| course |
course_year |
question_number |
tags |
title |
year |
Methods |
IB |
36 |
|
4.I.6C |
2004 |
Chebyshev polynomials $T_{n}(x)$ satisfy the differential equation
$$\left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+n^{2} y=0 \quad \text { on } \quad[-1,1],$$
where $n$ is an integer.
Recast this equation into Sturm-Liouville form and hence write down the orthogonality relationship between $T_{n}(x)$ and $T_{m}(x)$ for $n \neq m$.
By writing $x=\cos \theta$, or otherwise, show that the polynomial solutions of ( $\dagger$ ) are proportional to $\cos \left(n \cos ^{-1} x\right)$.