See also: [[Legendre Polynomials]] # Legendre Differential Equation Motivation: Understanding the [[Legendre Polynomials]] The Legendre Differential Equation is the second-order ordinary differential equation: $(1-x^2)\frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + l(l+1)y = 0 $ or $\frac{d}{dx} \left[(1-x^2)\frac{dy}{dx} \right] + l(l+1)y = 0 $ The solutions to this equation are known as [[Legendre Polynomials]]