If $y=\sum_kc_kx^k$, then the initial conditions determine $c_0=2$ and $c_1=1$. The differential equation yields
$\sum_kk(k-1)c_kx^{k-2}-\sum_kc_kx^{k+1}=x^2\;.$
The only constant term in this equation is from the first sum, so we must have $c_2=0$. The coefficients of the linear terms are $6c_3$ from the first sum, $-c_0=-2$ from the second and nothing on the right, so $c_3=1/3$. The coefficients of the quadratic terms are $12c_4$ from the first sum, $-c_1=-1$ from the second and $1$ on the right, so $c_4=1/6$. The right-hand side doesn't contribute to any of the remaining terms, so for $k\ge5$ we have $k(k-1)c_k=c_{k-3}$. Thus the solution is
$y=2\left(1+\frac1{3!}x^3+\frac{1\cdot4}{6!}x^6+\dotso\right)+x+2\left(\frac2{4!}x^4+\frac{2\cdot5}{7!}x^7+\dotso\right)\;.$
The closed form given by Wolfram|Alpha doesn't look very encouraging.