Problem: Find the behavior to the second leading order of $\frac{d^4 y}{dx^4} = (\frac{d^2 y}{dx^2})^2 - \frac{d y}{dx}+ \frac{1}{x^3+1}, y(0)=0, y'(0)=1, y''(0)=0, y'''(0)=1$ in the limit $x \rightarrow 0$. Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them with a ;.
Solution: $\boxed{y = x + \frac{1}{6}x^3}$