Problem: Find the leading order behavior of $\frac{d^5 y}{dx^5} + x \frac{d^4 y}{dx^4}+ \frac{d^3 y}{dx^3}+ e^x\left(\frac{d^2 y}{dx^2}\right)^{2}-x^3y^3+x^4 \frac{dy}{dx}=0; [y(0) = 1, \frac{dy}{dx}(0) = 1, \frac{d^2 y}{dx^2}(0) = -1, \frac{d^3 y}{dx^3}(0)= 2, \frac{d^4 y}{dx^4}(0)= 1]$ 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: $\boxed{y(x)=1}$