# Definition:Legendre's Differential Equation

## Definition

Legendre's differential equation is a second order ODE of the form:

$\paren {1 - x^2} \dfrac {\d^2 y} {\d x^2} - 2 x \dfrac {\d y} {\d x} + p \paren {p + 1} y = 0$

The parameter $p$ may be any arbitrary real or complex number.

Solutions of this equation are called Legendre functions of order $p$.

## Also presented as

Legendre's differential equation can also be written in the form:

$\paren {1 - x^2} \ddot y - 2 x \dot y + p \paren {p + 1} y = 0$

## Also known as

Some sources give it as Legendre's equation, but this can then be confused with the Legendre Equation.

## Also see

• Results about Legendre's Differential Equation can be found here.

## Source of Name

This entry was named for Adrien-Marie Legendre.