I just have a quick question regarding the following ODE:
$$P(x) y'' + Q(x) y' + R(x)=0.$$
To find the singular points of this ODE, is it correct to set $P(x)$ to $0$ and just solve for the $x$ terms? Or is there another method for finding the singular points. I understand you can take the limit of both $Q(x)$ and $R(x)$ over $P(x)$ respectively to check whether a given point is singular/ordinary.
But my main question is, if I am asked to specifically FIND the point, is setting $P(x)$ equal to $0$ and solving for the x values the correct method?
How about for finding ordinary points? Is there a method to find the point, or do I just look at $P(x)$ and think to myself, what values of $x$ can I plug into $P(x)$ such that $P(x)$ will not equal $0$?
Thank you!
Best Answer
Your approach will work for some ODEs, but not all. For instance, the ODE
$$xy''+y'+y=0$$
has one singular point at $x=0$, which your method would find. However, the ODE
$$xy''+xy+\tan(x)y=0$$
has a singularities at $x=0$ and $x=\pi/2+\pi n$. Your method would not find the latter. Why?
The formal definition for singularities starts by rewriting your ODE in the form
$$y''+\frac{Q(x)}{P(x)}y'+\frac{R(x)}{P(x)}y=0$$
We then ask if the functions $\frac{Q(x)}{P(x)}$ and $\frac{R(x)}{P(x)}$ are analytic. If you aren't familiar with the term analytic, it essentially means that the functions have an infinite number of well-defined derivatives (you can take derivatives forever). If either of those functions are not analytic at some point $x_0$, we call that point a singular point. In the first example, we have
$$\frac{Q(x)}{P(x)}=\frac{R(x)}{P(x)}=\frac{1}{x}$$
which is not analytic at $x=0$ since it blows up to $\infty$. In the second example, we have
$$\frac{Q(x)}{P(x)}=1$$
and
$$\frac{R(x)}{P(x)}=\frac{\tan(x)}{x}$$
Note that $\tan(x)$ blows up to $\pm\infty$ at $x=\pi/2+\pi n$. Thus, there are singularities at $x=\pi/2+\pi n$. I suggest you read up on analytic functions if this isn't clear to you. Once you've found all the singular points, all other points are ordinary points.