# How is this an ordinary point? (Frobenius method for DE's)

1. Nov 17, 2009

### phil ess

1. The problem statement, all variables and given/known data

Find two linearly independent power series solutions for the differential equation y′′ − xy = 0 about the ordinary point x0 = 0. Your answer should include a general formula for the coefficients.

3. The attempt at a solution

Im having trouble seeing how x0 = 0 is an ordinary point (i assume ordinary point means regular singular point?).

For it to be an ordinary point (x-x0)p(x) and (x-x0)q(x) have to be analytic at 0 right?

(x-x0)p(x) = x(0/x) = 0

(x-x0)q(x) = x(-1/x2) = -1/x which is not analytic at 0?

So how can x0=0 be an ordinary point?
Please help!!

2. Nov 17, 2009

### lanedance

so after a quick google to refresh my memory

if the DE is in the form
$$y'' + p(x) y' + q(x) y = 0$$

if p(x) & q(x) are analytic at x0, its an ordinary point, if either diverges, its a singular point

if its a singular point, but both (x-x0)p(x) & (x-x0)2q(x) are analytic at x0 its a regular singular point

http://mathworld.wolfram.com/RegularSingularPoint.html

3. Nov 17, 2009

### HallsofIvy

Well, that's where you are wrong- an "ordinary point" is not a singular point at all! And that means you don't use Frobenius' method, just a regular series solution.

No. For an ordinary point p(x) and q(x) must be analytic.

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