Confusion about series solutions to differential equations

[timetraveller123]

i have used series solutions to differential equations many times but i never really stopped to think why it works i understand that the series solution approximates the solution at a local provided there is no singularity in which frobenius is used but i am not understanding how exactly it works because the series is only valid for local value of x then what is the use of using such a solution and more specifically how is this justified when using it to solve Legendre differential equation is it because we are only interested in the behavior of solution form -1 to 1
thanks for the help

 

[FactChecker]

The series solution is only valid up to the nearest singularity, but that is often enough. The local solution may be all you care about. If you care about the solution around other points beyond the validity of a local series solution, you can expand the series at that point and get a local solution there.

 

[timetraveller123]

ok considering one of the Legendre differential equations

so for this the normal first kind solution is only valid between -1 and 1 ?(where the singularities occur)

also speaking of locality, so in the case where there is no singularity the series solution becomes the global solution
say for something like
$\frac{d y }{d t} = t$
series solution gives us $y = e^t$ which also happens to be our global solution
or is this because the radius of convergence of e^t happens to be infinity that's why the local series soltuion become the global solution also which one is it? thanks you have already helped me clear up a lot

 

[FactChecker]

ok considering one of the Legendre differential equations
View attachment 224426
so for this the normal first kind solution is only valid between -1 and 1 ?(where the singularities occur)

Yes. Exactly correct.

also speaking of locality, so in the case where there is no singularity the series solution becomes the global solution
say for something like
$\frac{d y }{d t} = t$
series solution gives us $y = e^t$ which also happens to be our global solution
or is this because the radius of convergence of e^t happens to be infinity that's why the local series soltuion become the global solution also which one is it? thanks you have already helped me clear up a lot

The convergence of the series is up to the nearest singularity in the complex plane. So if there are no singularities in the complex plane, the series solution is good everywhere. A factor of $(1+x^2)$ in your first example would probably be a problem in the complex plane because of $x= \pm i$.

 

[timetraveller123]

oh wow i see thanks are most applications of legendre de only concerned between -1 to 1 like the angular solution of hydrogen

edit:
also what would happen
$(1 + x^2) \frac{d y }{d x} + y =0$ if i blindly apply series solution to this without considering the complex singularity would the solution be valid for just real numbers? thanks once again

 

[FactChecker]

also what would happen
$(1 + x^2) \frac{d y }{d x} + y =0$ if i blindly apply series solution to this without considering the complex singularity would the solution be valid for just real numbers? thanks once again

A power series will converge for numbers in a disk of the complex plane up to the closest singularity and diverge outside of the disk. So the singularities at $z = \pm i$ would mean that the expansion at $z=0$ would also not converge for real numbers outside of [-1, +1]. The reason for it not converging for the real numbers would not be apparent unless you consider the complex singularities.

 

[timetraveller123]

oh now i get it is it like two dimensional circle whose radius is the distance to the nearest singularity?

 

[FactChecker]

oh now i get it is it like two dimensional circle whose radius is the distance to the nearest singularity?

Yes. It's the part of the real line that is in the disk of convergence in the complex plane.

 

[timetraveller123]

thanks sir now there is one last thing i don't really know how the legendre polynomial of second kind is derived it would be of help if you can enlighten me on that
i doubt it would not be the series method how is it derived? thanks

 

[FactChecker]

They can be very easily derived from the power series. It's a very powerful technique. Also, the derivative values of a function at a point b are closely related to the coefficients of the power series expanded at point b, so there are multiple ways to reach the same result. The n'th coefficient, $a_n$, is $a_n = \frac{f^{(n)}(b)}{n!}$

 

[timetraveller123]

sorry for the late reply i am understanding about that but now something has been bugging why and how does frobenius method work and where does the definition of regular singular point come from
thanks