Chapter 16, Problem 16PS

Chebyshev’sEquation ConsiderChebyshev’s equation

$\left(1-x^2\right)y^{″}-x{y}^{\prime }+k^{2}y=0$

Polynomial solutions of this differential equation are called Chebyshev polynomials and are denoted by T_k(x). They satisfy the recursion equation

$T_{n+1}(x)=2x{T}_n(x)-T_{n-1}(x)$.

Given that $T_0(x)=1\text{and}{T}_1(x)=x$, determine the Chebyshevpolynomials $T_2(x)$, $T_3(x)$ and $T_4(x)$.

Verify that $T_0(x),T_1(x),T_2(x),T_3(x),\text{and}{T}_4(x)$ are solutions of the given differential equation.

Verify the following Chebyshev polynomials.

$T_5(x)=16x^5-20x^3+5x$

$T_6(x)=32x^6-48x^4+18x^2-1$

$T_7(x)=64x^7-112x^5+56x^3-7x$