Solving the Hermite Equation: What are the Hermite Polynomials?

[fisico30]

HEllo everyone,

a question about the 2nd order, homogeneous, linear diff .eqn. of order n, called the Hermite equation.

A ODE has a general solution. The BCs and the ICs specify, select a particular solution out of the general solution.

What are the Hermite polynomials? They are polynomials of different order n, that are solutions to the mentioned equation.
But under which BCs or ICs?

I can see how they solve the eqn, but I am not sure what type of solution they represent... general, particular...

thanks
fisico30

 

[physman88]

The Hermite polynomials which can be given by the Rodrigues Formula :

H_{n}(x)=(-1)^{n}e^{x^{2}}\frac{d^{n}}{dx^{n}}e^{-x^{2}}

Which will be a solution to the Hermite Equation:

y''-2xy'+2ny=0

Hope this helps. We just covered this in my math for scientists class.

 

[physman88]

wow. totally just realized that you weren't asking what the polynomials solved. As far as BC's and IC's, I am not sure.

 

[fisico30]

Thanks physman88.

I know that we can generate solutions via the Frobenius method. But my mind is stuck with the idea that a particular solution has to come out of a general solution once BCs are used...