data/nonlinear_ode.jsonl

{"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": "$\\boxed{y(x) = x}$", "parameters": "$x$", "type": "nonlinear_ode", "index": 0}
{"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": "$\\boxed{e^x}$", "parameters": "$x$", "type": "nonlinear_ode", "index": 1}
{"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 ;.", "solution": "$\\boxed{y(x)=1}$", "parameters": "$x$", "type": "nonlinear_ode", "index": 2}
{"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": "$\\boxed{y(x) = 3.069(9.976 - x)^{-4/5} + (1 - 3.069(9.976 - x)^{-4/5})}$", "parameters": "$x$", "type": "nonlinear_ode", "index": 3}
{"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": "$\\boxed{y = x + \\frac{1}{6}x^3}$", "parameters": "$x$", "type": "nonlinear_ode", "index": 4}
{"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 ;.", "solution": "$\\boxed{y=6(4.01-x)^{-1}}$", "parameters": "$x$", "type": "nonlinear_ode", "index": 5}
{"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": "$\\boxed{y = 6^{1/3} x \\ln(x)^{1/3}}$", "parameters": "$x$", "type": "nonlinear_ode", "index": 6}