The Chebyshev Differential Equation is a second-order linear differential equation that arises in the study of Chebyshev polynomials. It is expressed as y'' - n^2 \fracyx^2 = 0 , where n is a constant. This equation is significant in various fields, including approximation theory and numerical analysis, due to its connection with optimal polynomial approximations. Solutions to the Chebyshev Differential Equation are known as Chebyshev functions, which are important in mathematical analysis and engineering. These functions exhibit properties that make them useful for minimizing the maximum error in polynomial interpolation, thus playing a crucial role in numerical methods and signal processing.