The Hermite differential equation is a second-order linear differential equation commonly expressed as y'' - 2xy' + 2ny = 0 , where n is a non-negative integer. This equation arises in various fields, including physics and engineering, particularly in quantum mechanics and probability theory. Solutions to the Hermite differential equation are known as Hermite polynomials, which are a set of orthogonal polynomials. These polynomials play a significant role in the quantum harmonic oscillator model and are used in statistical mechanics and numerical analysis due to their useful properties in approximating functions.