erf - Maple Help

The Error Function (erf) ODE

Description

 • The general form of the erf ODE is given by
 > erf_ode := diff(y(x),x,x)+2*x*diff(y(x),x)-2*n*y(x) = 0;
 ${\mathrm{erf_ode}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{2}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{2}{}{n}{}{y}{}\left({x}\right){=}{0}$ (1)
 where n is an integer. See Abramowitz and Stegun, "Handbook of Mathematical Functions", section 7.2.2. The solution of this type of ODE can be expressed in terms of the WhittakerM and WhittakerW functions.

Examples

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