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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.
|
$\boxed{I(x)=(\frac{2}{\sqrt{x}}-\frac{1}{3x^{3/2}}+\frac{3}{20x^{5/2}} )}$
|
$x; t$
|
asympytotic_series
| 0
|
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.
|
$\boxed{I(x)=(1/x)(1-e^{-\frac{x \pi}{4}}) }$
|
$x; t$
|
asympytotic_series
| 1
|
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.
|
$\boxed{I(x)=\frac{\pi}{2} - \frac{\cos x}{x} + \frac{\sin x}{x^2}}$
|
$x; t$
|
asympytotic_series
| 2
|
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.
|
$\boxed{I(x) = e^{-x^2}(\frac{1}{2x(1+x^5)} - \frac{(1+6x^5)}{4x^3(1+x^5)^2})}$
|
$x; t$
|
asympytotic_series
| 3
|
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.
|
$\boxed{I(x) = \frac{\ln(x^3)\sin(x^3)}{3x^2} - \frac{\ln(x)\sin(1)}{3} -\frac{2(\ln(x^3)-1)\cos(x^3)}{9x^5} + \frac{2(\ln(x)-1)\cos(1)}{9}}$
|
$x; t$
|
asympytotic_series
| 4
|
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.
|
$\boxed{y = e^{\frac{x^3}{3}} + (2-e^{1/3})e^{-(1-x)/\epsilon}}$
|
$x; \epsilon$
|
boundary_layers
| 0
|
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.
|
$\boxed{y = A*e^{\frac{x^3}{3}} + (B-A*e^{1/3})e^{-(1-x)/\epsilon}}$
|
$x; \epsilon; A; B$
|
boundary_layers
| 1
|
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.
|
$\boxed{y_{unif}(x) \approx \left[ e^x - x Ei(x) + (1 - e + Ei(1)) x \right] - \left[e^{-1} + Ei(-1) - 1 + e - Ei(1)\right] e^{-(x+1)/\epsilon}}$
|
$x; \epsilon$
|
boundary_layers
| 2
|
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.
|
$\boxed{y = \sqrt{\frac{\sin x}{\sin 1}}}$
|
$x; \epsilon$
|
boundary_layers
| 3
|
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.
|
$\boxed{y = E^{-(x+1)/\epsilon}+ E^{-(1-x)/\epsilon}}$
|
$x; \epsilon$
|
boundary_layers
| 4
|
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.
|
$\boxed{y = e^{1-x} + \epsilon[(-x+1)e^{1-x} -e^{1-\frac{x}{\epsilon}}]}$
|
$x; \epsilon$
|
boundary_layers
| 5
|
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.
|
$\boxed{y_{uniform}(x) = - \ln(2+x) -x + (\ln(3) + 2)e^{\frac{-3(1-x)}{\epsilon}}}$
|
$x; \epsilon$
|
boundary_layers
| 6
|
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.
|
$ \boxed{y = \text{erfc}(\frac{x}{\sqrt{2\epsilon}})} $
|
$x; \epsilon$
|
boundary_layers
| 7
|
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.
|
$\boxed{y = 2 e^{\arctan(x) - \pi/4} + (1 - 2 e^{-pi/4}) e^{-x/\epsilon} }$
|
$x; \epsilon$
|
boundary_layers
| 8
|
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.
|
$\boxed{y(x) = \frac{10}{x^2+1} + e^{\frac{-x}{\epsilon}} - 10e^{\frac{-x}{\epsilon}}}$
|
$x; \epsilon$
|
boundary_layers
| 9
|
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.
|
$\boxed{y(x) \approx \frac{1}{\sqrt{\epsilon}}e^{\frac{-x^2}{2\epsilon}} \\i \sqrt{\frac{\pi}{2}}erfi(\frac{x}{\sqrt{2\epsilon}})+ e^{\frac{-x^2}{2\epsilon}}}$
|
$x; \epsilon$
|
boundary_layers
| 10
|
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.
|
$\boxed{y(x) = \frac{1}{\frac{1}{2}x^2 + 1} + \frac{1}{3} \exp(\frac{x-1}{\epsilon})}$
|
$x; \epsilon$
|
boundary_layers
| 11
|
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.
|
$\boxed{y(x)=e^{\frac{-x^3}{3}}}$
|
$x; \epsilon$
|
boundary_layers
| 12
|
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.
|
$\boxed{y(x)=\left(\frac{3-x}{2}\right)e^{-\frac{x^2}{2}}}$
|
$x; \epsilon$
|
boundary_layers
| 13
|
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.
|
$\boxed{y = \frac{1}{2} x^2 + \ln{(x+1)} + 1 + (\frac{1}{2} - \ln{2}) e^{-2(1-x) / \epsilon}}$
|
$x; \epsilon$
|
boundary_layers
| 14
|
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.
|
$\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)}$
|
$x; \epsilon$
|
boundary_layers
| 15
|
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.
|
$\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}}$
|
$x; \epsilon$
|
boundary_layers
| 16
|
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.
|
$ \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} }$
|
$x; \epsilon$
|
boundary_layers
| 17
|
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.
|
$\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}}$
|
$x; \epsilon$
|
boundary_layers
| 18
|
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.
|
$ \boxed{ y(x) = e^{\frac{x^2}{2} - \frac{1}{8}} } $
|
$x; \epsilon$
|
boundary_layers
| 19
|
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.
|
$ \boxed{ y(x) = e^{\frac{x^2}{2} - \frac{9}{8}} } $
|
$x; \epsilon$
|
boundary_layers
| 20
|
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.
|
$\boxed{y =e^{-x^2/2} \left[ 1 \right]+ (1 - e^{-1/2}) \left[1 \right] e^{-\frac{1 - x}{\epsilon}}}$
|
$x; \epsilon$
|
boundary_layers
| 21
|
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.
|
$\boxed{y(x) \approx 3x + 2 \exp\left(-\frac{x^3}{3\epsilon}\right)}$
|
$x; \epsilon$
|
boundary_layers
| 22
|
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.
|
$\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)}$
|
$x; \epsilon$
|
boundary_layers
| 23
|
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.
|
$$\boxed{y(x) = -1 + \frac{2(\sec(1) + \tan(1))}{\sec(x) + \tan(x)} + 2(1 - \sec(1) - \tan(1))e^{-x/\epsilon}}$$
|
$x; \epsilon$
|
boundary_layers
| 24
|
Find a uniformly valid approximation to the solution of $ \epsilon y''(x) + (x-1)^2 y'(x) - x(x-1)^2 y(x) = \epsilon x^2 \sin(\pi x) [1+y(x)] $ with boundary conditions $y(1/2)=3, y(3/2)=3$ in the limit $\epsilon \rightarrow 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.
|
$$\boxed{y(x) \approx 3 e^{x^2/2 - 9/8} + 3(1 - e^{-1}) e^{-(x-1/2)/(4*\epsilon)}}$$
|
$x; \epsilon$
|
boundary_layers
| 25
|
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.
|
$\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)}$
|
$x; \epsilon$
|
boundary_layers
| 26
|
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.
|
$\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}}}$
|
$x; \epsilon$
|
boundary_layers
| 27
|
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.
|
$\boxed{y(x)=8(1+x)^{-3}-7e^{-\frac{x}{\epsilon}}}$
|
$x; \epsilon$
|
boundary_layers
| 28
|
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.
|
$\boxed{y(x)=2(3+2\sqrt{2})^\sqrt{2}((\frac{\sqrt{2}-x}{\sqrt{2}+x})^\sqrt{2}-e^{-\frac{2x}{\epsilon}})}$
|
$x; \epsilon$
|
boundary_layers
| 29
|
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.
|
$\boxed{y = \sin x + 2 - \sin(1) \, \mathrm{erf}\left(\frac{x}{\sqrt{2\epsilon}}\right)}$
|
$x; \epsilon$
|
boundary_layers
| 30
|
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.
|
$\boxed{y = e^{-(x+1)/\epsilon} + e^{(x-1)/\epsilon}}$
|
$x; \epsilon$
|
boundary_layers
| 31
|
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.
|
$$\boxed{y(x)=e^{-\frac{x^3}{3}}}$$
|
$x; \epsilon$
|
boundary_layers
| 32
|
Find a uniformly valid approximation to the solution of $\epsilon y'' + (\cosh x)y' + y = 0$ with boundary conditions $y(-1) = 0, y(1) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.
|
$$\boxed{y(x) = \exp (2(\arctan(e)-\arctan(e^{x})))-\exp(2(\arctan(e)-\arctan(e^{-1})))e^{-\cosh(1)\frac{x+1}{\epsilon}}}$$
|
$x; \epsilon$
|
boundary_layers
| 33
|
Find a uniformly valid approximation to the solution of $\epsilon y''(x) + \cosh(x)\,y'(x) - y(x) = 0$ with boundary conditions $y(0)=1, y(1)=1$ in the limit $\epsilon \ll 1$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.
|
$\boxed{y(x) = \exp (2[\arctan(e^x) - \arctan(e)]) + (1 - \exp (2[\arctan(1) - \arctan(e)]))e^{-\frac{x}{\epsilon}}}$
|
$x; \epsilon$
|
boundary_layers
| 34
|
Find a uniformly valid approximation to the solution of $\epsilon\,y'' + (x^2+1)\,y' - x^3\,y = 0$ with boundary conditions $y(0)=1, y(1)=1$ in the limit $\epsilon \ll 1$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.
|
$$\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}}$$
|
$x; \epsilon$
|
boundary_layers
| 35
|
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.
|
$\boxed{y(x) = \frac{\sqrt{2\epsilon}}{1-x + \sqrt{2\epsilon}}}$
|
$x; \epsilon$
|
boundary_layers
| 36
|
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.
|
$\boxed{y(x) = -\left(1 + \frac{1-x}{\sqrt{6\epsilon}}\right)^{-2}}$
|
$x; \epsilon$
|
boundary_layers
| 37
|
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.
|
$\boxed{y(x) = 1 + \left( x^{-2} + 2\varepsilon(x^{-3} - x^{-2}) - 1 \right) e^{-2\varepsilon^2 / x}}$
|
$x; \varepsilon$
|
boundary_layers
| 38
|
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.
|
$\boxed{y(x) = e^{-x} + 2 - e^{-1} - (2 - e^{-1})e^{-x/\epsilon}}$
|
$x; \epsilon$
|
boundary_layers
| 39
|
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.
|
$\boxed{y(t) = t + 2 \ln(2-t) + 1 - 2 \ln(2) - (2 - 2 \ln(2)) e^{-\frac{1-t}{\epsilon}}}$
|
$t; \epsilon$
|
boundary_layers
| 40
|
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.
|
$\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)}$
|
$t; \epsilon$
|
boundary_layers
| 41
|
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.
|
$\boxed{y(x)=\frac{2}{1+2x^2}-e^{-\frac{x}{\sqrt{\epsilon}}}+\frac{1}{3}e^{\frac{\sqrt{3}(x-1)}{\sqrt{\epsilon}}}}$
|
$x; \epsilon$
|
boundary_layers
| 42
|
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.
|
$\boxed{y = e^{-2 \tan(1) (1-x)/\epsilon} + e^{-2 \tan(1) (x+1)/\epsilon}}$
|
$x; \epsilon$
|
boundary_layers
| 43
|
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.
|
$\boxed{y(x) = \sqrt{\frac{\sin(x)}{\sin(1)}}}$
|
$x; \epsilon$
|
boundary_layers
| 44
|
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.
|
$\boxed{y(x) = \frac{162}{(1+2x)^4} - 161 e^{-x/ \epsilon}}$
|
$x; \epsilon$
|
boundary_layers
| 45
|
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.
|
$\boxed{\frac{75}{(2+3x)^2}-\frac{71}{4}e^{-2x/ \epsilon}}$
|
$x; \epsilon$
|
boundary_layers
| 46
|
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.
|
$\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)}$
|
$x; \epsilon$
|
boundary_layers
| 47
|
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.
|
$\boxed{y(x) = \frac{192}{(1+3x)^3} - 190 e^{-x/ \epsilon}}$
|
$x; \epsilon$
|
boundary_layers
| 48
|
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.
|
$\boxed{y(x,\epsilon) = -x - 31 \frac{\int_0^{x^3/(3\epsilon)} t^{-2/3} e^{-t} dt}{\Gamma(1/3)}}$
|
$x; \epsilon$
|
boundary_layers
| 49
|
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.
|
$\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}}}$
|
$x; \epsilon$
|
boundary_layers
| 50
|
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.
|
$\boxed{y(x,\epsilon) = e^{-x/\epsilon}}$
|
$x; \epsilon$
|
boundary_layers
| 51
|
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.
|
$\boxed{y(x) = (1 + \frac{\epsilon}{2})e^{-2x} - \frac{\epsilon}{2} e^{-\frac{2x}{\epsilon}}}$
|
$x;\epsilon$
|
boundary_layers
| 52
|
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.
|
$\boxed{y(x) = -\ln(e^{3}-x) + \ln(e^{3}-1)e^{\frac{x-1}{\epsilon}}} $
|
$x; \epsilon$
|
boundary_layers
| 53
|
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.
|
$\boxed{y(x, \epsilon = e^{(\frac{1}{1+x} - \frac{1}{2})} + (1-e^{1/2})e^{-\frac{x}{\epsilon}}}$
|
$x; \epsilon$
|
boundary_layers
| 54
|
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.
|
$\boxed{y(x)=\frac{9}{15-6x-\ln(\frac{3x+1}{4})}-\frac{9}{15+\ln(4)}e^{-x/\epsilon}}$
|
$x; \epsilon$
|
boundary_layers
| 55
|
Find a uniformly valid leading order approximation to the solution of $$ \epsilon y'' + 2y' + y = \cos\left(\frac{\pi x}{2}\right)$$ with boundary conditions in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.
|
$$ \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}} $$
|
$x; \epsilon$
|
boundary_layers
| 56
|
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.
|
$\boxed{y = \frac{1}{2} \text{erf}\left(\frac{x}{\sqrt{2\epsilon}}\right) + \frac{3}{2}}$
|
$x; \epsilon$
|
boundary_layers
| 57
|
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.
|
$\boxed{y_{unif}(x, \epsilon) = \frac{1}{2} \text{erf}\left(x \sqrt{\frac{\pi}{4\epsilon}}\right) + \frac{1}{2}}$
|
$x; \epsilon$
|
boundary_layers
| 58
|
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.
|
$\boxed{y_{unif}(x, \epsilon) = \frac{1}{2} \text{erf}\left(\frac{x}{\sqrt{2\epsilon}}\right) + \frac{1}{2}}$
|
$x; \epsilon$
|
boundary_layers
| 59
|
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
|
$\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}}$
|
$x; \epsilon$
|
boundary_layers
| 60
|
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.
|
$\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}}$
|
$x; \epsilon$
|
boundary_layers
| 61
|
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.
|
$\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}}$
|
$x; \epsilon$
|
boundary_layers
| 62
|
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.
|
$\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}}$
|
$x; \epsilon$
|
boundary_layers
| 63
|
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.
|
$\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}}$
|
$x; \epsilon$
|
boundary_layers
| 64
|
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.
|
$\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}}$
|
$x; \epsilon$
|
boundary_layers
| 65
|
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.
|
$\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}}$
|
$x; \epsilon$
|
boundary_layers
| 66
|
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.
|
$\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}}$
|
$x; \epsilon$
|
boundary_layers
| 67
|
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.
|
$\boxed{y(x) = \frac{\cos(x)- e^{-x/\epsilon}}{\cos(1)}}$
|
$x; \epsilon$
|
boundary_layers
| 68
|
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.
|
$\boxed{y(x,\epsilon) \approx 2 + \sin x - \sin 1 \erf \left(\frac{x}{\sqrt{2\epsilon}}\right)}$
|
$x; \epsilon$
|
boundary_layers
| 69
|
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.
|
$\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}}$
|
$x; \epsilon$
|
boundary_layers
| 70
|
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.
|
$\boxed{y_{uniform}(x) = \frac{3}{x} -\frac{x^2}{2} -\frac{3}{2} + 3e^{\frac{-4(2-x)}{\epsilon}}}$
|
$x; \epsilon$
|
boundary_layers
| 71
|
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.
|
$\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})}$
|
$x; \epsilon$
|
boundary_layers
| 72
|
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.
|
$\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}}}$
|
$x; \epsilon$
|
boundary_layers
| 73
|
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.
|
$\boxed{y_{uniform}(x) = \frac{5}{e^2+1}(e^{2x} + 1) + e^{\frac{-x}{\epsilon}}(\frac{1}{5}-\frac{10}{e^2+1})}$
|
$x; \epsilon$
|
boundary_layers
| 74
|
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.
|
$\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}}}$
|
$x; \epsilon$
|
boundary_layers
| 75
|
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.
|
$\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)} }$
|
$x; \epsilon; s$
|
boundary_layers
| 76
|
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.
|
$\boxed{\pi - \sqrt{2\pi} \int_0^{\frac{x}{\sqrt{\epsilon}}} e^{-s^2/2} \, ds}$
|
$x; \epsilon; s$
|
boundary_layers
| 77
|
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.
|
$\boxed{y(x)=1+\frac{\cosh(2)}{2\cosh(x)}(1+\text{erf}(\frac{x}{\sqrt{2\epsilon}}))}$
|
$x; \epsilon$
|
boundary_layers
| 78
|
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.
|
$\boxed{\frac{x-2\text{erf}(\frac{x}{\sqrt{2\epsilon}})}{\cosh(x)}}$
|
$x; \epsilon$
|
boundary_layers
| 79
|
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.
|
$\boxed{y(x) = \frac{1-e^{\frac{x}{\epsilon}}}{1-e^{\frac{1}{\epsilon}}}}$
|
$x;\epsilon$
|
boundary_layers
| 80
|
Find a uniformly valid leading order approximation to the solution of $$\epsilon y'' - y' = \sin(\pi x)$$ with boundary conditions $ y(0) = 0, y(1) = 0$ . Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.
|
$$\boxed{y(x) = \frac{\cos(\pi x) - 1}{\pi} + \frac{2}{\pi}e^{\frac{x-1}{\epsilon}}}$$
|
$x;\epsilon$
|
boundary_layers
| 81
|
Find the lowest-order uniform approximation to the boundary-value problem: $$ \epsilon y'' + y' \sin x + y \sin(2x) = 0 $$ with boundary conditions:$$ y(0) = \pi, \quad y(\pi) = 0 $$.
|
$$ \boxed{y(x) \approx \pi \, \text{erfc}\left(\frac{x}{\sqrt{2\epsilon}}\right)} $$
|
$x;\epsilon$
|
boundary_layers
| 82
|
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.
|
$\boxed{\sqrt{\frac{2 \pi}{2 \epsilon}}}$
|
$x; \epsilon$
|
integrals
| 0
|
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.
|
$\boxed{I(x) = \frac{\pi}{4}-\frac{x}{2}\ln(2)}$
|
$t;x$
|
integrals
| 1
|
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.
|
$\boxed{I(x) \approx \frac{85}{62}\sqrt{\frac{2\pi}{(2+\frac{\pi^2}{4})x}}}$
|
$t;x$
|
integrals
| 2
|
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.
|
$\boxed{I(x) = 1 - x - \frac{x^2}{6} + \frac{x^4}{120} + \frac{x^5}{6} - \frac{x^6}{5040} }$
|
$x$
|
integrals
| 3
|
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.
|
$\boxed{\int_{x}^{\infty} e^{-a t^b} \, dt \sim \frac{e^{-a x^b}}{a b x^{b-1}}}$
|
$x;a;b$
|
integrals
| 4
|
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.
|
$\boxed{I(x) \sim \sqrt{\frac{\pi}{2x}} e^{-x}}$
|
$t;x$
|
integrals
| 5
|
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.
|
$\boxed{-\frac{1}{x \ln x}}$
|
$t;x$
|
integrals
| 6
|
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.
|
$\boxed{I(\epsilon) = \frac{1}{\epsilon^{3/2}} \cdot 10}$
|
$\epsilon$
|
integrals
| 7
|
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.
|
$\boxed{I(x) \sim -\frac{1}{12x^2} + \frac{1}{120x^4} - \frac{1}{252x^6}}$
|
$x; t$
|
integrals
| 8
|
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.
|
$\boxed{I(\epsilon) = 10\cdot\epsilon^{-13/7}}$
|
$\epsilon$
|
integrals
| 9
|
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.
|
$\boxed{I(\epsilon) = \frac{\sqrt[11]{-1 + 2^{\frac{7}{13}}}}{\epsilon^{\frac{136}{77}}}}$
|
$\epsilon$
|
integrals
| 10
|
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.
|
$\boxed{y(x)= \frac{2e^{-9x}}{x}}$
|
$x$
|
integrals
| 11
|
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HARDMath2 Benchmark Dataset
This repository contains a collection of mathematical benchmark problems designed for evaluating Large Language Models (LLMs) on mathematical reasoning tasks.
Building
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.
The .csv file should have a descriptive name for the types of problems in the file, with underscores instead of spaces.
Data Format
Each benchmark problem in the dataset is structured as a JSON object containing the following fields:
Fields
Prompt: The input string that is fed to the LLM
Solution: A LaTeX-formatted string representing the mathematical formula that solves the question posed in the prompt
Parameters: A list of independent tokens that should be treated as single variables in the LaTeX response string. These include:
- Single variables (e.g.,
$A$,$x$) - Greek letters (e.g.,
$\epsilon$) - Complex strings with subscripts (e.g.,
$\delta_{i,j}$)
Each parameter should be separated by a semicolon (;).
- Single variables (e.g.,
Example
{
"prompt": "What is the derivative of f(x) = x^2?",
"solution": "\\frac{d}{dx}(x^2) = 2x",
"parameters": "x"
}
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