Consider a first order differential equation. What do ordinary and singular points mean? What do they represent? (I cannot understand their formal definitions so please explain with examples. Thank you.)
UPDATE
After some research I noticed something. Consider the direction field of the $x\dot{y}=y-1$:
I am pretty sure that an ordinary point should have one and only one (existence and uniqueness) line element (arrows in the picture).
Therefore all points except P(0,1) are ordinary points. On the other hand, P(0,1) is part of more than one integral curve and there are different line elements for each curve. This point violates uniqueness condition.
I think this point is a singular point. However, I don't know if it is regular or irregular. Please correct me if I am wrong and tell me if P(0,1) is irregular or not.
Best Answer
We read from the wikipedia arcticle http://en.wikipedia.org/wiki/Regular_singular_point.
"Consider an ordinary linear differential equation of n-th order: $$\sum_{i=0}^n p_i(z)f^{(i)}(z)=0$$ with $p_i$ meromorphic functions and $p_n(z)=1$."
A meromorphic function is a function that is holomorphic on $\mathbb{C}\setminus D$ where $D$ is a discrete set. And holomorphic or analytic at a point $x_0$ means equivalently: it admit complex derivative at $x_0$, its Taylor series converges to the function in a neighbourhood of $x_0$. What happens in $D$? Those points are called poles and the function goes to infinity like $\frac{1}{z^n}$ for some $n$, and that $n$ is the order of the pole in $x_0$.
Now given an ordinary linear differential equation of order n, a point is $\textbf{ordinary}$ if $p_i$ are analytic around $x_0$, it is a $\textbf{regular singular point}$ if $p_{n-i}$ has a pole of order at most $i$ in $x_0$, and it is a $\textbf{irregular point}$ if none of the above holds.
Is it somewhat clear? Or I just repeated some already know and already read notions?