The Laguerre differential equation is a second-order linear differential equation commonly encountered in mathematical physics and engineering. It is expressed in the form x y'' + (1 - x) y' + n y = 0 , where n is a non-negative integer. This equation arises in problems involving quantum mechanics, particularly in the context of the hydrogen atom. Solutions to the Laguerre differential equation are known as Laguerre polynomials, which are a set of orthogonal polynomials. These polynomials play a significant role in various applications, including numerical analysis and approximation theory, due to their useful properties in representing functions and solving integrals.