1. ## Explaining Singular Points

I am a bit confused on the topic of singular and ordinary points. (Also, with regular and irregular singular points).

Say if I have the equation

$x^2(9-x^2)y^{''} + \frac{2}{x}y^{'} +4y =0$

Find all singular points of the given equation and determine whether each one is regular or irregular

Thanks for the help!

2. Originally Posted by Oblivionwarrior
I am a bit confused on the topic of singular and ordinary points. (Also, with regular and irregular singular points).

Say if I have the equation

$x^2(9-x^2)y^{''} + \frac{2}{x}y^{'} +4y =0$

Find all singular points of the given equation and determine whether each one is regular or irregular

Thanks for the help!
This ordinary differencial equation has form $p(x)y'' + q(x)y'+r(x)y=0$ where $p(x) = x^2(9-x^2)$, $q(x) = \tfrac{2}{x}$, $4$.

The singular points points are when $p(x) = 0$ and that happens when $x=0,\pm 3$.

Now if $x_0$ is a singular point then it is a regular singular point if and only if $\lim_{x\to x_0}(x-x_0)q(x), \lim_{x\to x_0}(x-x_0)^2r(x)$ are finite limits.

Can you finish the problem now?

3. makes sense, thanks! But should that 4 be $r(x)$? What about irregular points? If the limit isn't finite?

4. Originally Posted by Oblivionwarrior
What about irregular points? If the limit isn't finite?
If the limit does not exist, like being infinite, then the point is an irregular singular point.