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;

$\mathrm{erf\_ode}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+2x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-2ny\left(x\right)=0$

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.

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

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

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

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

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

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