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);

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

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