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 polynomials of order $p$.

Also presented as

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

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

Also known as

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

Also see

• Solution to Legendre's Differential Equation
• Definition:Legendre Polynomial

• Results about Legendre's differential equation can be found here.

Source of Name

This entry was named for Adrien-Marie Legendre.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 18$: Basic Differential Equations and Solutions: $18.12$: Legendre's Equation
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 25$: Legendre's Differential Equation: $25.1$
• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 1$: Introduction: $(8)$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Legendre's differential equation
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Legendre's differential equation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Legendre's differential equation
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Legendre's differential equation