Question: The equation

is called the Chebyshev equation.

1. . Show that Eq.(i) can be written in the form
   \[-\left[\left(1-x^{2}\right)^{1 / 2} y^{\prime}\right]^{\prime}=\lambda\left(1-x^{2}\right)^{-1 / 2} y,-1
2. . Consider the boundary conditions:
   \[y, y^{\prime} \text { bounded as } x \rightarrow \pm 1\]
   Show that the boundary-value problem \(\{\) (ii) and (iii) \(\}\) is self-adjoint.
3. . It can be shown that the boundary-value problem \(\{\) (ii) and (iii) \(\}\) has eigenvalues \(\lambda_{0}=0, \lambda_{1}=1, \lambda_{2}=4, \ldots, \lambda_{n}=n^{2}, \ldots\) The corresponding

eigenfunctions are the Chebyshev polynomials \(T_{n}(x): T_{0}(x)=1\) \(T_{1}(x)=x, T_{2}(x)=1-2 x^{2}, \ldots .\) Show that

Note that this is a convergent improper integral.

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