singular points

[renegade05]

Question reads: Find the singular points of the following ODE and classify it as either regular or irregular.

\(\displaystyle cos(x)y''+y'+cot(x)y=0\)

So, I let \(\displaystyle p(x) = 1/cos(x)\) and know that this will have a singular point at \(\displaystyle x=\pi/2+n\pi, n\epsilon Z\)

I like the limit as \(\displaystyle x->\pi/2+n\pi\) of p(x) I find that if n is even the limit goes to -1, and if n is odd the limit goes to 1.

I know this is like beginner calculus stuff.. but I can't seem to draw the connection... Does this limit diverge, meaning that this is an IRREGULAR singular point? Because the limit DNE. or is it a REGULAR singular point because 1 and -1 are both finite values?

thanks!:cool:

 

[HallsofIvy]

Accidental double post.

 

[HallsofIvy]

IF you had limits at the same point, from different directions, giving different results, then you would have "no limit".

But that is not the case here. Here, you have an infinite number of singular points, \(\displaystyle \frac{\pi}{2}+ n\pi\) for infinitely many values of n. The limit at the different points exists but has different values at different points. You need to decide whether the singular point is "regular" or not for each point.

 

[Ishuda]

Re-write the equation as
\(\displaystyle y'' + p_1(x) y' + p_0(x) y=0\)
where
\(\displaystyle p_1(x) = \frac{1}{cos(x)}\)
and
\(\displaystyle p_0(x) = \frac{1}{sin(x)}\)
See
http://en.wikipedia.org/wiki/Regular_singular_point
for example

One distinguishes the following cases:

• Point a is an ordinary point when functions p₁(x) and p₀(x) are analytic at x = a.
• Point a is a regular singular point if p₁(x) has a pole up to order 1 at x = a and p₀ has a pole of order up to 2 at x = a.
• Otherwise point a is an irregular singular point.

The key is in the 'up to'. Suppose p₀ has a pole of order 1/2 at point a and p₁ is analytic at a. Then p₁ has a pole up to order 1 at x=a and p_o has a pole up to order 2 at x=a.

So expand cos(x) and sin(x) around their zeros and see what order poles they have.