Legendre's Equations

Legendre’s equations are a type of differential equation that often appear in physics and engineering, especially when dealing with problems involving spherical symmetry, such as gravitational and electromagnetic fields. They are written in the form \[(1 - x^2) \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + n(n+1) y= 0,\] where \(n\) is a constant. Solutions to these equations, called Legendre functions, are useful in expanding functions over spherical regions. They provide a systematic way to analyze physical phenomena where spherical coordinates and symmetry are important.