Hermite ODEs - Maple Programming Help

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Hermite ODEs

Description

 • 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);
 ${\mathrm{Hermite_ode}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{2}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{2}{}{n}{}{y}{}\left({x}\right)$ (1)
 where n is an integer. The solution of this type of ODE can be expressed in terms of hypergeometric or Whittaker functions.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (2)
 > $\mathrm{odeadvisor}\left(\mathrm{Hermite_ode}\right)$
 $\left[\left[{\mathrm{_2nd_order}}{,}{\mathrm{_with_linear_symmetries}}\right]\right]$ (3)
 > $\mathrm{dsolve}\left(\mathrm{Hermite_ode}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{KummerM}}{}\left(\frac{{1}}{{2}}{-}\frac{{n}}{{2}}{,}\frac{{3}}{{2}}{,}{{x}}^{{2}}\right){}{x}{+}{\mathrm{_C2}}{}{\mathrm{KummerU}}{}\left(\frac{{1}}{{2}}{-}\frac{{n}}{{2}}{,}\frac{{3}}{{2}}{,}{{x}}^{{2}}\right){}{x}$ (4)
 > $\mathrm{dsolve}\left(\mathrm{Hermite_ode},\left[\mathrm{hypergeometric}\right]\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{KummerM}}{}\left({-}\frac{{n}}{{2}}{,}\frac{{1}}{{2}}{,}{{x}}^{{2}}\right){+}{\mathrm{_C2}}{}{\mathrm{KummerU}}{}\left({-}\frac{{n}}{{2}}{,}\frac{{1}}{{2}}{,}{{x}}^{{2}}\right)$ (5)

References

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