| {"prompt": "Find the leading order WKB approximation for the specific differential equation: $$ \\epsilon^2 y''(x) + (1+x^2) y(x) = 0 $$ with initial conditions at $y(0) = 1, y'(0) = 0$ where $\\epsilon$ is a small positive parameter ($0 < \\epsilon \\ll 1$). Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\\boxed{}$ LaTeX environment.", "solution": "$\\boxed{ y_{WKB}(x) \\approx (1+x^2)^{-1/4} \\cos\\left( \\frac{1}{\\epsilon} \\left[ \\frac{1}{2} x \\sqrt{1+x^2} + \\frac{1}{2} arcsinh(x) \\right] \\right)} $", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 0} | |
| {"prompt": "Find the leading order WKB approximation for the specific differential equation: $\\epsilon^2 y'' = (x+1)y$ with initial conditions at $y(0)=0, y'(0)=1$ where $\\epsilon$ is a small positive parameter ($0 < \\epsilon \\ll 1$). Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\\boxed{}$ LaTeX environment.", "solution": "$\\boxed{y(x) \\sim \\epsilon (x+1)^{-1/4}\\sinh\\left[\\frac{2\\left((x+1)^{3/2}-1\\right)}{3\\epsilon}\\right]}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 1} | |
| {"prompt": "Find the leading order WKB approximation for the specific differential equation: $$ \\epsilon^2 y''(x) - (1+\\cos{x}) y(x) = 0 $$ with initial conditions at $y(0) = 0, y'(0) = \\frac{2^{5/4} \\cosh{(1)}}{\\epsilon}$ where $\\epsilon$ is a small positive parameter ($0 < \\epsilon \\ll 1$). Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\\boxed{}$ LaTeX environment.", "solution": "$\\boxed{y(x) \\sim 2\\cosh(1) (1+\\cos x)^{-1/4}\\sinh\\left[\\frac{2\\sqrt{2}\\sin(x/2)}{\\epsilon}\\right]}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 2} | |
| {"prompt": "Find the leading order WKB approximation for the specific differential equation: $\\epsilon^2 y'' - (1+x)^2 y = 0$ with initial conditions at $y(0) = 1, y(1) = e^{-1}$ where $\\epsilon$ is a small positive parameter ($0 < \\epsilon \\ll 1$). Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\\boxed{}$ LaTeX environment.", "solution": "$\\boxed{y(x)\\sim \\frac{(1+x)^{-1/2}}{e^{3/(2\\epsilon)}-e^{-3/(2\\epsilon)}} [(\\sqrt{2}e^{-1}-e^{-3/(2\\epsilon)})\\exp((x + x^2/2)/\\epsilon) +(e^{3/(2\\epsilon)}-\\sqrt{2}e^{-1})\\exp(-(x + x^2/2)/\\epsilon)]}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 3} | |
| {"prompt": "Find the leading order WKB approximation for the specific differential equation: $\\epsilon^2 y'' = (2+x+3x^2)^2y$ with initial conditions at $y(0) = 0, y'(0)=1$ where $\\epsilon$ is a small positive parameter ($0 < \\epsilon \\ll 1$", "solution": "$\\boxed{y(x) = \\frac{\\epsilon}{\\sqrt{2(2+x+3x^2)}} \\sinh[\\frac{2x+\\frac{1}{2}x^2+x^3}{\\epsilon}]}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 4} | |
| {"prompt": "Find the leading order WKB approximation for the specific differential equation: $\\epsilon^2 y'' = x y$ with initial conditions at $y(1)=1,y'(1)=1$ where $\\epsilon$ is a small positive parameter ($0 < \\epsilon \\ll 1$", "solution": "$\\boxed{\\frac{\\left(\\left(4 - 5 \\epsilon\\right) e^{\\frac{4}{3 \\epsilon}} + \\left(5 \\epsilon + 4\\right) e^{\\frac{4 x^{\\frac{3}{2}}}{3 \\epsilon}}\\right) e^{- \\frac{2 \\left(x^{\\frac{3}{2}} + 1\\right)}{3 \\epsilon}}}{8 \\sqrt[4]{x}}}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 5} | |
| {"prompt": "Find the leading order WKB approximation for the specific differential equation: $\\epsilon^2 y'' = e^x y$ with initial conditions at $y(0)=0, y'(0)=1$ where $\\epsilon$ is a small positive parameter ($0 < \\epsilon \\ll 1$", "solution": "$\\boxed{y(x) =\\epsilon e^{-x/4}\\sinh\\left[\\frac{2\\left(e^{x/2}-1\\right)}{\\epsilon}\\right]}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 6} | |
| {"prompt": "Find the leading order WKB approximation for the lowest order normalized eigenfunction of the differential equation: $y'' = -E_1 (x+\\pi)^4 y$ with boundary conditions at $y(0)=0, y(\\pi)=0$ where $E$ is a positive real value. Normalization is: $\\int_0^\\pi [y(x)^2 (x+\\pi)^4] dx = 1$.Put your final answer in a LaTeX \\boxed{} environment.", "solution": "$\\boxed{y(x)=\\sqrt{\\frac{6}{7\\pi^3 }}\\frac{1}{(x +\\pi)} \\sin(\\frac{x^3 + 3 \\pi x^2 + 3x \\pi^2}{7\\pi^2}) } $", "parameters": "$x; E_1$", "type": "wkb", "index": 7} | |
| {"prompt": "Find the leading order WKB approximation of $\\epsilon^2 y'' = (1 + x \\sin(x)) y$ subject to boundary conditions $y(0) = 0$, $y(1)= 1$ in the limit $\\epsilon \\rightarrow 0^+$. Approximate any integrals with a polynomial in $x$, up to third order in $x$. Put your final answer in a LaTeX \\boxed{} environment.", "solution": "$\\boxed{y = (1 + \\sin(1))^{1/4} \\left(\\sinh\\left(\\frac{7}{6\\epsilon}\\right)\\right)^{-1}(1 + x\\sin(x))^{-1/4} \\sinh\\left(\\frac{(x + x^3/6)}{\\epsilon}\\right)}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 8} | |
| {"prompt": "Find the leading order WKB approximation for the specific differential equation: $\\epsilon y''(x) + (1+x)^2 y(x) = 0$ with initial conditions at $ y(0) = 0, y(1) = 1$ where $\\epsilon$ is a small positive parameter ($0 < \\epsilon \\ll 1$). Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\\boxed{}$ LaTeX environment.", "solution": "$\\boxed{y(x) \\approx \\frac{\\sqrt{2}}{\\sin\\left(\\frac{3}{2\\sqrt{\\epsilon}}\\right)} \\frac{1}{\\sqrt{1+x}} \\sin\\left(\\frac{x + x^2/2}{\\sqrt{\\epsilon}}\\right)}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 9} | |
| {"prompt": "Find the leading order behavior of $\\epsilon^2 y''(x) = [1 + sin(x)^2]y$ subject to boundary conditions $y(0) = 1, y'(0)=1$ in the limit of large x.", "solution": "$\\boxed{y(x) = \\frac{1+\\epsilon}{2} (1 + sin^2(x))^{-\\frac{1}{4}} e^{1.2160 * x / \\epsilon}}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 10} | |
| {"prompt": "Find the leading order behavior of $\\epsilon^2 y'' = (1+x)^2 y$ subject to boundary conditions $y(0) = 0$, $y'(0)= 1$ in the limit of large x.", "solution": "$\\boxed{y(x) \\sim \\epsilon (1+x)^{-1/2}\\sinh[\\frac{2x+x^2}{2\\epsilon}]}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 11} | |
| {"prompt": "Find the leading order behavior of $\\epsilon^2 y'' = (1+x^2)^2 y$ subject to boundary conditions $y(0) = 0$, $y'(0)= 1$ in the limit of large x.", "solution": "$$\\boxed{y(x) \\sim \\epsilon (1+x^2)^{-1/2} \\sinh\\left(\\frac{x+x^3/3}{\\epsilon}\\right)}$$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 12} | |
| {"prompt": "Find the leading order behavior of $\\epsilon^2 y'' = (1/x^2) y$ subject to boundary conditions $y(1) = 1$, $y'(1)= 0$ in the limit of large x.", "solution": "$\\boxed{y(x) \\sim x^{1/2}[(\\frac{1}{2}-\\frac{\\epsilon}{4})x^{1/\\epsilon}+(\\frac{1}{2}+\\frac{\\epsilon}{4})x^{-1/\\epsilon}]}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 13} | |
| {"prompt": "Find the leading order behavior of $\\epsilon y'' + y = 0$ subject to boundary conditions $y(0) = 0$, $y(1) = 1$ in the limit of large x.", "solution": "$\\boxed{y(x) = \\frac{sin(\\frac{x}{\\sqrt{\\epsilon}})}{sin(\\frac{1}{\\sqrt{\\epsilon}})}}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 14} | |
| {"prompt": "Find the leading order behavior of $\\epsilon y'' + cosh(x)y = -1$ subject to boundary conditions $y(-1) = 1$, $y(1) = 1$ in the limit of large x. Write your answer as a single expression for y(x) without defining any functions or variables beyond what is given in the problem. Place this answer in a single $\\boxed{}$ LaTeX environment.", "solution": "$\\boxed{y = (1+\\frac{1}{\\cosh(1)}) \\sec(\\frac{2.1633}{2\\sqrt{\\epsilon}})\\cos(\\frac{0.00128074x^4 + 0.07861894x^3 - 0.00834220x^2 + 0.99478970x - 0.00117990}{\\sqrt{\\epsilon}})- \\frac{1}{\\cosh(x)}}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 15} | |
| {"prompt": "Find the leading order behavior of $x^4y'''=y$ subject to boundary conditions $y(1)=1,y'(1)=0,y''(1)=0$ in the limit of $x\\to\\0^+$.", "solution": "$\\boxed{y(x) =-0.47991x^{4/3}\\exp{\\left( \\frac{3}{2} x^{-1/3} \\right)}\\cos\\left( \\frac{3\\sqrt{3}}{2} x^{-1/3} -1.0750 \\right) }$", "parameters": "$x$", "type": "wkb", "index": 16} | |
| {"prompt": "Find the leading order behavior of $\\epsilon y'' = e^xy$ subject to boundary conditions $[y(0) = 0, y'(0) = 1]$ in the limit of $x\\to\\0^+$.", "solution": "$\\boxed{y(x) = \\sqrt{\\epsilon} e^{-\\frac{x}{4}}\\sinh(\\frac{2e^{\\frac{x}{2}}-2}{\\sqrt{\\epsilon}})}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 17} | |
| {"prompt": "Find the leading order WKB approximation for the specific differential equation: $\\epsilon^2 y'' = \\frac{e^{2x}}{(1+e^x)^4} y$ with initial conditions at $y(0)=0, y'(0)=1$ where $\\epsilon$ is a small positive parameter ($0 < \\epsilon \\ll 1)$. Give your final answer in a \\boxed latex environment.", "solution": "$\\boxed{y(x) \\sim 4\\epsilon \\cosh(x/2)\\sinh\\left[\\frac{e^x-1}{2\\epsilon(1+e^x)}\\right]}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 18} | |
| {"prompt": "Find the leading order WKB approximation for the specific differential equation: $\\epsilon^2y''=x^2y$ with initial conditions at $y(1)=0; y(2)=1$ where $\\epsilon$ is a small positive parameter ($0 < \\epsilon \\ll 1)$. Give your final answer in a \\boxed latex environment.", "solution": "$\\boxed{y=\\frac{\\sqrt{2}}{\\sqrt{x}}(exp(-\\frac{2}{\\epsilon})exp(\\frac{x^2}{2\\epsilon})-exp(-\\frac{1}{\\epsilon})exp(-\\frac{x^2}{2\\epsilon}))}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 19} | |
| {"prompt": "Find the leading order WKB approximation for the specific differential equation: $\\epsilon y'' + y = 0$ with initial conditions at $y(0) = 0, y(1) = 1$ where $\\epsilon$ is a small positive parameter ($0 < \\epsilon \\ll 1)$. Give your final answer in a \\boxed latex environment.", "solution": "$\\boxed{ \\frac{\\sin \\left(\\frac{x}{\\sqrt{\\epsilon}}\\right)}{\\sin \\left(\\frac{1}{\\sqrt{\\epsilon}}\\right)}}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 20} | |
| {"prompt": "Find the leading order WKB approximation for the specific differential equation: $\\epsilon y''' + y = 0$ with initial conditions at $y(0) = 1, y(\\epsilon^{1/3}) = e^{-1}, y(-\\epsilon^{1/3}) = e^{1} $ where $\\epsilon$ is a small positive parameter ($0 < \\epsilon \\ll 1)$. Give your final answer in a \\boxed latex environment.", "solution": "$\\boxed{e^{\\frac{-x}{\\epsilon^{1/3}}}}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 21} | |
| {"prompt": "Find the leading order behavior of $\\epsilon^2 y'' + y = 0$ subject to boundary conditions $y(0)=0, y(1)=1$ in the limit of small $\\epsilon$. Give your final answer in a \\boxed latex environment.", "solution": "$\\boxed{sin(x/\\epsilon)/sin(1/\\epsilon)}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 22} | |
| {"prompt": "Find the leading order approximation of $y(x)$ from the differential equation $\\epsilon^2 y''(x)=(\\sin x) y$ subject to boundary conditions $y(\\frac{\\pi}{2}) = 1, y'(\\frac{\\pi}{2}) = 0$ in the limit of $\\epsilon \\to 0$. The answer should be in terms of the incomplete beta function $B_z(a, b)$ where $B_z(a, b)=\\int_0^z t^{a-1}(1-t)^{b-1} d t, 0 \\leq z \\leq 1$", "solution": "$\\boxed{\\frac{1}{(\\sin x)^{\\frac{1}{4}}} \\cosh \\left(\\frac{1}{2 \\epsilon}\\left[B_{(\\sin(x))^2}\\left(\\frac{3}{4}, \\frac{1}{2}\\right)-B\\left(\\frac{3}{4}, \\frac{1}{2}\\right)\\right]\\right)}$.$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 23} | |
| {"prompt": "Find the leading order behavior of $y''=(\\cot x)^4 y$ in the limit of small x that satisfies the conditions $y(1) = 1; y'(1) = 1$. Leading order does not necessarily mean only one term. Use only the variables and constants given in the problem; do not define additional constants. Give your final answer in a \\boxed latex environment.", "solution": "$\\boxed{y(x) \\sim \\frac{e}{2} x e^{-1/x} + \\frac{1}{2e} x e^{1/x}}$", "parameters": "$x$", "type": "wkb", "index": 24} | |
| {"prompt": "Find the leading order WKB approximation for the specific differential equation: $\\epsilon^2 y'' = y \\cosh^2 x$ with initial conditions at $y(0) = 0; y'(0) = 1$ where $\\epsilon$ is a small positive parameter ($0 < \\epsilon \\ll 1)$. Give your final answer in a \\boxed latex environment.", "solution": "$\\boxed{y(x) = \\epsilon (\\cosh x)^{-1/2} \\sinh\\left( \\frac{1}{\\epsilon} \\sinh x \\right)}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 25} | |
| {"prompt": "Find the leading order WKB approximation for the specific differential equation: $\\epsilon^2 y'' + \\cosh(x) y' + \\sinh(x) y = 0$ with initial conditions at $y(0) = 0$, $y(1) = 1$. where $\\epsilon$ is a small positive parameter ($0 < \\epsilon \\ll 1)$. Give your final answer in a \\boxed latex environment.", "solution": "$\\boxed{y(x) = \\cosh(1)(\\frac{1}{\\cosh(x)} - \\cosh(x) e^{-\\sinh(x)/\\epsilon^2}) }$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 26} | |
| {"prompt": "Find the WKB approximation up to the second leading order for the specific differential equation as $x\\to 0$ $3x^5y'''=y$ with initial conditions at $y(1)=1, y'(1)=0, y''(1)=0$ as $x\\to 0$. Give ONE final answer in terms of y and x and numbers (not arbitrary constants) in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{y(x) \\approx x^{5/3} \\left( 1.9461391296842614 e^{-\\frac{3^{2/3}}{2}x^{-2/3}} + e^{\\frac{3^{2/3}}{4}x^{-2/3}}\\left(-1.4641592737823024 \\cos\\left(\\frac{3^{7/6}}{4}x^{-2/3}\\right) + 1.3969897071670665 \\sin\\left(\\frac{3^{7/6}}{4}x^{-2/3}\\right) \\right) \\right)}$", "parameters": "$x$", "type": "wkb", "index": 27} | |
| {"prompt": "Find the WKB approximation up to the second leading order for the specific differential equation as $x\\to 0$ $x^6 y''' + y = 0$ with initial conditions at $y(1)=1, y'(1)=0, y''(1)=0$ as $x\\to 0$. Give ONE final answer in terms of y and x and numbers in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{y(x) \\approx x^2 \\left( 0.6131324019838598 e^{x^{-1}} + e^{-x^{-1}/2}\\left(-0.712093006628941 \\cos\\left(\\frac{\\sqrt{3}}{2}x^{-1}\\right) -0.8372865406850002 \\sin\\left(\\frac{\\sqrt{3}}{2}x^{-1}\\right) \\right) \\right)}$", "parameters": "$x$", "type": "wkb", "index": 28} | |