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JVRoggeveen
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HARDMath2

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int64
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82
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