The general form of the Hermite ODE is given by the following.

Hermite_ode := diff(y(x),x,x) = 2*x*diff(y(x),x)-2*n*y(x);

$\mathrm{Hermite\_ode}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=2x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-2ny\left(x\right)$

where n is an integer. The solution of this type of ODE can be expressed in terms of hypergeometric or Whittaker functions.

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{odeadvisor}\right)$

$\left[\mathrm{odeadvisor}\right]$

$\mathrm{odeadvisor}\left(\mathrm{Hermite\_ode}\right)$

$\left[\left[\mathrm{\_2nd\_order}\,\mathrm{\_with\_linear\_symmetries}\right]\right]$

$\mathrm{dsolve}\left(\mathrm{Hermite\_ode}\right)$

$y\left(x\right)=\mathrm{c\_\_1}x\mathrm{KummerM}\left(\frac{1}{2}-\frac{n}{2}\,\frac{3}{2}\,x^2\right)+\mathrm{c\_\_2}x\mathrm{KummerU}\left(\frac{1}{2}-\frac{n}{2}\,\frac{3}{2}\,x^2\right)$

$\mathrm{dsolve}\left(\mathrm{Hermite\_ode}\,\left[\mathrm{hypergeometric}\right]\right)$

$y\left(x\right)=\mathrm{c\_\_1}\mathrm{KummerM}\left(-\frac{n}{2}\,\frac{1}{2}\,x^2\right)+\mathrm{c\_\_2}\mathrm{KummerU}\left(-\frac{n}{2}\,\frac{1}{2}\,x^2\right)$

Abramowitz, M., and Stegun, I. Handbook of Mathematical Functions, section 22.6.21. Dover Publications.