Definition:Legendre's Differential Equation

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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

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


Source of Name

This entry was named for Adrien-Marie Legendre.


Sources