Definition:Legendre's Differential Equation

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

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

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


 * $\left({1 - x^2}\right) \ddot y - 2 x \dot y + p \left({p + 1}\right) y = 0$

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