prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' - x y' + x^3 y = 0$ with boundary conditions $y(0) = 1$, $y(1) = 2$ in the limit $\epsilon \ll 1$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y = e^{\frac{x^3}{3}} + (2-e^{1/3})e^{-(1-x)/\epsilon}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' - x y' + x^3 y = 0$ with boundary conditions $y(0) = A$, $y(1) = B$ in the limit $\epsilon \ll 1$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y = A*e^{\frac{x^3}{3}} + (B-A*e^{1/3})e^{-(1-x)/\epsilon}}$

parameters: $x; \epsilon; A; B$


prompt: Find a single uniformly valid approximation to the solution of $\epsilon y'' + x y' - y = -e^x$ with boundary conditions $y(-1)=0, y(1)=1$ in the limit $\epsilon \ll 1$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y_{unif}(x) \approx \left[ e^x - x Ei(x) + (1 - e + Ei(1)) x \right] - \left[e^{-1} + Ei(-1) - 1 + e - Ei(1)\right] e^{-(x+1)/\epsilon}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''-2 tan(x) y'+y=0$ with boundary conditions $y(-1)=0, y(1)=1$ in the limit $\epsilon = 0$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y = \sqrt{\frac{\sin x}{\sin 1}}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''-x y'-(3+x)$ with boundary conditions $y(-1)=1, y(1)=1$ in the limit $\epsilon = 0$ from the positive direction. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y = E^{-(x+1)/\epsilon}+ E^{-(1-x)/\epsilon}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation, with error of order $\epsilon^2$, to the solution of $\epsilon y'' + y' +y = 0$ with boundary conditions $y(0) = e, y(1) = 1$ in the limit $\epsilon = 0$ from the positive direction. Notice that there is no boundary layer in leading order, but one does appear in next order. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y = e^{1-x} + \epsilon[(-x+1)e^{1-x} -e^{1-\frac{x}{\epsilon}}]}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' - (x+2)y' - (3+x) = 0$ with boundary conditions $y(-1) = 1, y(1) = 1$ in the limit $\epsilon = 0$ from the positive direction. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y_{uniform}(x) = - \ln(2+x) -x + (\ln(3) + 2)e^{\frac{-3(1-x)}{\epsilon}}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $ \epsilon y'' + y' \sin(x) + y \sin(\2x) = 0$ with boundary conditions $ y(0) = \pi, y(\pi) = 0 $ in the limit $\epsilon = 0$ from the positive direction. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $ \boxed{y = \text{erfc}(\frac{x}{\sqrt{2\epsilon}})} $

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' + (1 + x^2) y' - y = 0$ with boundary conditions $y(0) = 1, y(1) = 2$ in the limit $\epsilon = 0$ from the positive direction. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y = 2 e^{\arctan(x) - \pi/4} + (1 - 2 e^{-pi/4}) e^{-x/\epsilon} }$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' + (x^2 +1)y'+2xy=0$ with boundary conditions $y(0)=1, y(1)=5$ in the limit $\epsilon = 0$ from the positive direction. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) = \frac{10}{x^2+1} + e^{\frac{-x}{\epsilon}} - 10e^{\frac{-x}{\epsilon}}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' + x y' + y = 0$ with boundary conditions $y(0)=1, y(1)=1$ in the limit $\epsilon = 0$ from the positive direction. Denote the square root of -1 as I. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) \approx \frac{1}{\sqrt{\epsilon}}e^{\frac{-x^2}{2\epsilon}} \\i \sqrt{\frac{\pi}{2}}erfi(\frac{x}{\sqrt{2\epsilon}})+ e^{\frac{-x^2}{2\epsilon}}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' - y'/x - y^2 = 0$ with boundary conditions $y(0) = 1, y(1) = 1$ in the limit $\epsilon = 0$ from the positive direction. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment. Your response should have the form of a single analytical expression.

solution: $\boxed{y(x) = \frac{1}{\frac{1}{2}x^2 + 1} + \frac{1}{3} \exp(\frac{x-1}{\epsilon})}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $$\epsilon y''+\frac{y'}{x^2}+y=0 with boundary conditions $y(0)=0, y(1)=e^{-\frac{1}{3}}$ in the limit $\epsilon \rightarrow 0+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''+\frac{y'}{x}+y=0$ with boundary conditions $[y(-1)=2e^{-1/2}, y(1)=e^{-1/2}]$ in the limit $\epsilon \rightarrow 0+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x)=\left(\frac{3-x}{2}\right)e^{-\frac{x^2}{2}}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' - (x+1) y' + x^2 + x + 1 = 0$ with boundary conditions $y(0) = 1, y(1) = 2$ in the limit $\epsilon \ll 0+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y = \frac{1}{2} x^2 + \ln{(x+1)} + 1 + (\frac{1}{2} - \ln{2}) e^{-2(1-x) / \epsilon}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' + (\cosh(x))(x^2 + 1)y' - x^3 y = 0$ with boundary conditions $y(0) = 1, y(1) = 1$ in the limit $\epsilon \ll 0+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) = (1-\exp\left(\int_1^0 \frac{t^3}{\cosh(t)(t^2 + 1)}\ dt\right))e^{-x/\epsilon} + \exp\left(\int_1^x \frac{t^3}{\cosh(t)(t^2 + 1)}\ dt\right)}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' - (x^2+4)y' - y^3 = 0$ with boundary conditions $y(0)=1, y(1)=\sqrt{5}$ in the limit $\epsilon \ll 0+$. Use only the variables and constants given in the problem; do not define additional constants. Solve any integrals in the final solution. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x)=\frac{1}{\sqrt{\arctan\left(\frac{x}{2}\right)+1}}+\left(\sqrt{5}-\frac{1}{\sqrt{\arctan\left(\frac{1}{2}\right)+1}}\right)e^{-5(1-x)/\epsilon}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' - (x^2+1)y' - y^3 = 0$ with boundary conditions $y(0)=1, y(1)=1/2$ in the limit $\epsilon \ll 0+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $ \boxed{y(x) \sim \frac{1}{\sqrt{2\arctan(x) + 1}} + \left( \frac{1}{2} - \frac{1}{\sqrt{\pi/2 + 1}} \right) e^{-2(1-x)/\epsilon} }$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' + (x^2-12)y' - y^3 = 0$ with boundary conditions $y(0)=1, y'(1)=1/2$ in the limit $\epsilon \ll 0+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) \approx \left( 1 - \frac{1}{2\sqrt{3}} \ln\left( \frac{2\sqrt{3}-x}{x+2\sqrt{3}} \right) \right)^{-1/2} + \frac{\epsilon}{11} \left[ \frac{1}{2} + \frac{1}{11} \left( 1 - \frac{1}{2\sqrt{3}} \ln\left( \frac{2\sqrt{3}-1}{2\sqrt{3}+1} \right) \right)^{-3/2} \right] e^{-11(1-x)/\epsilon}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' + (\ln x) y' - x(\ln x) y = 0$ with boundary conditions $y(1/2)=1, y(3/2)=1$ in the limit $\epsilon \ll 0+$ for $x<1$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $ \boxed{ y(x) = e^{\frac{x^2}{2} - \frac{1}{8}} } $

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' + (\ln x) y' - x(\ln x) y = 0$ with boundary conditions $y(1/2)=1, y(3/2)=1$ in the limit $\epsilon \ll 0+$ for $x>1$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $ \boxed{ y(x) = e^{\frac{x^2}{2} - \frac{9}{8}} } $

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' - \frac{1}{x} y' - y = 0$ with boundary conditions $y(0) = 1, y(1) = 1$ in the limit $\epsilon \ll 0+$ to leading order. Use only the variables and constants given in the problem; do not define additional constants; in your final solution, only $\epsilon$ and $x$ should remain as variables. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y =e^{-x^2/2} \left[ 1 \right]+ (1 - e^{-1/2}) \left[1 \right] e^{-\frac{1 - x}{\epsilon}}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' + x^2y' - xy = 0$ with boundary conditions $y(0) = 2, y(1) = 3$ in the limit $\epsilon \ll 0+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) \approx 3x + 2 \exp\left(-\frac{x^3}{3\epsilon}\right)}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' - y'/(x^2-1.01) + ye^{-x} + sin(\epsilon)(x+cos(\epsilon)) y' = 0$ with boundary conditions $y(-1) = 1, y(1) = 1$ in the limit $\epsilon \ll 0+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) \approx \exp(3.99 e^{-1} - (x^2 + 2x + 0.99) e^{-x}) + \left(1 - \exp(3.99 e^{-1} + 0.01 e)\right) \exp\left(-\frac{100(x+1)}{\epsilon}\right)}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' + \cos(x)y' + y = -1$ with boundary conditions $y(0) = 1$, $y(1) = 1$ in the limit $\epsilon \rightarrow 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $$\boxed{y(x) = -1 + \frac{2(\sec(1) + \tan(1))}{\sec(x) + \tan(x)} + 2(1 - \sec(1) - \tan(1))e^{-x/\epsilon}}$$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $ \epsilon y''(x) + (x-1)^2 y'(x) - x(x-1)^2 y(x) = \epsilon x^2 \sin(\pi x) [1+y(x)] $ with boundary conditions $y(1/2)=3, y(3/2)=3$ in the limit $\epsilon \rightarrow 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $$\boxed{y(x) \approx 3 e^{x^2/2 - 9/8} + 3(1 - e^{-1}) e^{-(x-1/2)/(4*\epsilon)}}$$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' + (\ln x)y' - x(\ln x)y = 0$ with boundary conditions $y(1/2) = 1, y(3/2) = 1$ in the limit $\epsilon \to 0$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{\frac{1}{2} \left( e^{-\frac{1}{8} + \frac{x^2}{2}} + e^{-\frac{9}{8} + \frac{x^2}{2}} \right) + \frac{1}{2} \left( e^{-\frac{9}{8} + \frac{x^2}{2}} - e^{-\frac{1}{8} + \frac{x^2}{2}}\right) * erf\left(\frac{x-1}{\sqrt{2\epsilon}}\right)}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' + \frac{cos(x)}{3}y' - (\ln x)y = 0$ with boundary conditions $y(0) = 0, y(1) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) = e^{\int_{1}^{x}\frac{3\ln t}{\cos(t)}dt} - e^{\int_{1}^{0}\frac{3\ln t}{\cos(t)}dt}e^{- \frac{x}{3\epsilon}}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''(x) + (1 + x) y'(x) + 3 y(x) = 0$ with boundary conditions $y(0) = 1, y(1) = 1$ in the limit $\epsilon \to 0$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x)=8(1+x)^{-3}-7e^{-\frac{x}{\epsilon}}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''(x) + (2 - x^2) y'(x) + 4 y(x) = 0$ with boundary conditions $y(0) = 0, y(1) = 2$, in the limit $\epsilon \to 0$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x)=2(3+2\sqrt{2})^\sqrt{2}((\frac{\sqrt{2}-x}{\sqrt{2}+x})^\sqrt{2}-e^{-\frac{2x}{\epsilon}})}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' + x y' = x \cos x$ with boundary conditions $y(-1) = 2, y(1) = 2$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y = \sin x + 2 - \sin(1) \, \mathrm{erf}\left(\frac{x}{\sqrt{2\epsilon}}\right)}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' - x y' - (3 + x)y = 0$ with boundary conditions $y(-1) = 1, y(1) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y = e^{-(x+1)/\epsilon} + e^{(x-1)/\epsilon}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' + \frac{y'}{x^2} + y = 0$ with boundary conditions $y(0) = 0, y(1) = e^{-1/3}$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' + (\cosh x)y' + y = 0$ with boundary conditions $y(-1) = 0, y(1) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $$\boxed{y(x) = \exp (2(\arctan(e)-\arctan(e^{x})))-\exp(2(\arctan(e)-\arctan(e^{-1})))e^{-\cosh(1)\frac{x+1}{\epsilon}}}$$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''(x) + \cosh(x)\,y'(x) - y(x) = 0$ with boundary conditions $y(0)=1, y(1)=1$ in the limit $\epsilon \ll 1$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) = \exp (2[\arctan(e^x) - \arctan(e)]) + (1 - \exp (2[\arctan(1) - \arctan(e)]))e^{-\frac{x}{\epsilon}}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon\,y'' + (x^2+1)\,y' - x^3\,y = 0$ with boundary conditions $y(0)=1, y(1)=1$ in the limit $\epsilon \ll 1$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $$\boxed{y(x, \epsilon) = \sqrt{2}e^{-1/2} \frac{e^{x^2/2}}{\sqrt{x^2+1}} + \left( 1 - \sqrt{2}e^{-1/2} \right) e^{-x/\epsilon}}$$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon^2 y'' + \epsilon y' - y = 0$ with boundary conditions $y(0) = 0$ and $y(1) = 1$ in the limit $\epsilon = 0$ from the positive direction. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) = \frac{\sqrt{2\epsilon}}{1-x + \sqrt{2\epsilon}}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' + \epsilon (x+1) y' + y^2 = 0$ with boundary conditions $y(0) = 1, y(1) = -1$ in the limit $\epsilon \ll 0+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) = -\left(1 + \frac{1-x}{\sqrt{6\epsilon}}\right)^{-2}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $ \varepsilon y'' + \left(1 + \frac{2\varepsilon}{x} - \frac{2\varepsilon^3}{x^2}\right) y' + \frac{2y}{x} = 0 $ with boundary conditions $y(0)=1, y(1)=1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) = 1 + \left( x^{-2} + 2\varepsilon(x^{-3} - x^{-2}) - 1 \right) e^{-2\varepsilon^2 / x}}$

parameters: $x; \varepsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''(x) + y'(x) = -e^{-x}$ with boundary conditions $y(0) = 1, y(1) = 2$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) = e^{-x} + 2 - e^{-1} - (2 - e^{-1})e^{-x/\epsilon}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''(t) + (t-2) y'(t) = t$ with boundary conditions $y(0) = 1, y(1) = 0$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(t) = t + 2 \ln(2-t) + 1 - 2 \ln(2) - (2 - 2 \ln(2)) e^{-\frac{1-t}{\epsilon}}}$

parameters: $t; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' + (t-2) y' = t^2$ with boundary conditions $y(0) = 0, y(1) = e^{-1/3}$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{ y(x) = \frac{t^2}{2} + 2t + 4\ln \left( \frac{2-t}{2} \right) + \left( e^{-1/3} -\frac{5}{2} + 4\ln 2 \right)\exp\left( \frac{t-1}{\epsilon}\right)}$

parameters: $t; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''-(1+2x^2)y+2=0$ with boundary conditions $y(0)=y(1)=1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x)=\frac{2}{1+2x^2}-e^{-\frac{x}{\sqrt{\epsilon}}}+\frac{1}{3}e^{\frac{\sqrt{3}(x-1)}{\sqrt{\epsilon}}}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' - 2 \tan(x) y' + y = 0$ with boundary conditions $y(-1) = y(1) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y = e^{-2 \tan(1) (1-x)/\epsilon} + e^{-2 \tan(1) (x+1)/\epsilon}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y'' + 2 \tan(x) y' - y = 0$ with boundary conditions $y(-1) = y(1) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) = \sqrt{\frac{\sin(x)}{\sin(1)}}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''(x)+(1+2x) y'(x)+8y(x)=0$ with boundary conditions $y(0)=1, y(1)=2$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) = \frac{162}{(1+2x)^4} - 161 e^{-x/ \epsilon}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''(x)+(2+3x)y'(x)+6y(x)=0$ with boundary conditions $y(0)=1, y(1)=3$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{\frac{75}{(2+3x)^2}-\frac{71}{4}e^{-2x/ \epsilon}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''(x) - 2y(x) = e^{-x}$ with boundary conditions $y(0)=0, y(1)=1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) = -\frac{1}{2} e^{-x} + \frac{1}{2} \exp\left(-\sqrt{\frac{2}{\epsilon}}x\right) + \left(1 + \frac{1}{2} e^{-1}\right) \exp\left(-\sqrt{\frac{2}{\epsilon}}(1-x)\right)}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''(x)+(1+3x)y'(x)+9y(x)=0$ with boundary conditions $y(0)=2,y(1)=3$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) = \frac{192}{(1+3x)^3} - 190 e^{-x/ \epsilon}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''(x) + x^2y' + x^2 = 0$ with boundary conditions $y(0) = 0, y(1) = -32$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x,\epsilon) = -x - 31 \frac{\int_0^{x^3/(3\epsilon)} t^{-2/3} e^{-t} dt}{\Gamma(1/3)}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''(x) - (1 + \sin x)\, y'(x) - y(x) = 0$ with boundary conditions $y(0) = 1, y(1) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x,\epsilon)=\exp\left( -\int_0^x \frac{dt}{1 + \sin t} \right)+\left(1 - 0.493\right) e^{-(1 + \sin 1)\, \frac{1 - x}{\epsilon}}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''(x) + y' + x(y) = 0$ with boundary conditions $y(0) = 1, y(1) = 0$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x,\epsilon) = e^{-x/\epsilon}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''(x) + 2y' (x)+ 4y(x) = 0$ with boundary conditions $y(0) = 1, y'(0) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) = (1 + \frac{\epsilon}{2})e^{-2x} - \frac{\epsilon}{2} e^{-\frac{2x}{\epsilon}}}$

parameters: $x;\epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''(x) - y'(x) + e^{y(x)} = 0$ with boundary conditions $y(0) = -3, y(1) = 0$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) = -\ln(e^{3}-x) + \ln(e^{3}-1)e^{\frac{x-1}{\epsilon}}} $

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y"(x) + (1 + x)^2 y'(x) + y(x) = 0$ with boundary conditions $y(0) = 1, y(1) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x, \epsilon = e^{(\frac{1}{1+x} - \frac{1}{2})} + (1-e^{1/2})e^{-\frac{x}{\epsilon}}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid approximation to the solution of $\epsilon y''(x) + \frac{3x+1}{2x+1}y'(x) - y(x)^{2} = 0$ with boundary conditions $y(0)=0, y(1)=1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x)=\frac{9}{15-6x-\ln(\frac{3x+1}{4})}-\frac{9}{15+\ln(4)}e^{-x/\epsilon}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $$ \epsilon y'' + 2y' + y = \cos\left(\frac{\pi x}{2}\right)$$ with boundary conditions in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $$ \boxed{y = \frac{1}{1+\pi^2}\left(\cos\left(\frac{\pi x}{2}\right)+\pi\sin\left(\frac{\pi x}{2}\right)\right) - \frac{\pi \sqrt{e}}{1+\pi^2} e^{-x/2} + \frac{\pi(1+e)}{1+\pi^2} e^{-2(x+1)/\epsilon}} $$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' + x y' = 0$ with boundary conditions $y(-1) = 1$, $y(1) = 2$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y = \frac{1}{2} \text{erf}\left(\frac{x}{\sqrt{2\epsilon}}\right) + \frac{3}{2}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' + \sin\left(\frac{\pi x}{2}\right) y' = 0$ with boundary conditions $y(-1) = 0$, $y(1) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y_{unif}(x, \epsilon) = \frac{1}{2} \text{erf}\left(x \sqrt{\frac{\pi}{4\epsilon}}\right) + \frac{1}{2}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' + (e^x - 1) y' = 0$ with boundary conditions $y(-1) = 0$, $y(1) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y_{unif}(x, \epsilon) = \frac{1}{2} \text{erf}\left(\frac{x}{\sqrt{2\epsilon}}\right) + \frac{1}{2}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' + x y' + x y = 0$ with boundary conditions $y(-1) = 1$, $y(1) = 2$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment

solution: $\boxed{y = e^{-(x+1)} \frac{1-\text{erf}(x/\sqrt{2\epsilon})}{2} + 2e^{1-x} \frac{1+\text{erf}(x/\sqrt{2\epsilon})}{2}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of \epsilon y'' + x y' + x y = x, with boundary conditions $y(-1) = 1$, $y(1) = 2$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y_{unif}(x, \epsilon) = \frac{1-\text{erf}(x/\sqrt{2\epsilon})}{2} + (1+e^{1-x}) \frac{1+\text{erf}(x/\sqrt{2\epsilon})}{2}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of \epsilon y'' + x y' + x y = x^2 with boundary conditions $y(-1) = 1$, $y(1) = 3$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. The solution should be smooth, single-form expression which is valid across the entire domain. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y_{unif}(x, \epsilon) = \left(x - 1 + 3e^{-(x+1)}\right) \frac{1-\text{erf}(x/\sqrt{2\epsilon})}{2} + \left(x - 1 + 3e^{1-x}\right) \frac{1+\text{erf}(x/\sqrt{2\epsilon})}{2}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' + x y' + x y = x$ with boundary conditions $y(-1) = 0$, $y(1) = 0$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y_{unif}(x, \epsilon) = \left(1 - e^{-(x+1)}\right) \frac{1-\text{erf}(x/\sqrt{2\epsilon})}{2} + \left(1 - e^{1-x}\right) \frac{1+\text{erf}(x/\sqrt{2\epsilon})}{2}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of \epsilon y'' + x y' + x y = x(x-1) with boundary conditions $y(-1) = 0$, $y(1) = 0$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. The solution should be smooth, single-form expression which is valid across the entire domain. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y_{unif}(x, \epsilon) = \left(x - 2 + 3e^{-(x+1)}\right) \frac{1-\text{erf}(x/\sqrt{2\epsilon})}{2} + \left(x - 2 + e^{1-x}\right) \frac{1+\text{erf}(x/\sqrt{2\epsilon})}{2}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of \epsilon y'' + x y' + x y = x with boundary conditions $y(-1) = 0$, $y(1) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. The solution should be smooth, single-form expression which is valid across the entire domain. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y_{unif}(x, \epsilon) = \left(1 - e^{-(x+1)}\right) \frac{1-\text{erf}(x/\sqrt{2\epsilon})}{2} + \frac{1+\text{erf}(x/\sqrt{2\epsilon})}{2}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' + x y' + 2x^2 y = 0$ with boundary conditions $y(-1) = 1$, $y(1) = 2$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y_{unif}(x, \epsilon) = e^{1-x^2} \frac{1-\text{erf}(x/\sqrt{2\epsilon})}{2} + 2e^{1-x^2} \frac{1+\text{erf}(x/\sqrt{2\epsilon})}{2}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' + x y' + x^2 y = x^2$ with boundary conditions $y(-1) = 1$, $y(1) = 2$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y_{unif}(x, \epsilon) = \frac{1-\text{erf}(x/\sqrt{2\epsilon})}{2} + \left(1 + e^{(1-x^2)/2}\right) \frac{1+\text{erf}(x/\sqrt{2\epsilon})}{2}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' + \cos(x) y ' + \sin(x) y= 0$ with boundary conditions $y(0) = 0$, $y(1)= 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) = \frac{\cos(x)- e^{-x/\epsilon}}{\cos(1)}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' + xy' = x \cos{x}$ with boundary conditions $y(1) = 2; y(-1) = 2$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x,\epsilon) \approx 2 + \sin x - \sin 1 \erf \left(\frac{x}{\sqrt{2\epsilon}}\right)}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' + (1+x^2)y' - y = 0$ with boundary conditions $y(1) = 1; y(-1) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x,\epsilon) \approx \exp\left(\tan^{-1}(x) - \frac{\pi}{4}\right) + \left(1 - e^{-\pi/2}\right) e^{- 2(x+1)/ \epsilon}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' - x^2y' - (3+x^3) = 0$ with boundary conditions $y(1) = 1; y(2) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y_{uniform}(x) = \frac{3}{x} -\frac{x^2}{2} -\frac{3}{2} + 3e^{\frac{-4(2-x)}{\epsilon}}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' + \sinh(\pi x)y' - y = 0$ with boundary conditions $y(1) = 1; y(2) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x)=(\frac{\tanh(\frac{\pi x}{2})}{\tanh(\pi)})^{\frac{1}{\pi}} + (1 - (\frac{\tanh(\frac{\pi}{2})}{\tanh(\pi)})^{\frac{1}{\pi}}) \exp(\frac{\sinh(\pi)(1-x)}{\epsilon})}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' - \tanh(\pi x)y' - y = 0$ with boundary conditions $y(1) = 1; y(2) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) = [\frac{\sinh(\pi)}{\sinh(\pi x)}]^\frac{1}{\pi} + (1-[\frac{\sinh(\pi)}{\sinh(2\pi)}]^\frac{1}{\pi})e^{\tanh(2\pi)\frac{-(2-x)}{\epsilon}}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' + \cosh(x)y' - e^xy = 0$ with boundary conditions $y(0) = \frac{1}{5}; y(1) = 5$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y_{uniform}(x) = \frac{5}{e^2+1}(e^{2x} + 1) + e^{\frac{-x}{\epsilon}}(\frac{1}{5}-\frac{10}{e^2+1})}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' - \tanh(x^2)y' - xy = 0$ with boundary conditions $y(1) = 1; y(2) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) = [\frac{\sinh(1)}{\sinh(x^2)}]^\frac{1}{2} + (1-[\frac{\sinh(1)}{\sinh(4)}]^\frac{1}{2})e^{\tanh(4)\frac{-(2-x)}{\epsilon}}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y + \sqrt(x) y' - y = 0$ with boundary conditions $y(0)=0, y(1)=e^2$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{ e^{2\sqrt{x}} - 1 + \frac{\int_0^{\frac{x}{\epsilon^{2/3}}} e^{-\frac{2}{3}s^{3/2}} \, ds}{\left(\frac{2}{3}\right)^{1/3} \Gamma\left(\frac{2}{3}\right)} }$

parameters: $x; \epsilon; s$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' + y' \sin(x) + y \sin(2x) = 0$ with boundary conditions $y(0) = \pi, y(\pi) = 0$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{\pi - \sqrt{2\pi} \int_0^{\frac{x}{\sqrt{\epsilon}}} e^{-s^2/2} \, ds}$

parameters: $x; \epsilon; s$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' + \tanh(x)y' + tanh^2(x)y=tanh^2(x)$ with boundary conditions $y(-2)=1, y(2)=2$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x)=1+\frac{\cosh(2)}{2\cosh(x)}(1+\text{erf}(\frac{x}{\sqrt{2\epsilon}}))}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' + \tanh^2(x)y + \tanh(x)y'=\tanh(x)\text{sech}(x)$ with boundary conditions $y(-2)=0, y(2)=0$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{\frac{x-2\text{erf}(\frac{x}{\sqrt{2\epsilon}})}{\cosh(x)}}$

parameters: $x; \epsilon$


prompt: Find a uniformly valid solution of $ \epsilon y'' - y' = 0$ with boundary conditions $ y(0) = 0, y(1) = 1$ in the limit $\epsilon = 0$ from the positive direction. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) = \frac{1-e^{\frac{x}{\epsilon}}}{1-e^{\frac{1}{\epsilon}}}}$

parameters: $x;\epsilon$


prompt: Find a uniformly valid leading order approximation to the solution of $$\epsilon y'' - y' = \sin(\pi x)$$ with boundary conditions $ y(0) = 0, y(1) = 0$ . Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $$\boxed{y(x) = \frac{\cos(\pi x) - 1}{\pi} + \frac{2}{\pi}e^{\frac{x-1}{\epsilon}}}$$

parameters: $x;\epsilon$


prompt: Find the lowest-order uniform approximation to the boundary-value problem: $$ \epsilon y'' + y' \sin x + y \sin(2x) = 0 $$ with boundary conditions:$$ y(0) = \pi, \quad y(\pi) = 0 $$.

solution: $$ \boxed{y(x) \approx \pi \, \text{erfc}\left(\frac{x}{\sqrt{2\epsilon}}\right)} $$

parameters: $x;\epsilon$