2011-47.md

course	course_year	question_number	tags	title	year
Methods	IB	47	IB 2011 Methods	Paper 2, Section I, A	2011

The Legendre equation is

$$\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+n(n+1) y=0$$

for $-1 \leqslant x \leqslant 1$ and non-negative integers $n$.

Write the Legendre equation as an eigenvalue equation for an operator $L$ in SturmLiouville form. Show that $L$ is self-adjoint and find the orthogonality relation between the eigenfunctions.