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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 environm... | $\boxed{y = e^{\frac{x^3}{3}} + (2-e^{1/3})e^{-(1-x)/\epsilon}}$ | $x; \epsilon$ | 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 environm... | $\boxed{y = A*e^{\frac{x^3}{3}} + (B-A*e^{1/3})e^{-(1-x)/\epsilon}}$ | $x; \epsilon; A; B$ | 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 environ... | $\boxed{y_{unif}(x) \approx \left[ e^x - x Ei(x) + (1 - e + Ei(1)) x \right] - \left[e^{-1} + Ei(-1) - 1 + e - Ei(1)\right] e^{-(x+1)/\epsilon}}$ | $x; \epsilon$ | 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 $\boxed{}$ L... | $\boxed{y = E^{-(x+1)/\epsilon}+ E^{-(1-x)/\epsilon}}$ | $x; \epsilon$ | 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 onl... | $\boxed{y = e^{1-x} + \epsilon[(-x+1)e^{1-x} -e^{1-\frac{x}{\epsilon}}]}$ | $x; \epsilon$ | 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... | $\boxed{y_{uniform}(x) = - \ln(2+x) -x + (\ln(3) + 2)e^{\frac{-3(1-x)}{\epsilon}}}$ | $x; \epsilon$ | 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 fi... | $ \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... | $\boxed{y = 2 e^{\arctan(x) - \pi/4} + (1 - 2 e^{-pi/4}) e^{-x/\epsilon} }$ | $x; \epsilon$ | 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 $\box... | $\boxed{y(x) = \frac{10}{x^2+1} + e^{\frac{-x}{\epsilon}} - 10e^{\frac{-x}{\epsilon}}}$ | $x; \epsilon$ | 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... | $\boxed{y(x) \approx \frac{1}{\sqrt{\epsilon}}e^{\frac{-x^2}{2\epsilon}} \\i \sqrt{\frac{\pi}{2}}erfi(\frac{x}{\sqrt{2\epsilon}})+ e^{\frac{-x^2}{2\epsilon}}}$ | $x; \epsilon$ | 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 $\b... | $\boxed{y(x) = \frac{1}{\frac{1}{2}x^2 + 1} + \frac{1}{3} \exp(\frac{x-1}{\epsilon})}$ | $x; \epsilon$ | 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... | $\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 $\boxe... | $\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... | $\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{}... | $\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 yo... | $\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 environm... | $ \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 enviro... | $\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 $\box... | $ \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 $\box... | $ \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 $\eps... | $\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 you... | $\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{}$ La... | $$\boxed{y(x) = -1 + \frac{2(\sec(1) + \tan(1))}{\sec(x) + \tan(x)} + 2(1 - \sec(1) - \tan(1))e^{-x/\epsilon}}$$ | $x; \epsilon$ | boundary_layers | 24 |
Find a uniformly valid approximation to the solution of $ \epsilon y''(x) + (x-1)^2 y'(x) - x(x-1)^2 y(x) = \epsilon x^2 \sin(\pi x) [1+y(x)] $ with boundary conditions $y(1/2)=3, y(3/2)=3$ in the limit $\epsilon \rightarrow 0^+$. Use only the variables and constants given in the problem; do not define additional const... | $$\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{}$ LaT... | $\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{... | $\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{}$ LaT... | $\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{}$ ... | $\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 environm... | $\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 envi... | $\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{}$ L... | $$\boxed{y(x)=e^{-\frac{x^3}{3}}}$$ | $x; \epsilon$ | boundary_layers | 32 |
Find a uniformly valid approximation to the solution of $\epsilon y'' + (\cosh x)y' + y = 0$ with boundary conditions $y(-1) = 0, y(1) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX envi... | $$\boxed{y(x) = \exp (2(\arctan(e)-\arctan(e^{x})))-\exp(2(\arctan(e)-\arctan(e^{-1})))e^{-\cosh(1)\frac{x+1}{\epsilon}}}$$ | $x; \epsilon$ | boundary_layers | 33 |
Find a uniformly valid approximation to the solution of $\epsilon y''(x) + \cosh(x)\,y'(x) - y(x) = 0$ with boundary conditions $y(0)=1, y(1)=1$ in the limit $\epsilon \ll 1$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX e... | $\boxed{y(x) = \exp (2[\arctan(e^x) - \arctan(e)]) + (1 - \exp (2[\arctan(1) - \arctan(e)]))e^{-\frac{x}{\epsilon}}}$ | $x; \epsilon$ | boundary_layers | 34 |
Find a uniformly valid approximation to the solution of $\epsilon\,y'' + (x^2+1)\,y' - x^3\,y = 0$ with boundary conditions $y(0)=1, y(1)=1$ in the limit $\epsilon \ll 1$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX envir... | $$\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 solut... | $\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{}$ LaT... | $\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 add... | $\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 enviro... | $\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 enviro... | $\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 env... | $\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 enviro... | $\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 enviro... | $\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 environ... | $\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 environm... | $\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 environme... | $\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 envi... | $\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... | $\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 environm... | $\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 en... | $\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 ... | $\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{}$ La... | $\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 $\box... | $\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)$$ with boundary conditions in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\b... | $$ \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... | $\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... | $\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{... | $\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{}$... | $\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{}$ ... | $\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-for... | $\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{}$... | $\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-... | $\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 ... | $\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... | $\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 $\boxe... | $\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{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{}$ ... | $\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{}... | $\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{... | $\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 $\box... | $\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 $\box... | $\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 i... | $\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 $\boxe... | $\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{}$ L... | $\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 i... | $\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... | $\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 fin... | $\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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