_0)p(x),\underset{x\to x_0}{\lim }(x-x_0{)}^{2}q(x)$ are exist

Irregular Singular point

otherwise, an irregular singular point