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Browse files- README.md +37 -0
- data/asympytotic_series.jsonl +5 -0
- data/boundary_layers.jsonl +83 -0
- data/integrals.jsonl +14 -0
- data/nonlinear_ode.jsonl +7 -0
- data/nonlinear_pde.jsonl +67 -0
- data/other.jsonl +6 -0
- data/wkb.jsonl +29 -0
README.md
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---
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license: mit
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---
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# HARDMath2 Benchmark Dataset
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This repository contains a collection of mathematical benchmark problems designed for evaluating Large Language Models (LLMs) on mathematical reasoning tasks.
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## Building
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Save `.csv` file exported from Google Sheet to `raw_csv` folder and run `csv_to_yaml.py` to convert all of the `.csv` file sto `.yaml`. Then push the changes to remote and the `.yaml` file will automatically be converted to `.jsonl` and pushed to an anonymized HF repository.
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The `.csv` file should have a descriptive name for the types of problems in the file, with underscores instead of spaces.
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## Data Format
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Each benchmark problem in the dataset is structured as a JSON object containing the following fields:
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### Fields
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- **Prompt**: The input string that is fed to the LLM
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- **Solution**: A LaTeX-formatted string representing the mathematical formula that solves the question posed in the prompt
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- **Parameters**: A list of independent tokens that should be treated as single variables in the LaTeX response string. These include:
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- Single variables (e.g., `$A$`, `$x$`)
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- Greek letters (e.g., `$\epsilon$`)
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- Complex strings with subscripts (e.g., `$\delta_{i,j}$`)
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Each parameter should be separated by a semicolon (;).
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## Example
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```json
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{
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"prompt": "What is the derivative of f(x) = x^2?",
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"solution": "\\frac{d}{dx}(x^2) = 2x",
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"parameters": "x"
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}
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```
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data/asympytotic_series.jsonl
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{"prompt": "Find the first three terms in the asymptotic series of $I(x)=\\int_0^{\\pi/2} \\frac{\\cos{t}}{\\sqrt{x\\sin{t}+log(1+t^2)}}dt$ in the limit $x \\to \\infty$. Provide your answer in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{I(x)=(\\frac{2}{\\sqrt{x}}-\\frac{1}{3x^{3/2}}+\\frac{3}{20x^{5/2}} )}$", "parameters": "$x; t$", "type": "asympytotic_series", "index": 0}
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{"prompt": "Find the first two terms in the asymptotic series of $I(x)=\\int_0^{\\pi/4} e^{-x(\\tan{t}-\\frac{t^3}{6})}\\sqrt{1+\\sin^2(t)}dt$ in the limit $x \\to \\infty$. Provide your answer in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{I(x)=(1/x)(1-e^{-\\frac{x \\pi}{4}}) }$", "parameters": "$x; t$", "type": "asympytotic_series", "index": 1}
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{"prompt": "Find a single expression with the first three terms in the asymptotic series of I(x) = \\int\\limits_{0}^{x} \\frac{\\sin t}{t} \\ dt in the limit $x \\to \\infty$. Provide your answer in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{I(x)=\\frac{\\pi}{2} - \\frac{\\cos x}{x} + \\frac{\\sin x}{x^2}}$", "parameters": "$x; t$", "type": "asympytotic_series", "index": 2}
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{"prompt": "Write the first two term asymptotic series of $I(x) = \\int^\\infty_x \\frac{e^{-t^2}}{1+t^5} dt$ in the limit $x \\rightarrow \\infty$. Do not approximate the denominator. Provide your answer in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{I(x) = e^{-x^2}(\\frac{1}{2x(1+x^5)} - \\frac{(1+6x^5)}{4x^3(1+x^5)^2})}$", "parameters": "$x; t$", "type": "asympytotic_series", "index": 3}
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{"prompt": "Write the first two term asymptotic series of $I(x) = \\int^x_1 \\ln(xt^2)\\cos(t^3) dt$ in the limit $x \\rightarrow \\infty$. Provide your answer in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{I(x) = \\frac{\\ln(x^3)\\sin(x^3)}{3x^2} - \\frac{\\ln(x)\\sin(1)}{3} -\\frac{2(\\ln(x^3)-1)\\cos(x^3)}{9x^5} + \\frac{2(\\ln(x)-1)\\cos(1)}{9}}$", "parameters": "$x; t$", "type": "asympytotic_series", "index": 4}
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data/boundary_layers.jsonl
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' - x y' + x^3 y = 0$ with boundary conditions $y(0) = 1$, $y(1) = 2$ in the limit $\\epsilon \\ll 1$. 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": "$\\boxed{y = e^{\\frac{x^3}{3}} + (2-e^{1/3})e^{-(1-x)/\\epsilon}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 0}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' - x y' + x^3 y = 0$ with boundary conditions $y(0) = A$, $y(1) = B$ in the limit $\\epsilon \\ll 1$. 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": "$\\boxed{y = A*e^{\\frac{x^3}{3}} + (B-A*e^{1/3})e^{-(1-x)/\\epsilon}}$", "parameters": "$x; \\epsilon; A; B$", "type": "boundary_layers", "index": 1}
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{"prompt": "Find a single uniformly valid approximation to the solution of $\\epsilon y'' + x y' - y = -e^x$ with boundary conditions $y(-1)=0, y(1)=1$ in the limit $\\epsilon \\ll 1$. 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": "$\\boxed{y_{unif}(x) \\approx \\left[ e^x - x Ei(x) + (1 - e + Ei(1)) x \\right] - \\left[e^{-1} + Ei(-1) - 1 + e - Ei(1)\\right] e^{-(x+1)/\\epsilon}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 2}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''-2 tan(x) y'+y=0$ with boundary conditions $y(-1)=0, y(1)=1$ in the limit $\\epsilon = 0$. 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": "$\\boxed{y = \\sqrt{\\frac{\\sin x}{\\sin 1}}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 3}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''-x y'-(3+x)$ with boundary conditions $y(-1)=1, y(1)=1$ in the limit $\\epsilon = 0$ from the positive direction. 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": "$\\boxed{y = E^{-(x+1)/\\epsilon}+ E^{-(1-x)/\\epsilon}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 4}
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{"prompt": "Find a uniformly valid approximation, with error of order $\\epsilon^2$, to the solution of $\\epsilon y'' + y' +y = 0$ with boundary conditions $y(0) = e, y(1) = 1$ in the limit $\\epsilon = 0$ from the positive direction. Notice that there is no boundary layer in leading order, but one does appear in next order. 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": "$\\boxed{y = e^{1-x} + \\epsilon[(-x+1)e^{1-x} -e^{1-\\frac{x}{\\epsilon}}]}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 5}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' - (x+2)y' - (3+x) = 0$ with boundary conditions $y(-1) = 1, y(1) = 1$ in the limit $\\epsilon = 0$ from the positive direction. 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": "$\\boxed{y_{uniform}(x) = - \\ln(2+x) -x + (\\ln(3) + 2)e^{\\frac{-3(1-x)}{\\epsilon}}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 6}
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{"prompt": "Find a uniformly valid approximation to the solution of $ \\epsilon y'' + y' \\sin(x) + y \\sin(\\2x) = 0$ with boundary conditions $ y(0) = \\pi, y(\\pi) = 0 $ in the limit $\\epsilon = 0$ from the positive direction. 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 = \\text{erfc}(\\frac{x}{\\sqrt{2\\epsilon}})} $", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 7}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' + (1 + x^2) y' - y = 0$ with boundary conditions $y(0) = 1, y(1) = 2$ in the limit $\\epsilon = 0$ from the positive direction. 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": "$\\boxed{y = 2 e^{\\arctan(x) - \\pi/4} + (1 - 2 e^{-pi/4}) e^{-x/\\epsilon} }$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 8}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' + (x^2 +1)y'+2xy=0$ with boundary conditions $y(0)=1, y(1)=5$ in the limit $\\epsilon = 0$ from the positive direction. 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": "$\\boxed{y(x) = \\frac{10}{x^2+1} + e^{\\frac{-x}{\\epsilon}} - 10e^{\\frac{-x}{\\epsilon}}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 9}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' + x y' + y = 0$ with boundary conditions $y(0)=1, y(1)=1$ in the limit $\\epsilon = 0$ from the positive direction. Denote the square root of -1 as I. 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": "$\\boxed{y(x) \\approx \\frac{1}{\\sqrt{\\epsilon}}e^{\\frac{-x^2}{2\\epsilon}} \\\\i \\sqrt{\\frac{\\pi}{2}}erfi(\\frac{x}{\\sqrt{2\\epsilon}})+ e^{\\frac{-x^2}{2\\epsilon}}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 10}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' - y'/x - y^2 = 0$ with boundary conditions $y(0) = 1, y(1) = 1$ in the limit $\\epsilon = 0$ from the positive direction. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\\boxed{}$ LaTeX environment. Your response should have the form of a single analytical expression.", "solution": "$\\boxed{y(x) = \\frac{1}{\\frac{1}{2}x^2 + 1} + \\frac{1}{3} \\exp(\\frac{x-1}{\\epsilon})}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 11}
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{"prompt": "Find a uniformly valid approximation to the solution of $$\\epsilon y''+\\frac{y'}{x^2}+y=0 with boundary conditions $y(0)=0, y(1)=e^{-\\frac{1}{3}}$ in the limit $\\epsilon \\rightarrow 0+$. 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)=e^{\\frac{-x^3}{3}}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 12}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''+\\frac{y'}{x}+y=0$ with boundary conditions $[y(-1)=2e^{-1/2}, y(1)=e^{-1/2}]$ in the limit $\\epsilon \\rightarrow 0+$. 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)=\\left(\\frac{3-x}{2}\\right)e^{-\\frac{x^2}{2}}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 13}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' - (x+1) y' + x^2 + x + 1 = 0$ with boundary conditions $y(0) = 1, y(1) = 2$ in the limit $\\epsilon \\ll 0+$. 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 = \\frac{1}{2} x^2 + \\ln{(x+1)} + 1 + (\\frac{1}{2} - \\ln{2}) e^{-2(1-x) / \\epsilon}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 14}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' + (\\cosh(x))(x^2 + 1)y' - x^3 y = 0$ with boundary conditions $y(0) = 1, y(1) = 1$ in the limit $\\epsilon \\ll 0+$. 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) = (1-\\exp\\left(\\int_1^0 \\frac{t^3}{\\cosh(t)(t^2 + 1)}\\ dt\\right))e^{-x/\\epsilon} + \\exp\\left(\\int_1^x \\frac{t^3}{\\cosh(t)(t^2 + 1)}\\ dt\\right)}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 15}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' - (x^2+4)y' - y^3 = 0$ with boundary conditions $y(0)=1, y(1)=\\sqrt{5}$ in the limit $\\epsilon \\ll 0+$. Use only the variables and constants given in the problem; do not define additional constants. Solve any integrals in the final solution. Place your final solution in a $\\boxed{}$ LaTeX environment.", "solution": "$\\boxed{y(x)=\\frac{1}{\\sqrt{\\arctan\\left(\\frac{x}{2}\\right)+1}}+\\left(\\sqrt{5}-\\frac{1}{\\sqrt{\\arctan\\left(\\frac{1}{2}\\right)+1}}\\right)e^{-5(1-x)/\\epsilon}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 16}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' - (x^2+1)y' - y^3 = 0$ with boundary conditions $y(0)=1, y(1)=1/2$ in the limit $\\epsilon \\ll 0+$. 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}{\\sqrt{2\\arctan(x) + 1}} + \\left( \\frac{1}{2} - \\frac{1}{\\sqrt{\\pi/2 + 1}} \\right) e^{-2(1-x)/\\epsilon} }$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 17}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' + (x^2-12)y' - y^3 = 0$ with boundary conditions $y(0)=1, y'(1)=1/2$ in the limit $\\epsilon \\ll 0+$. 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 \\left( 1 - \\frac{1}{2\\sqrt{3}} \\ln\\left( \\frac{2\\sqrt{3}-x}{x+2\\sqrt{3}} \\right) \\right)^{-1/2} + \\frac{\\epsilon}{11} \\left[ \\frac{1}{2} + \\frac{1}{11} \\left( 1 - \\frac{1}{2\\sqrt{3}} \\ln\\left( \\frac{2\\sqrt{3}-1}{2\\sqrt{3}+1} \\right) \\right)^{-3/2} \\right] e^{-11(1-x)/\\epsilon}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 18}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' + (\\ln x) y' - x(\\ln x) y = 0$ with boundary conditions $y(1/2)=1, y(3/2)=1$ in the limit $\\epsilon \\ll 0+$ for $x<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) = e^{\\frac{x^2}{2} - \\frac{1}{8}} } $", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 19}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' + (\\ln x) y' - x(\\ln x) y = 0$ with boundary conditions $y(1/2)=1, y(3/2)=1$ in the limit $\\epsilon \\ll 0+$ for $x>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) = e^{\\frac{x^2}{2} - \\frac{9}{8}} } $", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 20}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' - \\frac{1}{x} y' - y = 0$ with boundary conditions $y(0) = 1, y(1) = 1$ in the limit $\\epsilon \\ll 0+$ to leading order. Use only the variables and constants given in the problem; do not define additional constants; in your final solution, only $\\epsilon$ and $x$ should remain as variables. Place your final solution in a $\\boxed{}$ LaTeX environment.", "solution": "$\\boxed{y =e^{-x^2/2} \\left[ 1 \\right]+ (1 - e^{-1/2}) \\left[1 \\right] e^{-\\frac{1 - x}{\\epsilon}}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 21}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' + x^2y' - xy = 0$ with boundary conditions $y(0) = 2, y(1) = 3$ in the limit $\\epsilon \\ll 0+$. 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 3x + 2 \\exp\\left(-\\frac{x^3}{3\\epsilon}\\right)}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 22}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' - y'/(x^2-1.01) + ye^{-x} + sin(\\epsilon)(x+cos(\\epsilon)) y' = 0$ with boundary conditions $y(-1) = 1, y(1) = 1$ in the limit $\\epsilon \\ll 0+$. 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 \\exp(3.99 e^{-1} - (x^2 + 2x + 0.99) e^{-x}) + \\left(1 - \\exp(3.99 e^{-1} + 0.01 e)\\right) \\exp\\left(-\\frac{100(x+1)}{\\epsilon}\\right)}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 23}
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{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' + \\cos(x)y' + y = -1$ with boundary conditions $y(0) = 1$, $y(1) = 1$ in the limit $\\epsilon \\rightarrow 0^+$. 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) = -1 + \\frac{2(\\sec(1) + \\tan(1))}{\\sec(x) + \\tan(x)} + 2(1 - \\sec(1) - \\tan(1))e^{-x/\\epsilon}}$$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 24}
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{"prompt": "Find a uniformly valid approximation to the solution of $ \\epsilon y''(x) + (x-1)^2 y'(x) - x(x-1)^2 y(x) = \\epsilon x^2 \\sin(\\pi x) [1+y(x)] $ with boundary conditions $y(1/2)=3, y(3/2)=3$ in the limit $\\epsilon \\rightarrow 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\\boxed{}$ LaTeX environment.", "solution": "$$\\boxed{y(x) \\approx 3 e^{x^2/2 - 9/8} + 3(1 - e^{-1}) e^{-(x-1/2)/(4*\\epsilon)}}$$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 25}
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| 27 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' + (\\ln x)y' - x(\\ln x)y = 0$ with boundary conditions $y(1/2) = 1, y(3/2) = 1$ in the limit $\\epsilon \\to 0$. 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{\\frac{1}{2} \\left( e^{-\\frac{1}{8} + \\frac{x^2}{2}} + e^{-\\frac{9}{8} + \\frac{x^2}{2}} \\right) + \\frac{1}{2} \\left( e^{-\\frac{9}{8} + \\frac{x^2}{2}} - e^{-\\frac{1}{8} + \\frac{x^2}{2}}\\right) * erf\\left(\\frac{x-1}{\\sqrt{2\\epsilon}}\\right)}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 26}
|
| 28 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' + \\frac{cos(x)}{3}y' - (\\ln x)y = 0$ with boundary conditions $y(0) = 0, y(1) = 1$ in the limit $\\epsilon \\to 0^+$. 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) = e^{\\int_{1}^{x}\\frac{3\\ln t}{\\cos(t)}dt} - e^{\\int_{1}^{0}\\frac{3\\ln t}{\\cos(t)}dt}e^{- \\frac{x}{3\\epsilon}}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 27}
|
| 29 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''(x) + (1 + x) y'(x) + 3 y(x) = 0$ with boundary conditions $y(0) = 1, y(1) = 1$ in the limit $\\epsilon \\to 0$. 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)=8(1+x)^{-3}-7e^{-\\frac{x}{\\epsilon}}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 28}
|
| 30 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''(x) + (2 - x^2) y'(x) + 4 y(x) = 0$ with boundary conditions $y(0) = 0, y(1) = 2$, in the limit $\\epsilon \\to 0$. 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)=2(3+2\\sqrt{2})^\\sqrt{2}((\\frac{\\sqrt{2}-x}{\\sqrt{2}+x})^\\sqrt{2}-e^{-\\frac{2x}{\\epsilon}})}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 29}
|
| 31 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' + x y' = x \\cos x$ with boundary conditions $y(-1) = 2, y(1) = 2$ in the limit $\\epsilon \\to 0^+$. 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 = \\sin x + 2 - \\sin(1) \\, \\mathrm{erf}\\left(\\frac{x}{\\sqrt{2\\epsilon}}\\right)}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 30}
|
| 32 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' - x y' - (3 + x)y = 0$ with boundary conditions $y(-1) = 1, y(1) = 1$ in the limit $\\epsilon \\to 0^+$. 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 = e^{-(x+1)/\\epsilon} + e^{(x-1)/\\epsilon}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 31}
|
| 33 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' + \\frac{y'}{x^2} + y = 0$ with boundary conditions $y(0) = 0, y(1) = e^{-1/3}$ in the limit $\\epsilon \\to 0^+$. 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)=e^{-\\frac{x^3}{3}}}$$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 32}
|
| 34 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' + (\\cosh x)y' + y = 0$ with boundary conditions $y(-1) = 0, y(1) = 1$ in the limit $\\epsilon \\to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\\boxed{}$ LaTeX environment.", "solution": "$$\\boxed{y(x) = \\exp (2(\\arctan(e)-\\arctan(e^{x})))-\\exp(2(\\arctan(e)-\\arctan(e^{-1})))e^{-\\cosh(1)\\frac{x+1}{\\epsilon}}}$$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 33}
|
| 35 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''(x) + \\cosh(x)\\,y'(x) - y(x) = 0$ with boundary conditions $y(0)=1, y(1)=1$ in the limit $\\epsilon \\ll 1$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\\boxed{}$ LaTeX environment.", "solution": "$\\boxed{y(x) = \\exp (2[\\arctan(e^x) - \\arctan(e)]) + (1 - \\exp (2[\\arctan(1) - \\arctan(e)]))e^{-\\frac{x}{\\epsilon}}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 34}
|
| 36 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon\\,y'' + (x^2+1)\\,y' - x^3\\,y = 0$ with boundary conditions $y(0)=1, y(1)=1$ in the limit $\\epsilon \\ll 1$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\\boxed{}$ LaTeX environment.", "solution": "$$\\boxed{y(x, \\epsilon) = \\sqrt{2}e^{-1/2} \\frac{e^{x^2/2}}{\\sqrt{x^2+1}} + \\left( 1 - \\sqrt{2}e^{-1/2} \\right) e^{-x/\\epsilon}}$$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 35}
|
| 37 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon^2 y'' + \\epsilon y' - y = 0$ with boundary conditions $y(0) = 0$ and $y(1) = 1$ in the limit $\\epsilon = 0$ from the positive direction. 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) = \\frac{\\sqrt{2\\epsilon}}{1-x + \\sqrt{2\\epsilon}}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 36}
|
| 38 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' + \\epsilon (x+1) y' + y^2 = 0$ with boundary conditions $y(0) = 1, y(1) = -1$ in the limit $\\epsilon \\ll 0+$. 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) = -\\left(1 + \\frac{1-x}{\\sqrt{6\\epsilon}}\\right)^{-2}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 37}
|
| 39 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $ \\varepsilon y'' + \\left(1 + \\frac{2\\varepsilon}{x} - \\frac{2\\varepsilon^3}{x^2}\\right) y' + \\frac{2y}{x} = 0 $ with boundary conditions $y(0)=1, y(1)=1$ in the limit $\\epsilon \\to 0^+$. 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) = 1 + \\left( x^{-2} + 2\\varepsilon(x^{-3} - x^{-2}) - 1 \\right) e^{-2\\varepsilon^2 / x}}$", "parameters": "$x; \\varepsilon$", "type": "boundary_layers", "index": 38}
|
| 40 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''(x) + y'(x) = -e^{-x}$ with boundary conditions $y(0) = 1, y(1) = 2$ in the limit $\\epsilon \\to 0^+$. 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) = e^{-x} + 2 - e^{-1} - (2 - e^{-1})e^{-x/\\epsilon}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 39}
|
| 41 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''(t) + (t-2) y'(t) = t$ with boundary conditions $y(0) = 1, y(1) = 0$ in the limit $\\epsilon \\to 0^+$. 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(t) = t + 2 \\ln(2-t) + 1 - 2 \\ln(2) - (2 - 2 \\ln(2)) e^{-\\frac{1-t}{\\epsilon}}}$", "parameters": "$t; \\epsilon$", "type": "boundary_layers", "index": 40}
|
| 42 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' + (t-2) y' = t^2$ with boundary conditions $y(0) = 0, y(1) = e^{-1/3}$ in the limit $\\epsilon \\to 0^+$. 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) = \\frac{t^2}{2} + 2t + 4\\ln \\left( \\frac{2-t}{2} \\right) + \\left( e^{-1/3} -\\frac{5}{2} + 4\\ln 2 \\right)\\exp\\left( \\frac{t-1}{\\epsilon}\\right)}$", "parameters": "$t; \\epsilon$", "type": "boundary_layers", "index": 41}
|
| 43 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''-(1+2x^2)y+2=0$ with boundary conditions $y(0)=y(1)=1$ in the limit $\\epsilon \\to 0^+$. 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)=\\frac{2}{1+2x^2}-e^{-\\frac{x}{\\sqrt{\\epsilon}}}+\\frac{1}{3}e^{\\frac{\\sqrt{3}(x-1)}{\\sqrt{\\epsilon}}}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 42}
|
| 44 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' - 2 \\tan(x) y' + y = 0$ with boundary conditions $y(-1) = y(1) = 1$ in the limit $\\epsilon \\to 0^+$. 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 = e^{-2 \\tan(1) (1-x)/\\epsilon} + e^{-2 \\tan(1) (x+1)/\\epsilon}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 43}
|
| 45 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y'' + 2 \\tan(x) y' - y = 0$ with boundary conditions $y(-1) = y(1) = 1$ in the limit $\\epsilon \\to 0^+$. 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) = \\sqrt{\\frac{\\sin(x)}{\\sin(1)}}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 44}
|
| 46 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''(x)+(1+2x) y'(x)+8y(x)=0$ with boundary conditions $y(0)=1, y(1)=2$ in the limit $\\epsilon \\to 0^+$. 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) = \\frac{162}{(1+2x)^4} - 161 e^{-x/ \\epsilon}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 45}
|
| 47 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''(x)+(2+3x)y'(x)+6y(x)=0$ with boundary conditions $y(0)=1, y(1)=3$ in the limit $\\epsilon \\to 0^+$. 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{\\frac{75}{(2+3x)^2}-\\frac{71}{4}e^{-2x/ \\epsilon}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 46}
|
| 48 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''(x) - 2y(x) = e^{-x}$ with boundary conditions $y(0)=0, y(1)=1$ in the limit $\\epsilon \\to 0^+$. 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) = -\\frac{1}{2} e^{-x} + \\frac{1}{2} \\exp\\left(-\\sqrt{\\frac{2}{\\epsilon}}x\\right) + \\left(1 + \\frac{1}{2} e^{-1}\\right) \\exp\\left(-\\sqrt{\\frac{2}{\\epsilon}}(1-x)\\right)}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 47}
|
| 49 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''(x)+(1+3x)y'(x)+9y(x)=0$ with boundary conditions $y(0)=2,y(1)=3$ in the limit $\\epsilon \\to 0^+$. 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) = \\frac{192}{(1+3x)^3} - 190 e^{-x/ \\epsilon}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 48}
|
| 50 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''(x) + x^2y' + x^2 = 0$ with boundary conditions $y(0) = 0, y(1) = -32$ in the limit $\\epsilon \\to 0^+$. 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,\\epsilon) = -x - 31 \\frac{\\int_0^{x^3/(3\\epsilon)} t^{-2/3} e^{-t} dt}{\\Gamma(1/3)}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 49}
|
| 51 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''(x) - (1 + \\sin x)\\, y'(x) - y(x) = 0$ with boundary conditions $y(0) = 1, y(1) = 1$ in the limit $\\epsilon \\to 0^+$. 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,\\epsilon)=\\exp\\left( -\\int_0^x \\frac{dt}{1 + \\sin t} \\right)+\\left(1 - 0.493\\right) e^{-(1 + \\sin 1)\\, \\frac{1 - x}{\\epsilon}}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 50}
|
| 52 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''(x) + y' + x(y) = 0$ with boundary conditions $y(0) = 1, y(1) = 0$ in the limit $\\epsilon \\to 0^+$. 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,\\epsilon) = e^{-x/\\epsilon}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 51}
|
| 53 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''(x) + 2y' (x)+ 4y(x) = 0$ with boundary conditions $y(0) = 1, y'(0) = 1$ in the limit $\\epsilon \\to 0^+$. 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) = (1 + \\frac{\\epsilon}{2})e^{-2x} - \\frac{\\epsilon}{2} e^{-\\frac{2x}{\\epsilon}}}$", "parameters": "$x;\\epsilon$", "type": "boundary_layers", "index": 52}
|
| 54 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''(x) - y'(x) + e^{y(x)} = 0$ with boundary conditions $y(0) = -3, y(1) = 0$ in the limit $\\epsilon \\to 0^+$. 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) = -\\ln(e^{3}-x) + \\ln(e^{3}-1)e^{\\frac{x-1}{\\epsilon}}} $", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 53}
|
| 55 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y\"(x) + (1 + x)^2 y'(x) + y(x) = 0$ with boundary conditions $y(0) = 1, y(1) = 1$ in the limit $\\epsilon \\to 0^+$. 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, \\epsilon = e^{(\\frac{1}{1+x} - \\frac{1}{2})} + (1-e^{1/2})e^{-\\frac{x}{\\epsilon}}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 54}
|
| 56 |
+
{"prompt": "Find a uniformly valid approximation to the solution of $\\epsilon y''(x) + \\frac{3x+1}{2x+1}y'(x) - y(x)^{2} = 0$ with boundary conditions $y(0)=0, y(1)=1$ in the limit $\\epsilon \\to 0^+$. 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)=\\frac{9}{15-6x-\\ln(\\frac{3x+1}{4})}-\\frac{9}{15+\\ln(4)}e^{-x/\\epsilon}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 55}
|
| 57 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $$ \\epsilon y'' + 2y' + y = \\cos\\left(\\frac{\\pi x}{2}\\right)$$ with boundary conditions in the limit $\\epsilon \\to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\\boxed{}$ LaTeX environment.", "solution": "$$ \\boxed{y = \\frac{1}{1+\\pi^2}\\left(\\cos\\left(\\frac{\\pi x}{2}\\right)+\\pi\\sin\\left(\\frac{\\pi x}{2}\\right)\\right) - \\frac{\\pi \\sqrt{e}}{1+\\pi^2} e^{-x/2} + \\frac{\\pi(1+e)}{1+\\pi^2} e^{-2(x+1)/\\epsilon}} $$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 56}
|
| 58 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y'' + x y' = 0$ with boundary conditions $y(-1) = 1$, $y(1) = 2$ in the limit $\\epsilon \\to 0^+$. 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 = \\frac{1}{2} \\text{erf}\\left(\\frac{x}{\\sqrt{2\\epsilon}}\\right) + \\frac{3}{2}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 57}
|
| 59 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y'' + \\sin\\left(\\frac{\\pi x}{2}\\right) y' = 0$ with boundary conditions $y(-1) = 0$, $y(1) = 1$ in the limit $\\epsilon \\to 0^+$. 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_{unif}(x, \\epsilon) = \\frac{1}{2} \\text{erf}\\left(x \\sqrt{\\frac{\\pi}{4\\epsilon}}\\right) + \\frac{1}{2}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 58}
|
| 60 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y'' + (e^x - 1) y' = 0$ with boundary conditions $y(-1) = 0$, $y(1) = 1$ in the limit $\\epsilon \\to 0^+$. 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_{unif}(x, \\epsilon) = \\frac{1}{2} \\text{erf}\\left(\\frac{x}{\\sqrt{2\\epsilon}}\\right) + \\frac{1}{2}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 59}
|
| 61 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y'' + x y' + x y = 0$ with boundary conditions $y(-1) = 1$, $y(1) = 2$ in the limit $\\epsilon \\to 0^+$. 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 = e^{-(x+1)} \\frac{1-\\text{erf}(x/\\sqrt{2\\epsilon})}{2} + 2e^{1-x} \\frac{1+\\text{erf}(x/\\sqrt{2\\epsilon})}{2}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 60}
|
| 62 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of \\epsilon y'' + x y' + x y = x, with boundary conditions $y(-1) = 1$, $y(1) = 2$ in the limit $\\epsilon \\to 0^+$. 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_{unif}(x, \\epsilon) = \\frac{1-\\text{erf}(x/\\sqrt{2\\epsilon})}{2} + (1+e^{1-x}) \\frac{1+\\text{erf}(x/\\sqrt{2\\epsilon})}{2}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 61}
|
| 63 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of \\epsilon y'' + x y' + x y = x^2 with boundary conditions $y(-1) = 1$, $y(1) = 3$ in the limit $\\epsilon \\to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. The solution should be smooth, single-form expression which is valid across the entire domain. Place your final solution in a $\\boxed{}$ LaTeX environment.", "solution": "$\\boxed{y_{unif}(x, \\epsilon) = \\left(x - 1 + 3e^{-(x+1)}\\right) \\frac{1-\\text{erf}(x/\\sqrt{2\\epsilon})}{2} + \\left(x - 1 + 3e^{1-x}\\right) \\frac{1+\\text{erf}(x/\\sqrt{2\\epsilon})}{2}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 62}
|
| 64 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y'' + x y' + x y = x$ with boundary conditions $y(-1) = 0$, $y(1) = 0$ in the limit $\\epsilon \\to 0^+$. 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_{unif}(x, \\epsilon) = \\left(1 - e^{-(x+1)}\\right) \\frac{1-\\text{erf}(x/\\sqrt{2\\epsilon})}{2} + \\left(1 - e^{1-x}\\right) \\frac{1+\\text{erf}(x/\\sqrt{2\\epsilon})}{2}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 63}
|
| 65 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of \\epsilon y'' + x y' + x y = x(x-1) with boundary conditions $y(-1) = 0$, $y(1) = 0$ in the limit $\\epsilon \\to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. The solution should be smooth, single-form expression which is valid across the entire domain. Place your final solution in a $\\boxed{}$ LaTeX environment.", "solution": "$\\boxed{y_{unif}(x, \\epsilon) = \\left(x - 2 + 3e^{-(x+1)}\\right) \\frac{1-\\text{erf}(x/\\sqrt{2\\epsilon})}{2} + \\left(x - 2 + e^{1-x}\\right) \\frac{1+\\text{erf}(x/\\sqrt{2\\epsilon})}{2}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 64}
|
| 66 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of \\epsilon y'' + x y' + x y = x with boundary conditions $y(-1) = 0$, $y(1) = 1$ in the limit $\\epsilon \\to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. The solution should be smooth, single-form expression which is valid across the entire domain. Place your final solution in a $\\boxed{}$ LaTeX environment.", "solution": "$\\boxed{y_{unif}(x, \\epsilon) = \\left(1 - e^{-(x+1)}\\right) \\frac{1-\\text{erf}(x/\\sqrt{2\\epsilon})}{2} + \\frac{1+\\text{erf}(x/\\sqrt{2\\epsilon})}{2}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 65}
|
| 67 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y'' + x y' + 2x^2 y = 0$ with boundary conditions $y(-1) = 1$, $y(1) = 2$ in the limit $\\epsilon \\to 0^+$. 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_{unif}(x, \\epsilon) = e^{1-x^2} \\frac{1-\\text{erf}(x/\\sqrt{2\\epsilon})}{2} + 2e^{1-x^2} \\frac{1+\\text{erf}(x/\\sqrt{2\\epsilon})}{2}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 66}
|
| 68 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y'' + x y' + x^2 y = x^2$ with boundary conditions $y(-1) = 1$, $y(1) = 2$ in the limit $\\epsilon \\to 0^+$. 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_{unif}(x, \\epsilon) = \\frac{1-\\text{erf}(x/\\sqrt{2\\epsilon})}{2} + \\left(1 + e^{(1-x^2)/2}\\right) \\frac{1+\\text{erf}(x/\\sqrt{2\\epsilon})}{2}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 67}
|
| 69 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y'' + \\cos(x) y ' + \\sin(x) y= 0$ with boundary conditions $y(0) = 0$, $y(1)= 1$ in the limit $\\epsilon \\to 0^+$. 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) = \\frac{\\cos(x)- e^{-x/\\epsilon}}{\\cos(1)}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 68}
|
| 70 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y'' + xy' = x \\cos{x}$ with boundary conditions $y(1) = 2; y(-1) = 2$ in the limit $\\epsilon \\to 0^+$. 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,\\epsilon) \\approx 2 + \\sin x - \\sin 1 \\erf \\left(\\frac{x}{\\sqrt{2\\epsilon}}\\right)}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 69}
|
| 71 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y'' + (1+x^2)y' - y = 0$ with boundary conditions $y(1) = 1; y(-1) = 1$ in the limit $\\epsilon \\to 0^+$. 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,\\epsilon) \\approx \\exp\\left(\\tan^{-1}(x) - \\frac{\\pi}{4}\\right) + \\left(1 - e^{-\\pi/2}\\right) e^{- 2(x+1)/ \\epsilon}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 70}
|
| 72 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y'' - x^2y' - (3+x^3) = 0$ with boundary conditions $y(1) = 1; y(2) = 1$ in the limit $\\epsilon \\to 0^+$. 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_{uniform}(x) = \\frac{3}{x} -\\frac{x^2}{2} -\\frac{3}{2} + 3e^{\\frac{-4(2-x)}{\\epsilon}}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 71}
|
| 73 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y'' + \\sinh(\\pi x)y' - y = 0$ with boundary conditions $y(1) = 1; y(2) = 1$ in the limit $\\epsilon \\to 0^+$. 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)=(\\frac{\\tanh(\\frac{\\pi x}{2})}{\\tanh(\\pi)})^{\\frac{1}{\\pi}} + (1 - (\\frac{\\tanh(\\frac{\\pi}{2})}{\\tanh(\\pi)})^{\\frac{1}{\\pi}}) \\exp(\\frac{\\sinh(\\pi)(1-x)}{\\epsilon})}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 72}
|
| 74 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y'' - \\tanh(\\pi x)y' - y = 0$ with boundary conditions $y(1) = 1; y(2) = 1$ in the limit $\\epsilon \\to 0^+$. 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) = [\\frac{\\sinh(\\pi)}{\\sinh(\\pi x)}]^\\frac{1}{\\pi} + (1-[\\frac{\\sinh(\\pi)}{\\sinh(2\\pi)}]^\\frac{1}{\\pi})e^{\\tanh(2\\pi)\\frac{-(2-x)}{\\epsilon}}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 73}
|
| 75 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y'' + \\cosh(x)y' - e^xy = 0$ with boundary conditions $y(0) = \\frac{1}{5}; y(1) = 5$ in the limit $\\epsilon \\to 0^+$. 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_{uniform}(x) = \\frac{5}{e^2+1}(e^{2x} + 1) + e^{\\frac{-x}{\\epsilon}}(\\frac{1}{5}-\\frac{10}{e^2+1})}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 74}
|
| 76 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y'' - \\tanh(x^2)y' - xy = 0$ with boundary conditions $y(1) = 1; y(2) = 1$ in the limit $\\epsilon \\to 0^+$. 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) = [\\frac{\\sinh(1)}{\\sinh(x^2)}]^\\frac{1}{2} + (1-[\\frac{\\sinh(1)}{\\sinh(4)}]^\\frac{1}{2})e^{\\tanh(4)\\frac{-(2-x)}{\\epsilon}}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 75}
|
| 77 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y + \\sqrt(x) y' - y = 0$ with boundary conditions $y(0)=0, y(1)=e^2$ in the limit $\\epsilon \\to 0^+$. 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{ e^{2\\sqrt{x}} - 1 + \\frac{\\int_0^{\\frac{x}{\\epsilon^{2/3}}} e^{-\\frac{2}{3}s^{3/2}} \\, ds}{\\left(\\frac{2}{3}\\right)^{1/3} \\Gamma\\left(\\frac{2}{3}\\right)} }$", "parameters": "$x; \\epsilon; s$", "type": "boundary_layers", "index": 76}
|
| 78 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y'' + y' \\sin(x) + y \\sin(2x) = 0$ with boundary conditions $y(0) = \\pi, y(\\pi) = 0$ in the limit $\\epsilon \\to 0^+$. 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{\\pi - \\sqrt{2\\pi} \\int_0^{\\frac{x}{\\sqrt{\\epsilon}}} e^{-s^2/2} \\, ds}$", "parameters": "$x; \\epsilon; s$", "type": "boundary_layers", "index": 77}
|
| 79 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y'' + \\tanh(x)y' + tanh^2(x)y=tanh^2(x)$ with boundary conditions $y(-2)=1, y(2)=2$ in the limit $\\epsilon \\to 0^+$. 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)=1+\\frac{\\cosh(2)}{2\\cosh(x)}(1+\\text{erf}(\\frac{x}{\\sqrt{2\\epsilon}}))}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 78}
|
| 80 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $\\epsilon y'' + \\tanh^2(x)y + \\tanh(x)y'=\\tanh(x)\\text{sech}(x)$ with boundary conditions $y(-2)=0, y(2)=0$ in the limit $\\epsilon \\to 0^+$. 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{\\frac{x-2\\text{erf}(\\frac{x}{\\sqrt{2\\epsilon}})}{\\cosh(x)}}$", "parameters": "$x; \\epsilon$", "type": "boundary_layers", "index": 79}
|
| 81 |
+
{"prompt": "Find a uniformly valid solution of $ \\epsilon y'' - y' = 0$ with boundary conditions $ y(0) = 0, y(1) = 1$ in the limit $\\epsilon = 0$ from the positive direction. 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) = \\frac{1-e^{\\frac{x}{\\epsilon}}}{1-e^{\\frac{1}{\\epsilon}}}}$", "parameters": "$x;\\epsilon$", "type": "boundary_layers", "index": 80}
|
| 82 |
+
{"prompt": "Find a uniformly valid leading order approximation to the solution of $$\\epsilon y'' - y' = \\sin(\\pi x)$$ with boundary conditions $ y(0) = 0, y(1) = 0$ . Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\\boxed{}$ LaTeX environment.", "solution": "$$\\boxed{y(x) = \\frac{\\cos(\\pi x) - 1}{\\pi} + \\frac{2}{\\pi}e^{\\frac{x-1}{\\epsilon}}}$$", "parameters": "$x;\\epsilon$", "type": "boundary_layers", "index": 81}
|
| 83 |
+
{"prompt": "Find the lowest-order uniform approximation to the boundary-value problem: $$ \\epsilon y'' + y' \\sin x + y \\sin(2x) = 0 $$ with boundary conditions:$$ y(0) = \\pi, \\quad y(\\pi) = 0 $$.", "solution": "$$ \\boxed{y(x) \\approx \\pi \\, \\text{erfc}\\left(\\frac{x}{\\sqrt{2\\epsilon}}\\right)} $$", "parameters": "$x;\\epsilon$", "type": "boundary_layers", "index": 82}
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data/integrals.jsonl
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{"prompt": "Consider the following integral:$\\int_0^{5} ( \\frac{e^{-x}}{1 + x^2}) e^{-\\epsilon (\\frac{\\sin^2(x)}{1 + x^4})} dx$In the limit$\\epsilon \\rightarrow \\infty$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{\\sqrt{\\frac{2 \\pi}{2 \\epsilon}}}$", "parameters": "$x; \\epsilon$", "type": "integrals", "index": 0}
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| 2 |
+
{"prompt": "Consider the following integral:$I(x) = \\int_0^1[\\frac{e^{-xt}}{1+t^2}]dt$In the limit$x \\rightarrow 0$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{I(x) = \\frac{\\pi}{4}-\\frac{x}{2}\\ln(2)}$", "parameters": "$t;x$", "type": "integrals", "index": 1}
|
| 3 |
+
{"prompt": "Consider the following integral:$I(x) = \\int_1^\\infty g(t) e^{-xf(t)}dt; g(x)=\\frac{85}{-t+t^6}; f(t) = (\\ln(t-1))^2 + \\cos(\\frac{\\pi}{2} t) + 1$In the limit$x \\rightarrow \\infty$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{I(x) \\approx \\frac{85}{62}\\sqrt{\\frac{2\\pi}{(2+\\frac{\\pi^2}{4})x}}}$", "parameters": "$t;x$", "type": "integrals", "index": 2}
|
| 4 |
+
{"prompt": "Consider the following integral:$I(x)=\\int_x^{1}cos(xt)dt$In the limit$x \\to 0+$, find approximate behavior of the integral up to and including the order x^6. Provide your answer in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{I(x) = 1 - x - \\frac{x^2}{6} + \\frac{x^4}{120} + \\frac{x^5}{6} - \\frac{x^6}{5040} }$", "parameters": "$x$", "type": "integrals", "index": 3}
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| 5 |
+
{"prompt": "Consider the following integral:$I(x) = \\int_{x}^{\\infty} e^{-at^b} dt$In the limit$x \\to +\\infty$, find approximate behavior of the integral up to and including the leading order in x. Provide an expression for the approximate behavior of the integral in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{\\int_{x}^{\\infty} e^{-a t^b} \\, dt \\sim \\frac{e^{-a x^b}}{a b x^{b-1}}}$", "parameters": "$x;a;b$", "type": "integrals", "index": 4}
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| 6 |
+
{"prompt": "Consider the following integral:$ I(x) = \\int_{x}^{\\infty} K_0(t) \\, dt $In the limit$x \\to +\\infty$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{I(x) \\sim \\sqrt{\\frac{\\pi}{2x}} e^{-x}}$", "parameters": "$t;x$", "type": "integrals", "index": 5}
|
| 7 |
+
{"prompt": "Consider the following integral:$\\int_{0}^{1/e} \\frac{e^{-xt}}{\\ln t} \\, dt$In the limit$x \\to +\\infty$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{-\\frac{1}{x \\ln x}}$", "parameters": "$t;x$", "type": "integrals", "index": 6}
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| 8 |
+
{"prompt": "Consider the following integral:$I(\\epsilon) = \\int_0^{10} \\frac{1}{(\\epsilon + 4x^3 + 2x^9)^{3/2}} dx$In the limit$\\epsilon \\rightarrow \\infty$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{I(\\epsilon) = \\frac{1}{\\epsilon^{3/2}} \\cdot 10}$", "parameters": "$\\epsilon$", "type": "integrals", "index": 7}
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| 9 |
+
{"prompt": "Consider the following integral:$I(x) = -\\int_{0}^{\\infty} \\left[ \\frac{1}{e^t - 1} - \\frac{1}{t} + \\frac{1}{2} \\right] e^{-xt} \\, dt$In the limit$x \\to +\\infty$, find the asymptotic expansion of the integral up to and including the first three leading orders in z. Provide your answer in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{I(x) \\sim -\\frac{1}{12x^2} + \\frac{1}{120x^4} - \\frac{1}{252x^6}}$", "parameters": "$x; t$", "type": "integrals", "index": 8}
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| 10 |
+
{"prompt": "Consider the following integral:$I(\\epsilon) = \\int_0^{10} \\frac{dx}{(\\epsilon + 9x^5 + x^{11})^\\frac{13}{7}}$In the limit$\\epsilon \\to \\infty$, find approximate behavior of the integral up to and including the first leading order in \\epsilon. Provide your answer in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{I(\\epsilon) = 10\\cdot\\epsilon^{-13/7}}$", "parameters": "$\\epsilon$", "type": "integrals", "index": 9}
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| 11 |
+
{"prompt": "Consider the following integral:$I(\\epsilon) = \\int_0^{10} \\frac{dx}{(\\epsilon + 9x^5 + x^{11})^\\frac{13}{7}}$In the limit$\\epsilon \\to 10^6$, find approximate behavior of the integral up to and including the first leading order in \\epsilon. Provide your answer in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{I(\\epsilon) = \\frac{\\sqrt[11]{-1 + 2^{\\frac{7}{13}}}}{\\epsilon^{\\frac{136}{77}}}}$", "parameters": "$\\epsilon$", "type": "integrals", "index": 10}
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| 12 |
+
{"prompt": "Consider the following integral:$I(x) = \\int_0^3 (\\cos(t^2) + 5 + 2t^3) e^{-x(2e^t + 7 + \\sin(t))} dt$In the limit$x\\to\\infty$, find approximate behavior of the integral up to and including the first leading order in x. Provide your answer in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{y(x)= \\frac{2e^{-9x}}{x}}$", "parameters": "$x$", "type": "integrals", "index": 11}
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| 13 |
+
{"prompt": "Consider the following integral:$I(x)=\\int_0^\\infty \\frac{t^{x-1}e^{-t}}{t+x}dt$In the limit$x\\to\\infty$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{I(x)\\sim \\frac{\\Gamma(x)}{2x}}$", "parameters": "$x$", "type": "integrals", "index": 12}
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| 14 |
+
{"prompt": "Consider the following integral:$I(x) = \\int_0^{π/4}\\sqrt{sin (t)}e^{-x^2t^2}dt$ In the limit$x \\rightarrow \\infty$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\\boxed{}$ latex environment.", "solution": "$\\boxed{I(x) = \\frac{1}{2}x^{-3/2}\\cdot\\Gamma(\\frac{3}{4})}$", "parameters": "$x$", "type": "integrals", "index": 13}
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data/nonlinear_ode.jsonl
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{"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}
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+
{"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}
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| 3 |
+
{"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}
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| 4 |
+
{"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}
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| 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}$", "parameters": "$x$", "type": "nonlinear_ode", "index": 4}
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| 6 |
+
{"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}
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| 7 |
+
{"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}
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data/nonlinear_pde.jsonl
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{"prompt": "Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$ \\frac{\\partial u}{\\partial t} = \\frac{\\partial^2 u}{\\partial x^2} + u^2 (1-u), \\lim_{x \\to \\infty} u(x, t) = 0 \\quad \\text{and} \\quad \\lim_{x \\to -\\infty} u(x, t) = 1 $$ in the limit . Please place your final solution in a $\\boxed{}$ LaTeX Environment.DONT USE \\operatorname in your boxed solution. There should be a free parameter $v$. Set $v=0.5$ YOU MUST DO THIS. If there are multiple solutions please separate them with a ;.", "solution": "$$ \\boxed{ u(x, t) = \\frac{1}{2} \\left[ 1 - \\tanh\\left( \\frac{x - t/\\sqrt{2}}{2\\sqrt{2}} \\right) \\right] } $$", "parameters": "$x; t;$", "type": "nonlinear_pde", "index": 0}
|
| 2 |
+
{"prompt": "Given the following PDE:$$\\frac{\\partial y}{\\partial t} = \\frac{\\partial^2y}{\\partial x^2} - y^5, \\quad y(x,t)>0$$For the $D\\frac{\\partial^2y}{\\partial x^2}$ and $ \\alpha y^5 t$ terms of the same order of magnitude, find the asymptotic behavior of $y$ at times after $y$ blows up. Please place your final solution in a $\\boxed{}$ LaTeX Environment. Don't use a * symbol in your notation.", "solution": "$\\boxed{y(x,t) = (3/4)^{1/4} x^{-\\frac{1}{2}}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 1}
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| 3 |
+
{"prompt": "Find a self similarity solution (soliton behavior) for the non-linear partial differential equation $$\\frac{\\partial u}{\\partial t} = \\frac{\\partial^2 u}{\\partial x^2} + 2v^2 u^2 (1 - u), \\quad \\lim_{x \\to \\infty} u(x, t) = 0, \\quad \\lim_{x \\to -\\infty} u(x, t) = 1$$ in the limit $t \\rightarrow \\infty$. Please place your final solution in a $\\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=0.5$. If there are multiple solutions please separate them with a \";\".", "solution": "$\\boxed{u(x, t) = \\frac{1}{1 + e^{0.5(x - 0.5 t)}}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 2}
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| 4 |
+
{"prompt": "Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$ \\frac{\\partial u}{\\partial t} = \\frac{\\partial^2 u}{\\partial x^2} + 2 u(1-u)(u-\\frac{1}{4}), \\quad \\lim_{x \\to -\\infty} u(x,t) = 1, \\quad \\lim_{x \\to \\infty} u(x,t) = 0 $$ in the limit $t \\rightarrow \\infty $. Please place your final solution in a $\\boxed{}$ LaTeX Environment.DONT USE \\operatorname in your boxed solution. There should be a free parameter $v$. Set $v=0.5$ YOU MUST DO THIS. If there are multiple solutions please separate them with a ;.", "solution": "$$ \\boxed{ u(x, t) = \\frac{1}{1 + e^{x - 0.5t}} } $$", "parameters": "$x; t;$", "type": "nonlinear_pde", "index": 3}
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| 5 |
+
{"prompt": "Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$ \\frac{\\partial u}{\\partial t} + 6 u^2 \\frac{\\partial u}{\\partial x} + \\frac{\\partial^3 u}{\\partial x^3} = 0, \\quad u(x, t) > 0, \\lim_{x \\to \\pm \\infty} u(x,t) = 0 $$ in the limit $t \\rightarrow \\infty $. Please place your final solution in a $\\boxed{}$ LaTeX Environment.DONT USE \\operatorname in your boxed solution. There should be a free parameter $v$. Set $v=0.5$ YOU MUST DO THIS. If there are multiple solutions please separate them with a ;.", "solution": "$ \\boxed{u_1(x, t) = \\frac{1}{\\sqrt{2}} \\sech\\left(\\frac{x - 0.5t}{\\sqrt{2}}\\right)} $", "parameters": "$x; t;$", "type": "nonlinear_pde", "index": 4}
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| 6 |
+
{"prompt": "Find a localized self similarity solution (soliton behvaiour) for the non-linear partial differential equation $\\partial_{xx} u + \\tanh(u \\partial_x u) \\sech(u \\partial_y u) + \\sin^2(\\partial_{xy} u) - e^{xy} = 0$, $u(0, y) = \\cosh(y)$, $\\partial_x(0, y) = \\sinh(y)$ with $u(0, 0) = 1$ as the maximum value. Localization means $u(x,t)$ and its derivatives vanish at $t= \\pm \\infty$. Please place your final solution in a $\\boxed{}$ LaTeX Environment.", "solution": "$\\boxed{u(x, y) = sech(\\sqrt{\\frac{1}{2}} (x-\\frac{y}{2}))}$", "parameters": "$x; y$", "type": "nonlinear_pde", "index": 5}
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| 7 |
+
{"prompt": "Find a localized self similarity solution (soliton behvaiour) for the non-linear partial differential equation $u_{tt}-u_{xx}-3\\bigl(u^{2}\\bigr)_{xx}-u_{xxxx}=0$ with $u(0, 0) = 1/4$ as the maximum value and $\\partial_t u(-2, 1) < 0$. Localization means $u(x,t)$ and its derivatives vanish at $t= \\pm \\infty$. Please place your final solution in a $\\boxed{}$ LaTeX Environment.", "solution": "$\\boxed{u(x, t) = \\frac{1}{4\\cosh^2(\\frac{x-\\sqrt{\\frac{3}{2}}t}{2\\sqrt{2}})}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 6}
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| 8 |
+
{"prompt": "Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $\\partial_t u +\\sqrt{u}\\partial_x u + \\partial_x^3 u=0, \\lim_{|x|\\to\\infty} u(x,t)=0$ in the limit $t\\rightarrow\\infty$. Please place your final solution in a $\\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=0.5$. If there are multiple solutions please separate them with a ;.", "solution": "$\\boxed{u(x,t) = \\frac{225}{256} \\sech^4( \\frac{\\sqrt{2}}{8} (x - 0.5t) )}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 7}
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| 9 |
+
{"prompt": "Find a self similarity solution (soliton behaviour) for the non-linear partial differential equation $$ \\partial_t u = \\partial_{xx} u - \\frac{(\\partial_x u)^2}{u} + u \\ln u \\left(1 - (\\ln u)^2\\right) $$ that connects the stable state $u = e$ and $u = e^{-1}$ in the limit $|t| \\rightarrow \\infty$. Please place your final solution in a \\boxed{} LaTeX environment. If there are multiple solutions please separate them with a ;.", "solution": "$\\boxed{u(x,t) = e^{\\tanh\\left(\\frac{x}{\\sqrt{2}}\\right)} ; u(x,t) = e^{-\\tanh\\left(\\frac{x}{\\sqrt{2}}\\right)}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 8}
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| 10 |
+
{"prompt": "Find an approximate solution describing the behavior for the non-linear partial differential equation $$ \\frac{\\partial u}{\\partial t} = u^2 \\frac{\\partial^2 u}{\\partial x^2} + \\tan(u);\\quad \\frac{\\partial u}{\\partial x}(0, t) = 0,\\quad u(x\\to\\pm\\infty, t) = 0.$$. The initial condition is $$ u(x, 0) = \\begin{cases} 1 & \\text{if } |x| < 2 \\\\ 0 & \\text{if } |x| \\ge 2 \\end{cases} $$ If the solution blows up in finite time $t^*$, seek the behavior for the limit $t \\rightarrow t^*$ around the blowup point. A local approximation is sufficient. Solve fully, such that the final solution contains only known functions (no undefined functions). Please place your final solution in a \\boxed{} LaTeX Environment. If there is any free parameter, set it equal to 5. $t^*$ should not be in the box; if there is a $t^*$, replace it with 0.5. The solution should not be in cases and should not contain any text. If there are multiple solutions please separate them with a ;. If you cannot find a solution, return zero. If you encounter a function like $\\Theta$, replace it with $Theta$ with no backslash.", "solution": "$$ \\boxed{u(x,t) \\approx \\frac{\\pi}{2} - \\sqrt{2(0.1-t)} - 5 \\frac{x^2}{\\sqrt{0.1-t}}} $$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 9}
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| 11 |
+
{"prompt": "Find an approximate solution describing the behavior for the non-linear partial differential equation $$ \\frac{\\partial u}{\\partial t} = u^4 \\frac{\\partial^2 u}{\\partial x^2} + \\tan(u);\\quad \\frac{\\partial u}{\\partial x}(0, t) = 0,\\quad u(x\\to\\pm\\infty, t) = 0.$$. The initial condition is $$ u(x, 0) = \\begin{cases} 1 & \\text{if } |x| < 2 \\\\ 0 & \\text{if } |x| \\ge 2 \\end{cases} $$ If the solution blows up in finite time $t^*$, seek the behavior for the limit $t \\rightarrow t^*$ around the blowup point. A local approximation is sufficient. Solve fully, such that the final solution contains only known functions (no undefined functions). Please place your final solution in a \\boxed{} LaTeX Environment. If there is any free parameter, set it equal to 5. $t^*$ should not be in the box; if there is a $t^*$, replace it with 0.5. The solution should not be in cases and should not contain any text. If there are multiple solutions please separate them with a ;. If you cannot find a solution, return zero. If you encounter a function like $\\Theta$, replace it with $Theta$ with no backslash.", "solution": "$$ \\boxed{u(x,t) \\approx \\frac{\\pi}{2} - \\sqrt{2(0.1-t)} - 5 \\frac{x^2}{\\sqrt{0.1-t}}} $$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 10}
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| 12 |
+
{"prompt": "Find an approximate analytical solution $u(x, t)$ to the following nonlinear partial differential equation $$\\frac{\\partial u}{\\partial t} + 3u^2 \\frac{\\partial u}{\\partial x} = 0.3 \\frac{\\partial^2 u}{\\partial x^2} - 1.5 u; \\quad u(0, t) = 1 \\quad \\text{for } t > 0.$$ The solution is sought for $x \\ge 0$ and $t \\ge 0$. The initial condition is $$u(x, 0) = 0 \\quad \\text{for } x > 0.$$ Derive an approximate analytical solution $u(x, t)$ that captures the dominant behavior in the limit $t \\rightarrow \\infty_-$. Solve fully, such that the final solution contains only known functions (no undefined functions). Please place your final solution in a \\boxed{} LaTeX Environment. If there are any free constants, approximate them numerically. There should be no variable (non-evaluated) constants or free constraints, however. Do not box more than one equation! The approximation should include the zeroth-order term and the first-order correction term accounting for the nonlinearity.", "solution": "$$\\boxed{u(x, t \\rightarrow \\infty) \\approx \\left(1 + \\frac{\\sqrt{5}}{4}\\right) e^{-\\sqrt{5}x} - \\frac{\\sqrt{5}}{4} e^{-3\\sqrt{5}x}}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 11}
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| 13 |
+
{"prompt": "Find a self-similarity solution (soliton behavior) for $\\partial_t u = \\partial_{xx} u + u (4 - u^2)$ that connects the $u = 0$ solution in the $t \\rightarrow -\\infty$ limit to $u = 2$ in the $t \\rightarrow \\infty$ limit, subject to the boundary condition of $u(0, 0) = 1$. Please place your final solution in a $\\boxed{}$ LaTeX Environment. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants.", "solution": "$$\\boxed{u(x, t) = 2(1 + e^{-\\sqrt{2}(x + 3\\sqrt{2}t)})^{-1}}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 12}
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| 14 |
+
{"prompt": "Find a self-similarity solution (soliton behavior) for $\\partial_{tt} u - \\partial_{xx} u + 2u ((\\partial_t u)^2 - (\\partial_x u)^2) = 2u^5 - u$ that travels at velocity $v = 1/\\sqrt{2}$, subject to the boundary condition of $u(0, 0) = 1$. Please place your final solution in a $\\boxed{}$ LaTeX Environment. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants.", "solution": "$$\\boxed{u(x, t) = 2(e^{\\sqrt{2}(x - (1/\\sqrt{2})t)} + e^{-\\sqrt{2}(x - (1/\\sqrt{2})t)})^{-1}}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 13}
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| 15 |
+
{"prompt": "Find a solution for the non-linear partial differential equation $\\frac{\\partial u}{\\partial t} = -5u\\frac{\\partial u}{\\partial x} -2.5u^2\\frac{\\partial u}{\\partial x} - 0.5\\frac{\\partial^3 u}{\\partial x^3},\\lim_{x \\to \\pm \\infty} u(x,t) = 0,u'(x,t) = 0,u''(x,t) = 0$. Please place your final solution in a $\\boxed{}$ LaTeX Environment.", "solution": "$\\boxed{u(x, t) = \\tanh\\left( \\frac{x }{\\sqrt{2}} \\right)}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 14}
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| 16 |
+
{"prompt": "Find a self-similarity solution for the non-linear partial differential equation $ \\partial_t u = \\partial_{xx} u - \\frac{(\\partial_x u)^2}{u} + u \\ln u \\left(1 - (\\ln u)^2\\right) - \\delta \\partial_x u $ where $\\delta$ is a real constant in the limit $|t| \\rightarrow \\infty$ Please place your final solution in a $\\boxed{}$ LaTeX environment. If there are multiple solutions please separate them with a semicolon.", "solution": "$\\boxed{u(x,t) = e^{\\tanh\\left(\\frac{x-\\delta t}{\\sqrt{2}}\\right)}; u(x,t) = e^{-\\tanh\\left(\\frac{x-\\delta t}{\\sqrt{2}}\\right)}}$", "parameters": "$x; t; \\delta$", "type": "nonlinear_pde", "index": 15}
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| 17 |
+
{"prompt": "Find a self-similar solution (soliton behaviour) for the non-linear partial differential equation $\\partial_t u - \\frac{10}{\\sqrt{30}} \\, \\partial_x u = \\frac{2}{5} \\, \\partial_x^2 u + 2 u (1 - u), \\quad \\lim_{x \\to -\\infty} u(x,t) = 1, \\quad \\lim_{x \\to \\infty} u(x,t) = 0$ in the limit $t \\rightarrow \\infty$. Please write the full equation and don't introduce new variables. Place your final solution in a $\\boxed{}$ LaTeX Environment. If there are multiple solutions, please separate them with a ;.", "solution": "$\\boxed{u(x,t) = \\frac{1}{\\left(1 + \\exp\\left[\\sqrt{\\frac{5}{6}}\\left(x + \\frac{10}{\\sqrt{30}} t \\right)\\right] \\right)^2}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 16}
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| 18 |
+
{"prompt": "Find a self similarity solution for the non-linear partial differential equation $ \\partial_t u = \\partial_{xx} u + (2u-\\sqrt{5})(1-u^2), \\lim_{x \\to -\\infty} u(x,t) = -1, \\lim_{x \\to \\infty} u(x,t) = 1 $ in the limit $t \\rightarrow \\infty$. Please place your final solution in a $\\boxed{}$ LaTeX Environment.", "solution": "$ \\boxed{ u(x,t)=\\tanh(x-\\sqrt{5}t) }$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 17}
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| 19 |
+
{"prompt": "Please solve the non-linear partial differential equation $\\frac{\\partial^2 u}{\\partial t^2}+ \\frac{\\partial^2 u}{\\partial x \\partial t}= \\left( \\frac{\\partial u}{\\partial t} \\right)^2+ \\frac{\\partial u}{\\partial x} \\cdot \\frac{\\partial u}{\\partial t}$ with initial conditions $u(x, 0) = 0, \\quad u_t(x, 0) = x^2$ . Please place your final solution in a $\\boxed{}$ LaTeX Environment.", "solution": "$\\boxed{u(x, t) = -\\ln\\left(1 - \\frac{t^3}{3} + x^2 t - x t^2\\right)}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 18}
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| 20 |
+
{"prompt": "Find a self-similarity solution (soliton behaviour) for the non-linear partial differential equation $$\\frac{\\partial u}{\\partial t} = \\frac{\\partial^2 u}{\\partial x^2} + 5u(1-u)$$ with boundary condition: $\\lim_{x \\to -\\infty} u(x,t) = 1,\\lim_{x \\to \\infty} u(x,t) = 0$. Please place your final solution in a $\\boxed{}$ LaTeX Environment. If there are multiple solutions, please separate them with a ;.", "solution": "$\\boxed{u(x,t) =\\frac{1}{1 + e^{\\frac{\\sqrt{5}}{2}(x - 2\\sqrt{5}\\, t)}}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 19}
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| 21 |
+
{"prompt": "Consider the PDE $\\u_t + \\frac{2x}{t} u_x = u_{xx} + (1-u^2) \\sinh(x), \\quad u(x,1) = \\frac{1}{4}e^{-x^2} -1, \\lim_{|x| \\to \\infty} u(x, t) = -1$. Find the solution in the limit $t \\to \\infty$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\\boxed{}$ LaTeX environment. Do not define additional parameters or constants.", "solution": "$\\boxed{ u(x,t) = \\tanh(-\\cosh(x) + x^2/t)}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 20}
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| 22 |
+
{"prompt": "For $0 < x < 1$, $t > 0$ and a small parameter $0 < \\epsilon \\ll 1$ consider the PDE $u_t = \\epsilon u_{xx} + u(1-u), \\quad u(0, t) = 1, u(1, t) = 1/2, u(x, 0) =1.$. Find the leading order solution as $t \\to \\infty$. Please place your final solution in a $\\boxed{}$ LaTeX Environment", "solution": "$\\boxed{u(x,t \\to \\infty) \\approx \\frac{3}{2} \\left(\\frac{(5+2\\sqrt{6}) e^{(1-x)/\\sqrt{\\epsilon}} - 1}{(5+2\\sqrt{6}) e^{(1-x)/\\sqrt{\\epsilon}} + 1}\\right)^2 - \\frac{1}{2}}$", "parameters": "$x; t; \\epsilon$", "type": "nonlinear_pde", "index": 21}
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| 23 |
+
{"prompt": "Find a self similarity solution (traveling wave front) for the reaction-diffusion equation $$ \\frac{\\partial u}{\\partial t} = \\frac{\\partial^2 u}{\\partial x^2} + 4u(1 - u^2) $$ that connects the stable state $u=-1$ as $x \\to -\\infty$ to the stable state $u=+1$ as $x \\to +\\infty$. The solution should satisfy $u(x,t) = 0$ when $x=ct$ (i.e., the center of the wave where $\\xi=0$). Please place your final solution $u(x,t)$ in a \\boxed{} LaTeX environment. If you have a free parameter, set it to 2.", "solution": "$\\boxed{u(x,t) = \\tanh(\\sqrt{2} x)}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 22}
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| 24 |
+
{"prompt": "Find the solution behavior of $\\frac{\\partial u}{\\partial t} = \\frac{\\partial}{\\partial x} \\left( u^2 \\frac{\\partial u}{\\partial x} \\right)$ in the limit $t \\to \\infty$ (similarity solution) with boundary conditions $\\lim_{|x| \\to \\infty} u(x,t) = 0 \\text{ and initial mass } \\int_{-\\infty}^{\\infty} u(x,0) dx = M$. Please place your final solution in a $\\boxed{}$ LaTeX Environment. When returning the answer, set M=6, center of mass to x=0. Return the expression only for where u(x,t) is nonzero. If you have another free parameter, set it to 3.", "solution": "$\\boxed{u(x,t) = \\frac{1}{(4t)^{1/4}} ( \\frac{12}{\\pi} - (\\frac{x}{(4t)^{1/4}})^2 )^{1/2}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 23}
|
| 25 |
+
{"prompt": "Solve the following nonlinear partial differential equation $$\\partial_t u + \\partial_x u = u \\partial_{xx}u + (x-t)^{-2}$$by finding a travelling wave solution, and determine its leading-order behavior as $|x-t| \\to \\infty$. Your final answer should contain only the variables $x$ and $t$. In particular, do not define any new functions or constants. Leave your final answer in a $\\boxed{}$ LaTeX environment, so that plugging in values of $x$ and $t$ yield a numerical answer. Replace any absolute value signs with the usual () brackets for easy evaluation.", "solution": "$\\boxed{(\\ln((x-t)^2))^{1/2}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 24}
|
| 26 |
+
{"prompt": "Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $u_{tt} - u_{xx} - 6 (u^2)_{xx} - u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = \\frac{1}{2}$.", "solution": "$\\boxed{u(x,t) = \\frac{1}{2} \\text{ sech}^2(\\frac{1}{\\sqrt{2}} (x - \\sqrt{3}t)); u(x,t) = \\frac{1}{2} \\text{ sech}^2(\\frac{1}{\\sqrt{2}} (x + \\sqrt{3}t))}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 25}
|
| 27 |
+
{"prompt": "Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $u_{tt} - u_{xx} - 2 (u^3)_{xx} - u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = \\frac{\\sqrt{3}}{3}$.", "solution": "$\\boxed{u(x,t) = \\frac{\\sqrt{3}}{3} \\text{ sech}(\\frac{1}{\\sqrt{3}} (x - \\frac{2}{\\sqrt{3}}t)); u(x,t) = \\frac{\\sqrt{3}}{3} \\text{ sech}(\\frac{1}{\\sqrt{3}} (x + \\frac{2}{\\sqrt{3}}t))}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 26}
|
| 28 |
+
{"prompt": "Find a self-similarity solution (soliton behaviour) for the non-linear partial differential equation $$\\frac{\\partial u}{\\partial t} + u^2 \\frac{\\partial u}{\\partial x} + \\frac{\\partial^3 u}{\\partial x^2 \\partial t} = 0$$ with boundary condition: $\\lim_{x \\to -\\infty} u(x,t) = 0,\\lim_{x \\to \\infty} u(x,t) = 0$. Please place your final solution in a $\\boxed{}$ LaTeX Environment. If there are multiple solutions, please separate them with a ;.", "solution": "$\\boxed{u(x,t)= \\sqrt{6} \\, \\mathrm{sech}(x-t)}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 27}
|
| 29 |
+
{"prompt": "Suppose we have the following Reaction-Diffusion type Partial Differential Equation, $ \\partial_t u = \\partial_{xx} u - \\alpha (u - \\frac{1}{5}) + \\beta (u - \\frac{1}{5})^3 $, for some function $u(x,t)$ where $\\alpha,\\beta > 0$. Please find a self similarity solution (solition behavior) that connects the state $u = \\frac{1}{5}$ in the limit $t \\rightarrow -\\infty $ to $u = \\frac{1}{5}$ in the limit $t \\rightarrow +\\infty$. Use the substitution $z=x-vt$ and express the answer in terms of $z$. Please place your final solution in a $\\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them with a ;.", "solution": "$\\boxed{u(x,t) = \\sqrt{\\frac{2 \\alpha}{\\beta}}\\text{sech}(\\sqrt{\\alpha} z) + \\frac{1}{5}; u(x,t) = -\\sqrt{\\frac{2\\alpha}{\\beta}}\\text{sech}(\\sqrt{\\alpha} z) + \\frac{1}{5}}$", "parameters": "$x; t;\\alpha;\\beta; z$", "type": "nonlinear_pde", "index": 28}
|
| 30 |
+
{"prompt": "Suppose we have the following Partial Differential Equation, $\\partial_{xxx}u + \\partial_xu(1+\\partial_xu) = \\partial_tu$, for some function $u(x,t)$. Please find a traveling wave solution that connects the steady states $u = 1$ in the limit $t \\rightarrow -\\infty $ and $u = 3$ in the limit $t \\rightarrow +\\infty$. Place your final solution in a $\\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them with a ;.", "solution": "$\\boxed{u(x,t) = 2+\\tanh(\\frac{1}{6}(x+\\frac{10}{9}t))}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 29}
|
| 31 |
+
{"prompt": "Find a traveling-wave solution to the nonlinear partial differential equation $$\\partial_t u + \\partial_x u = -u^3 \\partial_{xx}u + (x-t)^2 + 1$$ and determine its leading-order behavior as $|x-t| \\to \\infty$. Please place your final solution in a $\\boxed{}$ LaTeX Environment. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants. There should be no words inside the $\\boxed{}$ environment; only an expression that can be evaluated by subsituting in values of $x$ and $t$. If there are absolute values in your final answer, replace those with parentheses before putting the answer in the $\\boxed{}$ environment.", "solution": "$$\\boxed{u(x,t) = \\sqrt{2} (x-t) (\\ln(|x-t|))^{\\frac{1}{4}}}$$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 30}
|
| 32 |
+
{"prompt": "Find the solution that contains a logarithmic term to the nonlinear PDE for $x \\in (0, \\pi)$, $t > 0$, $ t u_t - x u_x - (3 - 2x \\cot(x) ) u \\log u = 0, \\quad u(0, t) = 1, u(\\pi, t) = 1, u(x, 0) = 1 $. Please place your final solution in a $\\boxed{}$ LaTeX Environment. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants. There should be no words inside the $\\boxed{}$ environment; only an expression that can be evaluated by subsituting in values of $x$ and $t$.", "solution": "$\\boxed{ u(x,t) = \\exp(\\sin^2(x) t^3 \\log(1 + xt)) }$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 31}
|
| 33 |
+
{"prompt": "Find the solution behavior to$u \\partial_{xt} u - \\partial_x u \\partial_t u + t u \\partial_x u + (1 + t^2) \\sin(2x) u^2 = 0, u(0, t) = e^{-t}, u_t(x, 0) = \\cos^2(x)$ in the limit $t \\to \\infty$Please place your final solution in a $\\boxed{}$ LaTeX Environment.", "solution": "$\\boxed{e^{\\cos(x)^2 t - \\log(1 + t^2)}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 32}
|
| 34 |
+
{"prompt": "Approximate a self similar traveling wave solution $ \\partial_t u = \\partial_{xx} u + u(1-u)(u-\\frac{1}{4}) + x \\left(\\partial_t u + \\frac{\\sqrt{2}}{4} \\partial_x u\\right)$ in the limit $t \\to \\infty$. Return one exact expression for u(x,t). Please place your final solution in a $\\boxed{}$ LaTeX Environment.If there are multiple solutions please separate them with a ;.", "solution": "$\\boxed{u(x,t) = \\frac{1}{1+\\exp{\\left(\\frac{x}{\\sqrt{2}}-\\frac{t}{4}\\right)}}}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 33}
|
| 35 |
+
{"prompt": "Find an approximate analytical solution $u(x, t)$ to the following nonlinear partial differential equation $$u_t + 0.1 (u_x)^2 = u_{xxxx} - 16 u; \\quad u(0, t) = 1, u(x, 0) = 0 $$ The solution is sought for $x \\ge 0$ and $t \\ge 0$. Derive an approximate analytical solution $u(x, t)$ that captures the dominant behavior in the limit $t \\to \\infty$. Solve fully, such that the final solution contains only known functions (no undefined functions). Please place your final solution in a \\boxed{} LaTeX Environment. If there are any free constants, approximate them numerically. There should be no variable (non-evaluated) constants or free constraints, however. Do not box more than one equation! The approximation should include the zeroth-order term and the first-order correction term accounting for the nonlinearity.", "solution": "$\\boxed{u(x, t) \\sim \\frac{599}{600}e^{-2x} + \\frac{1}{600}e^{-4x}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 34}
|
| 36 |
+
{"prompt": "Find a solution to the nonlinear partial differential equation $ \\partial_t u + \\partial_x u = -\\left(u + \\frac{1}{u}\\right) \\partial_{xx}u + (x-t)^2 + \\frac{1}{x-t}$ for its leading-order behavior as $|x-t| \\to \\infty$. Please place your final solution in a $\\boxed{}$ LaTeX Environment.If there are multiple solutions, please separate them with a ;. Do not define additional parameters or constants. There should be no words inside the $\\boxed{}$ environment; only an expression that can be evaluated by subsituting in values of $x$ and $t$.", "solution": "$\\boxed{u(x,t) \\sim \\frac{1}{\\sqrt{2}} (x-t)^2}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 35}
|
| 37 |
+
{"prompt": "Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $u_{tt} - u_{xx} - 8(u^2)_{xx} - u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = 2$.", "solution": "$\\boxed{\\frac{2}{\\cosh^2(2\\sqrt{\\frac{2}{3}}(x-\\sqrt{\\frac{35}{3}}t))}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 36}
|
| 38 |
+
{"prompt": "Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $u_{tt} - u_{xx} - 9(u^3)_{xx} - u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = 1$.", "solution": "$\\boxed{u(x,t)=\\frac{1}{\\cosh(\\frac{3}{\\sqrt{2}}(x+\\sqrt{\\frac{11}{2}}t))}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 37}
|
| 39 |
+
{"prompt": "Solve the following nonlinear partial differential equation $$ \\partial_t u + u^2 \\partial_x u = \\partial_{xx}u - (x-t)^{-2} $$ by finding a travelling wave solution of the form $u(x,t) = U(x-t)$, and determine its leading-order behavior as $x-t \\to \\infty$. Your final answer should contain only the variables $x$ and $t$. In particular, do not define any new functions or constants. Leave your final answer in a $\\boxed{}$ LaTeX environment, so that plugging in values of $x$ and $t$ yield a numerical answer. Replace any absolute value signs with the usual () brackets for easy evaluation.", "solution": "$\\boxed{u(x,t) = 1 + (x-t)^{-1/2}}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 38}
|
| 40 |
+
{"prompt": "Solve the following nonlinear partial differential equation $$\\partial_t u + \\frac{3}{2} \\partial_x u = u^2 \\partial_{xx}u + \\frac{2}{3}\\left(x-\\frac{3}{2}t\\right)^{-2}$$ by finding a travelling wave solution, and determine its leading-order behavior as $|x-\\frac{3}{2}t| \\to \\infty$. Your final answer should contain only the variables $x$ and $t$. In particular, do not define any new functions or constants. Leave your final answer in a $\\boxed{}$ LaTeX environment, so that plugging in values of $x$ and $t$ yield a numerical answer. Replace any absolute value signs with the usual () brackets for easy evaluation.", "solution": "$\\boxed{\\left(2\\ln\\left(x-\\frac{3}{2}t\\right)\\right)^{1/3}}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 39}
|
| 41 |
+
{"prompt": "solve$ \\partial_t u = (u^3-2(x-2t)) \\partial_x u - \\partial_x \\left( u \\partial_x u \\right) - (x-2t)^{-3} $as $t\\to\\infty, u(\\pm infty,t)\\to 5$ away from any divergences to a nonconstant function. Place one final solution in a $\\boxed{}$ LaTeX Environment", "solution": "$\\boxed{u(x,t)=-\\frac{1}{254 (x-2t)^2}-\\frac{2}{16129 (x-2t)}+\\frac{4 \\log (x-2t)}{2048383}-\\frac{4 \\log (127-2 (x-2t))}{2048383}+5}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 40}
|
| 42 |
+
{"prompt": "solve$ \\partial_t u = u^2 \\partial_x u - \\partial_x \\left( u \\partial_x u \\right) - (x-2t)^{-3/2} $as $t\\to\\infty, u(\\pm infty,t)\\to 5$ away from any divergences to a nonconstant function. Place one final solution in a $\\boxed{}$ LaTeX Environment", "solution": "$\\boxed{u(x,t)=-2(x-2t)^{-1/2}/(2+5^3)+5}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 41}
|
| 43 |
+
{"prompt": "Suppose we have the following Partial Differential Equation, $\\partial_{xxx}u + \\partial_{t}u(1-\\partial_xu)= 0$, for some function $u(x,t)$. Please find a travelling wave solution that connects the steady states $u = 1$ in the limit $t \\rightarrow -\\infty $ and $u = 3$ in the limit $t \\rightarrow +\\infty$. Place your final solution in a $\\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them with a ;.", "solution": "$\\boxed{u(x,t) = 2+\\tanh(\\frac{3}{2}(x-9t))}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 42}
|
| 44 |
+
{"prompt": "solve$ \\partial_t u = (u^3-2(x-2t)) \\partial_x u - \\partial_x \\left( (u^2+3) \\partial_x u \\right) - (x-2t)^{-3} $as $t\\to\\infty, u(\\pm infty,t)\\to 5$ away from any divergences to a nonconstant function. Place ONE final solution in a $\\boxed{}$ LaTeX Environment", "solution": "$\\boxed{u(x,t)=-\\frac{1}{254 (x-2t)^2}-\\frac{2}{16129 (x-2t)}+\\frac{4 \\log (x-2t)}{2048383}-\\frac{4 \\log (127-2 (x-2t))}{2048383}+5}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 43}
|
| 45 |
+
{"prompt": "Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $\\frac{\\partial u}{\\partial t} + u \\frac{\\partial u}{\\partial x} = \\frac{\\partial^2 u}{\\partial x^2} - u^2; x \\geq 0, t > 0; u(0, t) = \\frac{1}{\\sqrt{t}}, u(x,0) = 0$ in the limit $t \\rightarrow \\infty$. Please place your final solution in a $\\boxed{}$ LaTeX Environment.DONT USE \\operatorname in your boxed solution. If there are multiple solutions please separate them with a ;.", "solution": "$\\boxed{u(x,t) = \\frac{1}{\\sqrt{t}} e^{-x/\\sqrt{t}}}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 44}
|
| 46 |
+
{"prompt": "Find the asymptotic solution with first two leading terms for the non-linear partial differential equation $\\frac{\\partial u}{\\partial t} + u \\frac{\\partial u}{\\partial x} = \\frac{\\partial^2 u}{\\partial x^2} - u^2; x \\geq 0, t > 0; u(0, t) = 1, u(x,0) = 0$ in the limit $t \\rightarrow \\infty$. Please place your final solution in a $\\boxed{}$ LaTeX Environment.DONT USE \\operatorname in your boxed solution. If there are multiple solutions please separate them with a ;.", "solution": "$\\boxed{u(x,t) = \\frac{1}{1 + 2.5x} + 0.611 e^{-0.5 x} - \\frac{0.509}{1 + 2.5x} e^{-0.5 x}}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 45}
|
| 47 |
+
{"prompt": "Find an approximate analytical solution $u(x, t)$ to the following nonlinear partial differential equation $u_t +0.1(u_x)^2+0.05uu_{xx} = u_{xxxx}-16u+0.2e^{-3x}, \\quad u(0,t)=1, u(x,0)=0 $ The solution is sought for $x \\ge 0$ and $t \\ge 0$. Derive an approximate analytical solution $u(x, t)$ that captures the dominant behavior in the limit $t \\to \\infty$. Solve fully, such that the final solution contains only known functions (no undefined functions). Please place your final solution in a \\boxed{} LaTeX Environment. If there are any free constants, approximate them numerically. There should be no variable (non-evaluated) constants or free constraints, however. Do not box more than one equation! The approximation should include the zeroth-order term and the first-order correction term accounting for the nonlinearity.", "solution": "$\\boxed{$u(x,t)=0.9994170441764800e^{-2x}+0.003076923076923077e^{-3x}-0.002484639053254438e^{-4x}-0.000009318214941557116e^{-5x}-0.000000009985207100591716e^{-6x}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 46}
|
| 48 |
+
{"prompt": "Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $u_t - u_{xx} - u^3 = (1-2t)\\cos(x)e^{-t^2} - \\cos^3(x)e^{-3t^2}, u(x,0) = \\cos(x)$ in the limit N/A. Please place your final solution in a $\\boxed{}$ LaTeX Environment.DONT USE \\operatorname in your boxed solution. There should be a free parameter $v$. Set $v=0.5$ YOU MUST DO THIS. If there are multiple solutions please separate them with a ;.", "solution": "$\\boxed{u(x,t) = \\cos(x) e^{-t^2}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 47}
|
| 49 |
+
{"prompt": "Find a nontrivial self similarity solution (soliton behaviour) for the non-linear partial differential equation $$ \\partial_t u + \\left(\\frac{45}{16} u^{1/2} - \\frac{3}{2} u\\right)\\partial_x u + \\partial_x^3 u=0 $$ The wave is assumed to travel with velocity $v=0.5$. The boundary condition is $\\lim_{|x|\\to\\infty} u(x,t)=0$. Consider this in the limit \"$t\\rightarrow\\infty$\". Please place your final solution in a $\\boxed{}$ LaTeX Environment. If there are multiple solutions, please separate them with a \";\". If there exist forms in terms of exponential for functions in the output, then you should use them.", "solution": "$\\boxed{u(x,t) = \\frac{16}{\\left(6 + e^{\\frac{x-0.5t}{2\\sqrt{2}}} + e^{-\\left(\\frac{x-0.5t}{2\\sqrt{2}}\\right)}\\right)^2}}$", "parameters": "$x; t; t_0$", "type": "nonlinear_pde", "index": 48}
|
| 50 |
+
{"prompt": "Find a self-similar solution to $$u_t=(u^2)_{xx}$$ for $$x\\in \\mathbb{R}, t>0$$. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants. Box your final answer.", "solution": "$$\\boxed{u(x,t)=t^{-1/3}\\cdot\\max\\{1-\\frac{x^2}{12t^{2/3}},0\\}}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 49}
|
| 51 |
+
{"prompt": "For the nonlinear PDE $$u_t + 6u\\cdot u_x + u_{xxx} = 0$$ with $x\\in [-L,L] \\ \\text{(periodic)}, \\ t>0$, seek a traveling‐wave solution. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants. Box your final answer.", "solution": "$$\\boxed{u(x,t) = \\frac{1}{2}\\cdot\\text{sech}^2\\left(\\frac{1}{2}\\cdot(x - t)\\right)}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 50}
|
| 52 |
+
{"prompt": "Find a self similarity solution (soliton behavior) for the non-linear partial differential equation $ \\partial_t u + u^2 \\partial_x u + \\partial_x^3 u = 0 $ in the limit $t\\to\\infty$. Please place your final solution in a $\\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=6$. If there are multiple solutions please separate them with a ;.", "solution": "$$\\boxed{u(x,t) = 6 \\sech{\\left( \\sqrt{6} (x - 6t) \\right)}}$$", "parameters": "$x; t; v$", "type": "nonlinear_pde", "index": 51}
|
| 53 |
+
{"prompt": "Find a self similarity solution (soliton behavior) for the non-linear partial differential equation $ \\partial_t u + u^{1/2} \\partial_x u + 1.5\\partial_x^3 u = 0 $ in the limit $t\\to\\infty$. Please place your final solution in a $\\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=0.5$. If there are multiple solutions please separate them with a ;.", "solution": "$$\\boxed{u(x,t) = \\frac{225}{256} \\sech^4\\left( \\frac{\\sqrt{3}}{12}(x - 0.5t) \\right)}$$", "parameters": "$x; t; v$", "type": "nonlinear_pde", "index": 52}
|
| 54 |
+
{"prompt": "Find a self similarity solution (soliton behavior) for the non-linear partial differential equation $ \\partial_t u + u^{3/2} \\partial_x u + \\partial_x^3 u = 0 $ in the limit $t\\to\\infty$. Please place your final solution in a $\\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=0.5$. If there are multiple solutions please separate them with a ;.", "solution": "$$\\boxed{u(x,t) = ( \\frac{35}{16} )^{2/3} \\sech^{4/3}( \\frac{3\\sqrt{2}}{8}(x - 0.5t) )}$$", "parameters": "$x; t; v$", "type": "nonlinear_pde", "index": 53}
|
| 55 |
+
{"prompt": "Find a self similarity solution (soliton behavior) for the non-linear partial differential equation $ \\partial_t u + 2u^{1/2} \\partial_x u + \\partial_x^3 u = 0 $ in the limit $t\\to\\infty$. Please place your final solution in a $\\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=0.5$. If there are multiple solutions please separate them with a ;.", "solution": "$$\\boxed{u(x,t) = \\frac{225}{1024} \\sech^4\\left( \\frac{\\sqrt{2}}{8}(x - 0.5t) \\right) }$$", "parameters": "$x; t; v$", "type": "nonlinear_pde", "index": 54}
|
| 56 |
+
{"prompt": "Find the self-similar (soliton-like) solution $u(x,t)$ for the nonlinear partial-differential equation $$ \\frac{\\partial u}{\\partial t} + u\\frac{\\partial u}{\\partial x} = \\frac{\\partial^2 u}{\\partial x^2} - u^3, u(0,t)=1, u(x,0)=0, x\\ge 0, t>0 $$ in the limit $t\\to\\infty$", "solution": "$$ \\boxed{U(x)=\\frac{1}{1+\\frac{1}{2}x}} $$", "parameters": "$x$", "type": "nonlinear_pde", "index": 55}
|
| 57 |
+
{"prompt": "Find the self-similar (soliton-like) solution $u(x,t)$ for the nonlinear partial-differential equation $$\\partial_t u + u^{1/2}\\,\\partial_x u = \\partial_{xx}u - 2.5\\,u^2, u(0,t)=1, u(x,0)=0,\\;x\\ge0,\\;t>0$$ in the limit $t\\to\\infty$", "solution": "$$ \\boxed{U(x)=\\frac{1}{(1+0.5\\,x)^{2}}} $$", "parameters": "$x$", "type": "nonlinear_pde", "index": 56}
|
| 58 |
+
{"prompt": "Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$\\partial_t u = partial_{xx} u + u^2 - u; u(x;0) = 0.1 \\quad (|x| \\leq 3), u(x;0) = 0 \\quad (|x| \\gt 3); \\lim_{x \\to -\\infty} u(x,t) = 0, \\quad \\lim_{x \\to \\infty} u(x,t) = 0$$ in the limit $t\\to\\infty$. Please place your final solution in a $\\boxed{}$ LaTeX Environment.DONT USE \\operatorname in your boxed solution. There should be a free parameter $v$. Set $v=0.5$ YOU MUST DO THIS. If there are multiple solutions please separate them with a ;.", "solution": "$$\\boxed{\\frac{0.6}{\\sqrt{4 \\pi t}} e^{-t} e^{-\\frac{x^2}{4t}}}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 57}
|
| 59 |
+
{"prompt": "Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$\\partial_{t} u = \\partial_{xx} u - 1.5 \\, \\partial_{x} u + 0.2 \\, u, \\quad u(x,0) = \\frac{1}{\\sqrt{0.1}} e^{-\\frac{x^2}{0.4}}, \\quad u(+\\infty,t) = u(-\\infty,t) = 0$$.", "solution": "$$\\boxed{\\frac{1}{\\sqrt{t+0.1}} e^{-\\frac{(x-1.5t)^2}{4(t+0.1)} + 0.2t}}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 58}
|
| 60 |
+
{"prompt": "Find a nontrivial self similarity solution (soliton behaviour) for the non-linear partial differential equation $$ \\partial_t u + \\left(\\frac{15}{8} u^{1/2} + \\frac{9}{4} u\\right)\\partial_x u + \\partial_x^3 u=0 $$ The wave is assumed to travel with velocity $v=0.5$. The boundary condition is $\\lim_{|x|\\to\\infty} u(x,t)=0$. Consider this in the limit \"$t\\rightarrow\\infty$\". Please place your final solution in a $\\boxed{}$ LaTeX Environment. If there are multiple solutions, please separate them with a \";\". If there exist expressions in terms of exponential for functions in the output, then you should use them. Do not use $\\operatorname$ in the solution", "solution": "$$ \\boxed{u(x,t) = \\frac{40}{\\left(2\\sqrt{10}+5e^{\\frac{x-0.5t}{2\\sqrt{2}}} + 5e^{-\\left(\\frac{x-0.5t}{2\\sqrt{2}}\\right)}\\right)^2} ; u(x,t)=0} $$", "parameters": "$x; t; v$", "type": "nonlinear_pde", "index": 59}
|
| 61 |
+
{"prompt": "Find the self similarity solution for the nonlinear partial differential equation $$\\partial_t = \\partial_x (u^3 \\partial_xu) $$ Where the initial shape of the solution is $u(x, t)=u(x, 0)= 0.9 * \\exp(-x^2 /2)$ There should be a free parameter A in the solution which you should set to $A =1.1$. Please put your final solution in a $\\boxed{}$ LaTeX Environment.", "solution": "$$\\boxed{u(x,t) = \\frac{1}{(5t)^{\\frac{1}{5}}} \\sqrt[3]{-\\frac{3}{2}\\left(\\frac{x}{(5t)^{\\frac{1}{5}}}\\right)^2 + A}}$$", "parameters": "$x; t; A$", "type": "nonlinear_pde", "index": 60}
|
| 62 |
+
{"prompt": "Find the solution for the nonlinear partial differential equation $$\\partial_t u = \\partial_{xx}u + u^2 (1 - u^2)$$", "solution": "$$\\boxed{u(x,t)=\\tanh( \\frac{1}{\\sqrt{2}}(x - \\sqrt{2}t))}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 61}
|
| 63 |
+
{"prompt": "Find a traveling-wave solution to $u_t = -u_{xx} - 2 u u_x^2/(1 - u^2)$, such that $u \\rightarrow 0$ as $t \\rightarrow \\infty$, $u \\rightarrow 1$ as $t \\rightarrow -\\infty$, $u(0,0) - 0.76159 < 0.01$, and $u_x(0,0) - 0.41997 < 0.01$. Please place your final solution in a $\\boxed{}$ LaTeX Environment. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants.", "solution": "$$\\boxed{u(x,t) = \\tanh(e^{x - t})}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 62}
|
| 64 |
+
{"prompt": "Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $2u_{tt} - 3u_{xx} - 6(u^2)_{xx} - 5u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = 13$. Please place your final solution in a $\\boxed{}$ LaTeX Environment.", "solution": "$$ \\boxed{ u(x,t) = 13\\,\\mathrm{sech}^2\\left(\\sqrt{\\frac{13}{5}}\\left(x-\\sqrt{\\frac{55}{2}}t\\right)\\right) } $$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 63}
|
| 65 |
+
{"prompt": "Find a self-similar solution to $$u_t=(u^5)_{xx}$$ for $$x\\in \\mathbb{R}, t>0$$. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants. Please place your final solution in a $\\boxed{}$ LaTeX Environment.", "solution": "$$\\boxed{u(x,t)=t^{-1/6}\\cdot\\max\\{1-\\frac{x^2}{15t^{1/3}},0\\}}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 64}
|
| 66 |
+
{"prompt": "Find a self-similarity solution for the non-linear partial differential equation: $d_t u = d_{xx}u + u(u - \\alpha)(1 - u)$ where $\\alpha$ is a constant such that $0 < \\alpha < 1/2$. The solution connects $u=1$ ($t \\rightarrow -\\infty$) to $u=0$ ($t \\rightarrow +\\infty$). Determine the wave speed $c$ in terms of $\\alpha$. Then, find the explicit form of the solution $u(x,t)$ given the initial condition $u(0,0) = \\frac{1}{1+e^{\\sqrt{2}/2}}$ and assuming $\\alpha = 1/4$.", "solution": "$$ \\boxed{c(\\alpha) = \\frac{1-2\\alpha}{\\sqrt{2}}} $$;$$ \\boxed{u(x,t) = \\frac{1}{1+e^{\\frac{1}{\\sqrt{2}}(-x + \\frac{\\sqrt{2}}{4}t + 1)}}} $$", "parameters": "$x; t; \\alpha$", "type": "nonlinear_pde", "index": 65}
|
| 67 |
+
{"prompt": "Find a self-similarity solution for $$ \\partial_t u = \\partial_{xx}(u^2) - \\partial_{xxxx}u + (\\partial_x u)^2 $$ with maximum value at $u(0,0) = 1$.", "solution": "$$ \\boxed{ u(x,t) = (t+K_0^2)^{-1/2} f_s\\left( \\frac{x}{(t+K_0^2)^{1/4}} \\right) } $$", "parameters": "$x; t;K_0;f_s$", "type": "nonlinear_pde", "index": 66}
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{"prompt": "Approximate the expectation of the solution of SDE $$dX_t = \\Bigl(-\\frac{\\sin(t^{-1/2})}{t^2}-\\frac{1+\\tfrac1t}{2t^{3/2}}\\cos(t^{-1/2})-\\frac{B_t}{2t^{3/2}}+\\frac1{t+1}+B_t^2\\bigl(6t^{-3}-\\frac1{(t+1)^2}\\bigr)-3\\frac{B_t^4}{t^4}\\Bigr)\\,dt + \\Bigl(\\frac1{\\sqrt t}+\\frac{2B_t}{t+1}+\\frac{4B_t^3}{t^3}\\Bigr)\\,dB_t.$$ in the limit $t \\to \\infty$ Please provide your answer in LaTeX \\boxed{} environment. Final answer should be in closed form and parsable. I don't want any self-defined constants other than e and \\pi. I don't want to have any short-cut function you used throughout the solution in the final solution either. The solution should solely be a function of t.", "solution": "$\\boxed{\\frac{1}{1 + \\sqrt{t}}}$", "parameters": "$t$", "type": "other", "index": 0}
|
| 2 |
+
{"prompt": "Approximate the expectation function of the solution of SDE $dX_t = \\Bigl[\\bigl(\\frac1t-\\frac1{t^2}+\\frac2{t^3}\\bigr)+\\ln(t)\\bigl(\\frac1{t^2}-\\frac4{t^3}\\bigr)-\\tfrac12\\ln(t)\\bigl(1-\\frac1t+\\frac2{t^2}\\bigr)\\Bigr]\\cos(B_t)\\,dt - \\ln(t)\\bigl(1-\\frac1t+\\frac2{t^2}\\bigr)\\sin(B_t)\\,dB_t$ in the limit $ t > 0$ Please provide your answer in LaTeX \\boxed{} environment. Final answer should be in closed form and parsable. I don't want any self-defined constants other than e and \\pi. I don't want to have any short-cut function you used throughout the solution in the final solution either. The solution should solely be a function of t.", "solution": "$\\boxed{e^{-\\frac{t}{2}} \\ln(t) (1 - \\frac{1}{t} + \\frac{2}{t^2})}$", "parameters": "$t$", "type": "other", "index": 1}
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| 3 |
+
{"prompt": "Approximate the expectation function of the solution of SDE $$dX_t=\\exp\\bigl(\\sqrt{t}\\sin(2t)\\,B_t\\bigr)\\Bigl(-\\frac{2t}{(1+t^2)^2}+\\frac{B_t}{1+t^2}\\Bigl(\\frac{\\sin(2t)}{2\\sqrt{t}}+2\\sqrt{t}\\cos(2t)\\Bigr)+\\frac{(\\sqrt{t}\\sin(2t))^2}{2(1+t^2)}\\Bigr)\\,dt+\\frac{\\sqrt{t}\\sin(2t)}{1+t^2}\\exp\\bigl(\\sqrt{t}\\sin(2t)\\,B_t\\bigr)\\,dB_t$$ in the limit $t \\to \\infty$ Please provide your answer in LaTeX \\boxed{} environment. Final answer should be in closed form and parsable. I don't want any self-defined constants other than e and \\pi. I don't want to have any short-cut function you used throughout the solution in the final solution either. The solution should solely be a function of t.", "solution": "$\\boxed{u(x,t) = \\frac{1}{1 + t^2} e^{t^2 \\sin^2(2t)}}$", "parameters": "$t$", "type": "other", "index": 2}
|
| 4 |
+
{"prompt": "Approximate the expectation function of the solution of SDE $$dX_t=\\exp\\bigl(\\cos(t)B_t\\bigr)\\Bigl(-\\frac{2t}{(1+t^2)^2}-\\frac{B_t\\sin(t)}{1+t^2}+\\frac{\\cos^2(t)}{2(1+t^2)}\\Bigr)\\,dt+\\frac{\\cos(t)}{1+t^2}\\exp\\bigl(\\cos(t)B_t\\bigr)\\,dB_t.$$ in the limit $t \\to \\infty$ Please provide your answer in LaTeX \\boxed{} environment. Final answer should be in closed form and parsable. I don't want any self-defined constants other than e and \\pi. I don't want to have any short-cut function you used throughout the solution in the final solution either. The solution should solely be a function of t. Do not use \\exp, but just explicitly write e^.", "solution": "$\\boxed{\\frac{1}{1 + t^2} e^{\\frac{t}{2}\\cos^2(t)}}$", "parameters": "$t$", "type": "other", "index": 3}
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| 5 |
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{"prompt": "Approximate the expectation function of the solution of SDE $$dX_t =\\exp\\!\\Bigl(\\tfrac{\\sqrt2\\,\\sin(t)}{t^{1/4}}\\,B_t\\Bigr)\\Bigl(-\\frac{2t}{(1+t^2)^2}+\\frac{B_t}{1+t^2}\\Bigl(\\frac{\\sqrt2\\,\\cos(t)}{t^{1/4}}-\\frac{\\sqrt2\\,\\sin(t)}{4\\,t^{5/4}}\\Bigr)+\\frac{\\sin^2(t)}{t^{1/2}(1+t^2)}\\Bigr)\\,dt+\\frac{\\sqrt2\\,\\sin(t)}{t^{1/4}(1+t^2)}\\exp\\!\\Bigl(\\tfrac{\\sqrt2\\,\\sin(t)}{t^{1/4}}\\,B_t\\Bigr)\\,dB_t.$$ in the limit $t \\to \\infty$ Please provide your answer in LaTeX \\boxed{} environment. Final answer should be in closed form and parsable. I don't want any self-defined constants other than e and \\pi. I don't want to have any short-cut function you used throughout the solution in the final solution either. The solution should solely be a function of t. Do not use \\exp, but just explicitly write e^.", "solution": "$\\boxed{\\frac{1}{1 + t^2} e^{\\sqrt{t} \\sin^2(t)}}$", "parameters": "$t$", "type": "other", "index": 4}
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{"prompt": "Approximate the expectation function of the solution of SDE $$ dX_t=\\exp\\Bigl(\\tfrac{t}{\\sqrt3}\\sin\\bigl(\\tfrac{t}{3}\\bigr)B_t\\Bigr)\\Bigl(-\\tfrac{2B_t^2}{t^3(1+t^2)}-\\tfrac{2t(1+\\tfrac{B_t^2}{t^2})}{(1+t^2)^2}+\\tfrac{(1+\\tfrac{B_t^2}{t^2})B_t}{1+t^2}(\\tfrac{\\sin\\tfrac{t}{3}}{\\sqrt3}+\\tfrac{t\\cos\\tfrac{t}{3}}{3\\sqrt3})+\\tfrac12\\tfrac{1+\\tfrac{B_t^2}{t^2}}{1+t^2}\\tfrac{t^2}{3}\\sin^2\\bigl(\\tfrac{t}{3}\\bigr)\\Bigr)\\,dt+\\tfrac{1+\\tfrac{B_t^2}{t^2}}{1+t^2}\\tfrac{t}{\\sqrt3}\\sin\\bigl(\\tfrac{t}{3}\\bigr)\\exp\\Bigl(\\tfrac{t}{\\sqrt3}\\sin\\bigl(\\tfrac{t}{3}\\bigr)B_t\\Bigr)\\,dB_t.$$ in the limit $t \\to \\infty$ Please provide your answer in LaTeX \\boxed{} environment. Final answer should be in closed form and parsable. I don't want any self-defined constants other than e and \\pi. I don't want to have any short-cut function you used throughout the solution in the final solution either. The solution should solely be a function of t. Do not use \\exp, but just explicitly write e^. Do not use X_0. Do not use enlarge any parenthesis. Keep paranthesis () without modifying anything", "solution": "$\\boxed{\\frac{1}{1 + t^2} e^{\\frac{t^3}{6} \\sin^2(t/3)}}$", "parameters": "$t$", "type": "other", "index": 5}
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{"prompt": "Find the leading order WKB approximation for the specific differential equation: $$ \\epsilon^2 y''(x) + (1+x^2) y(x) = 0 $$ with initial conditions at $y(0) = 1, y'(0) = 0$ where $\\epsilon$ is a small positive parameter ($0 < \\epsilon \\ll 1$). Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\\boxed{}$ LaTeX environment.", "solution": "$\\boxed{ y_{WKB}(x) \\approx (1+x^2)^{-1/4} \\cos\\left( \\frac{1}{\\epsilon} \\left[ \\frac{1}{2} x \\sqrt{1+x^2} + \\frac{1}{2} arcsinh(x) \\right] \\right)} $", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 0}
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{"prompt": "Find the leading order WKB approximation for the specific differential equation: $\\epsilon^2 y'' = (x+1)y$ with initial conditions at $y(0)=0, y'(0)=1$ where $\\epsilon$ is a small positive parameter ($0 < \\epsilon \\ll 1$). Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\\boxed{}$ LaTeX environment.", "solution": "$\\boxed{y(x) \\sim \\epsilon (x+1)^{-1/4}\\sinh\\left[\\frac{2\\left((x+1)^{3/2}-1\\right)}{3\\epsilon}\\right]}$", "parameters": "$x; \\epsilon$", "type": "wkb", "index": 1}
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{"prompt": "Find the leading order WKB approximation for the specific differential equation: $$ \\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}
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| 4 |
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{"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}
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| 5 |
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{"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}
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| 6 |
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{"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}
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| 7 |
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{"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}
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| 8 |
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{"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}
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| 9 |
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{"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}
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| 10 |
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{"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}
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| 11 |
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{"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}
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| 12 |
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{"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}
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| 13 |
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{"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}
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| 14 |
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{"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}
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| 15 |
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{"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}
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| 16 |
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{"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}
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| 17 |
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{"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}
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| 18 |
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{"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}
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| 19 |
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{"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}
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| 20 |
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{"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}
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| 21 |
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{"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}
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| 22 |
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{"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}
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| 23 |
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{"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}
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| 24 |
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{"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}
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| 25 |
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{"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}
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| 26 |
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{"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}
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| 27 |
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{"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}
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| 28 |
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{"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}
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| 29 |
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{"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}
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