prompt: Find the behavior of $y(y") + y' + xy = x^2$ in the limit $ x \rightarrow \infty$ to leading order. Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them with a ;.

solution: $\boxed{y(x) = x}$


prompt: Find the behavior of $\frac{d^4y}{dx^4} = \cos(x^2 \frac{d^2y}{dx^2}) + \arctan(x^3 dy/dx) + e^x$ in the limit $ x \rightarrow \infty$ to leading order $x^4$. Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them with a ;.

solution: $\boxed{e^x}$


prompt: 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 ;.


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

solution: $\boxed{y(x) = 3.069(9.976 - x)^{-4/5} + (1 - 3.069(9.976 - x)^{-4/5})}$


prompt: 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: $\boxed{y = x + \frac{1}{6}x^3}$


prompt: Find the leading order behavior 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 \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them with a ;.


prompt: Find the first order behavior of $y'' = \frac{2xy}{x^3 + y^3}, y(0)=1,y'(0)=1$ in the limit $x \rightarrow \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them with a ;.

solution: $\boxed{y = 6^{1/3} x \ln(x)^{1/3}}$