How Do You Derive the Second Solution for Legendre Polynomials?

[h.a.y.l.e.y]

Hi,
I have a problem where I am given the Legendre equation and have been told 1 solution is u(x). It asks me to obtain an expression for the second solution v(x) corresponding to the same value of l.
I think it requires Sturm Liouville treatment but don't have a clue how to begin.
Please HELP!

 

[dextercioby]

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

You're looking for Lagrange associate polynomials of the first and second kind.

Daniel.

 

[HallsofIvy]

More generally, if you have a known solution, u(x) to a differential equation, Taking
y(x)= u(x)v(x), where v(x) is an unknown function, and plugging into the equation,you get an equation of order one lower for v. In particular, if you know one solution to a second order equation, this will give you a first order equation for an independent solution v(x).

 

[h.a.y.l.e.y]

OK so I've skimmed that page and its confirmed what I've got in my lecture notes (albeit in a more complex manner!)
The question remains though, how would I be expected to answer the question, given that this question is only worth 9 marks out of a possible 25 on my exam sheet?!
Surely its asking for something a lot more direct, and I'm still in the dark as to how to begin and what to do...

 

[dextercioby]

Take Halls' advice and do what he said.I'm sure u'll get the second indep.solution.

Daniel.

 

[h.a.y.l.e.y]

Yes, sorry I was typing that reply whilst Hall's was posted.
Thanks for your help, I think I know where to go from here
x

 

[tulips]

Can somebody help me in deriving legendre differential function using its generating function