Definition:Legendre's Differential Equation

Definition
Legendre's differential equation is a second order ODE of the form:
 * $\displaystyle \paren {1 - x^2} \frac {\d^2 y} {\d x^2} - 2 x \frac {\d y} {\d x} + p \paren {p + 1} y = 0$

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

 * Solution to Legendre's Differential Equation