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Differential Equations

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Series Solutions for Chebyshev Equations

DIFFEQ-IH2EVS

Consider the Chebyshev equation $(1-x^2)y''-xy'+n^2y=0$. Assume that $n$ is a positive integer.

Which of the following are TRUE?

Select ALL that apply.

A

$x=0$ is an ordinary point and the power series solution $y=\sum_{k=0}^{\infty}a_kx^k$ is deemed to have a radius of convergence $\rho=\infty$ since all coefficients are continuous.

B

$x=0$ is an ordinary point and the radius of convergence for the power series solution $y=\sum_{k=0}^{\infty}a_kx^k$ has a lower bound $1$ since the Taylor series of $\cfrac{x}{1-x^2}$ and $\cfrac{n^2}{1-x^2}$ about the center $x=0$ converge on $(-1,1)$.

C

There is a polynomial solution.

D

There are no polynomial solutions.