$${G_{2n}} = {c_1}{e^{\frac{{m{x^2}}}{2}}} + \frac{{\left( {2n - 1} \right)!!}}{{{m^n}}}{P_0} - \sum\limits_{k = 1}^n {\frac{{\left( {2n - 1} \right)!!}}{{\left( {2k - 1} \right)!!}}\frac{{{x^{2k - 1}}}}{{{m^{n - k + 1}}}}} $$
$${G_{2n + 1}} = {c_1}{e^{\frac{{m{x^2}}}{2}}} + \frac{{\left( {2n} \right)!!}}{{{m^{n + 1}}}}\left( {{e^{\frac{{m{x^2}}}{2}}} - 1} \right) - \sum\limits_{k = 1}^n {\frac{{\left( {2n} \right)!!}}{{\left( {2k} \right)!!}}\frac{{{x^{2k}}}}{{{m^{n - k + 1}}}}} $$
When finding the solutions to the simple ODE
$$ y'- mxy= x^n \text{ ; } y(0) = 0$$
I found the following:
Let $P_n$ be the particular solution for each integer exponent $n$. Then if we define
$$P_0(x) = exp\left(\displaystyle \frac{mx^2}{2}\right)\int_0^x exp\left(-\displaystyle \frac{mt^...
