# Hermites differential equation

• April 27th 2010, 09:54 AM
surjective
Hermites differential equation
Hello all,

I want to show that a function $w$ is a solution to Hermite's equation

$\frac{d^{2}w}{dx^{2}}-2x\frac{dw}{dx}+2\ell w =0$

if and only if $w$ is a solution to the Sturm-Liouville equation

$\left( e^{-x^{2}}w' \right)'+2\ell e^{-x^{2}}w=0$

Could someone assist me in the right direction.

Thanks.
• April 27th 2010, 11:06 AM
Black
$\left(e^{-x^2}w'\right)'+2\ell e^{-x^2}w=e^{-x^2}w''-2xe^{-x^2}w'+2\ell e^{-x^2}w=0$.

Divide both sides of the equation by the exponential and you get the first equation.