Power series when to use Frobenius method

[John777]

Hi, I'm new to the forum and need some help regarding my calc class. Any help you could provide would be greatly appreciated.

In doing a power series series solution when should I use the frobenius method and when should I use the simple power series method. The simple method seems a little faster, but I know there is a certain type of problem where you must use frobenius.

Frobenius being y=$$\Sigma$$A_nX^n+s

Regular method being y =$$\Sigma$$A_nXⁿ

 

[kof9595995]

These 2 are equivalent

 

[John777]

These 2 are equivalent

Don't take this the wrong way as I'm just trying to learn, but why do they teach both methods? There is no difference between them?

 

[Mute]

Don't take this the wrong way as I'm just trying to learn, but why do they teach both methods? There is no difference between them?

There is a difference between them, but for differential equations without a singularity at some value of x the difference disappears because you will be forced to conclude s = 0.

When you have a differential equation with a singularity at some value of x, you will find a non-trivial value of s when you do a power series around the singular point.

i.e., if you have a singularity at a point x = c, you would plug in a series

$$y = \sum_{n=0}^\infty A_n(x-c)^{n+s}$$

and you would get s = some non-zero number. If there were no singularity at x = c, you would find s = 0.

 

[Trifis]

Can you explain what correction does the x^s factor contribute exactly? I don't see why the Frobenius method improves the failing ordinary power series method.

 

[HallsofIvy]

You use "Frobenius" method when the point about which you are exanding (the "$x_0$" in $\sum a_n(x-x_0)^n$) is a "regular singular point". That means that the leading coefficient has a singularity there, but not "too bad" a singularity: essentially that is acts like $(x- x_0)^{-n}$ for nth order equations but no worse. Every DE text I have seen explains all that.