Hermite equation

A linear homogeneous second-order ordinary differential equation

$$w''-2zw'+\lambda w=0$$

or, in self-adjoint form,

$$\frac{d}{dz}\left(e^{-z^2}\frac{dw}{dz}\right)+\lambda e^{-z^2}w=0;$$

here $\lambda$ is a constant. The change of the unknown function $w=u\exp(z^2/2)$ transforms the Hermite equation into

$$u''+(\lambda+1-z^2)u=0$$

and after the change of variables

$$w=v\exp(t^2/4),\quad t=z\sqrt2$$

one obtains from the Hermite equation the Weber equation

$$v''+\left(\frac\lambda2+\frac12-\frac{t^2}{4}\right)v=0.$$

For $\lambda=2n$, where $n$ is a natural number, the Hermite equation has among its solutions the Hermite polynomial of degree $n$ (cf. Hermite polynomials),

$$H_n(z)=(-1)^ne^{z^2}\frac{d^n}{dz^n}(e^{-z^2}).$$

This explains the name of the differential equation. In general, the solutions of the Hermite equation can be expressed in terms of special functions: the parabolic cylinder functions or Weber–Hermite functions.

References

[a1]	E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1965)