Definition:Laguerre's Differential Equation

Definition

Laguerre's differential equation is the $2$nd order ODE:

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

Laguerre Polynomial

The Laguerre polynomials are the solutions to Laguerre's differential equation:

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

for $\alpha = n$.

They are of the form:

$\map {L_n} x = e^x \map {\dfrac {\d^n} {\d x^n} } {x^n e^{-x} }$

Also see

• Solution to Laguerre's Differential Equation

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

Source of Name

This entry was named for Edmond Nicolas Laguerre.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Laguerre's differential equation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Laguerre's differential equation