Sunday, 18 August 2013

Form of the Wronskian in the Legendre operator $Lu=(1-x^2)u''-2xu'+\nu (\nu+1)u$.

Form of the Wronskian in the Legendre operator $Lu=(1-x^2)u''-2xu'+\nu
(\nu+1)u$.

:)
The question was in two parts, just having problems with the second half.
Part 1: Consider a second-order, linear differential operator
$Lu=a_2u''-a_1u'+a_0u$. Prove that if $Lu_1=0=Lu_2$ then the Wronskian
$$W=\left\vert \begin{array}{c c} u_1 & u_1' \\ u_2 & u_2'
\end{array}\right\vert$$ satisfies $a_2W'+a_1W=0$.
Part 2: Hence find the form of the Wronskian in the case of the Legendre
oeprator, $$Lu=(1-x^2)u''-2xu'+\nu (\nu+1)u.$$
To be honest, I'm not really sure how to get a Wronskian. Our professor
has just sort of said "this is a Wronskian, let's move on" and since then
I've been trying to figure it out without much success. If anyone could
even point me towards a useful definition of the Wronskian I would be very
greatful. :)

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