Just as for a homogeneous differential equation take the ansatz $y_p = \sum_{k=0}^\infty a_k x^k.$ Plug this into the inhomogeneous differential equation and collect terms in powers of $x$, $\begin{equation*} (a_0+2 a_2) +\left(6 a_3-6\right) x +\left(12 a_4-3 a_2\right) x^2 +\left(20 a_5-8 a_3\right) x^3 +\ldots = 0.\tag{1} \end{equation*}$ To find the particular solution we can set $a_0 = a_1 = 0$. This amounts to casting out linear combinations of the homogeneous solutions. Thus, $y_p = x^3 + \frac{2}{5}x^5 + \ldots.$ We can also read off the homogeneous solutions from (1), $\begin{eqnarray*} y_1 &=& a_0\left(1 - \frac{1}{2}x^2 - \frac{1}{8} x^4 + \ldots\right) \\ y_2 &=& a_1 x. \end{eqnarray*}$ The solutions $y_1$, $y_2$, and $y_p$ can be found exactly following the steps laid out by @GEdgar in the comments. I recommend finding them and verifying that their Taylor expansions yield the series solutions given above.