Chebyshev equation

The linear homogeneous second-order ordinary differential equation

$$(1-x^2)\frac{d^2y}{dx^2}-x\frac{dy}{dx}+ay=0$$

or, in self-adjoint form,

$$\sqrt{1-x^2}\frac d{dx}\left(\sqrt{1-x^2}\frac{dy}{dx}\right)+ay=0,$$

where $a$ is a constant. Chebyshev's equation is a special case of the hypergeometric equation.

The points $x=-1$ and $x=1$ are regular singular points (cf. Regular singular point) of the Chebyshev equation. Substituting the independent variable

$$t=\arccos x\quad\text{for }|x|<1,$$

$$t=\operatorname{Arcosh}|x|\quad\text{for }|x|>1$$

reduces this equation to a corresponding linear equation with constant coefficients:

$$\frac{d^2y}{dt^2}+ay=0\quad\text{or}\quad\frac{d^2y}{dt^2}-ay=0,$$

so that Chebyshev's equation can be integrated in closed form. A fundamental system of solutions to Chebyshev's equation on the interval $-1<x<1$ with $a=n^2$, where $n$ is a natural number, consists of the Chebyshev polynomials (of the first kind) of degree $n$,

$$T_n(x)=\cos(n\arccos x),$$

and the functions $U_n(x)=\sin(n\arccos x)$, which are related to Chebyshev polynomials of the second kind. The polynomial $T_n(x)$ is a real solution to Chebyshev's equation on the entire real line, with $a=n^2$. Chebyshev's equation is also studied in complex domains.