Is the legendre equation an example of a frobenius equation?

[asdf1]

is the legendre equation an example of a frobenius equation?

 

[saltydog]

is the legendre equation an example of a frobenius equation?

Legendre's equation:

$$(1-x^2)y^{''}-2xy^{'}+n(n+1)y=0$$

is not singular at x=0 and thus is not in the form generally suited to the Frobenius Method. Usually equations of the form:

$$y^{''}+\frac{b(x)}{x}y^{'}+\frac{c(x)}{x^2}y=0$$

are solved using the Frobenius method. In the case of Legendre's equation, a solution of the form:

$$y(x)=\sum_{n=0}^{\infty}a_n x^n$$

is used, that is integer powers whereas the Frobenius method allows for non-integer powers.

 

[Tide]

I wasn't aware that there was such a thing as a "Frobenius Equation." It is properly referred to as the "method of Frobenius" which can be applied to a wide variety of linear differential equations including Legendre's DE. It may not be the most convenient, in specific cases, but the method is of general applicability.

 

[HallsofIvy]

Legendre's equation is singular at x= 1 and x= -1 and Frobenius' method could be used to find series solutions about those points.

 

[saltydog]

Yep, yep. That's very good. Thank you both for clearing that up for me. Asdf, I request you solve Legendre's equation about one of the singular points using the method of Frobenius (you brought it up). Please provide a complete report with plots by next Monday morning. Oh I could do it, that would be fun. I use big paper for things like that, write very neat, slowly, constantly referring to references in the matter, and take breaks during the process. And if I get stuck, put it up and work on simpler ones preferably some that are already worked out in some textbook then a few on my own.

Oh yea, I'd use Mathematica to check my work.

 

[asdf1]

ORZ... taylor series stuff are one annoying pain...