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$$Q_n(x)=P_n(x)\displaystyle\int^x \dfrac{dx}{(1-x^2)[P_n(x)]^2}$$

The form looks like Green's function, or general solution after the variation of parameters method. I couldnot figure it out. I checked every book I know but didnot find anything related, our lecturer asked this in the exam and I am very curious now.

How should we attack it?

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Hint: Let $Q=Pw$ is the second solution then find $Q'$, $Q''$ and after substitution in Legendre DE, you will find $$(1-x^2)(2P'w'+Pw'')-2xPw'=0$$ or after dividing on $Pw'$ $$2\dfrac{P'}{P}+\dfrac{w''}{w'}=\dfrac{2x}{1-x^2}$$ can you take it from here!

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  • $\begingroup$ thank a lot, it was very trivial, too pity I couldn $\endgroup$ – user2312512851 May 21 at 17:52
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    $\begingroup$ You are welcome :) $\endgroup$ – Nosrati May 21 at 17:58

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