Ordinary And Singular Points (Regular and Irregular) of Differential Equations

Let R(x, y, p) be a continuous functions in some neighborhood of the point P = (x0, y0, y1) except P. This point is called a singular point for the differential equation

A point x0 is called a singular point of the linear homogeneous differential equation of order n
y(n)+an−1(x)y(n−1)+⋯+a0(x)y(x)=0,(1)
if at least one of the coefficients 𝑎n-1, …, 𝑎0 is not a holomorphic function at that point. If at a point x0 all the coefficients of Eq.(1) are analytic, then the point is said to be an ordinary point.