How Do You Derive the Second Solution for Legendre Polynomials?

AI Thread Summary
To derive the second solution for Legendre polynomials, start with the known solution u(x) and apply the Sturm-Liouville theory by substituting y(x) = u(x)v(x) into the Legendre equation. This substitution will yield a first-order differential equation for the unknown function v(x). The discussion emphasizes the importance of understanding the relationship between known solutions and deriving independent solutions. There is also a concern regarding the exam's expectations, suggesting that the question may require a more straightforward approach. Overall, the conversation highlights the need for clarity in applying mathematical concepts to solve differential equations.
h.a.y.l.e.y
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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!
 
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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).
 
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...
 
Take Halls' advice and do what he said.I'm sure u'll get the second indep.solution.

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