prompt: Find the leading order WKB approximation for the specific differential equation: $$ \epsilon^2 y''(x) + (1+x^2) y(x) = 0 $$ with initial conditions at $y(0) = 1, y'(0) = 0$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1$). Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{ y_{WKB}(x) \approx (1+x^2)^{-1/4} \cos\left( \frac{1}{\epsilon} \left[ \frac{1}{2} x \sqrt{1+x^2} + \frac{1}{2} arcsinh(x) \right] \right)} $

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order WKB approximation for the specific differential equation: $\epsilon^2 y'' = (x+1)y$ with initial conditions at $y(0)=0, y'(0)=1$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1$). Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) \sim \epsilon (x+1)^{-1/4}\sinh\left[\frac{2\left((x+1)^{3/2}-1\right)}{3\epsilon}\right]}$

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order WKB approximation for the specific differential equation: $$ \epsilon^2 y''(x) - (1+\cos{x}) y(x) = 0 $$ with initial conditions at $y(0) = 0, y'(0) = \frac{2^{5/4} \cosh{(1)}}{\epsilon}$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1$). Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) \sim 2\cosh(1) (1+\cos x)^{-1/4}\sinh\left[\frac{2\sqrt{2}\sin(x/2)}{\epsilon}\right]}$

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order WKB approximation for the specific differential equation: $\epsilon^2 y'' - (1+x)^2 y = 0$ with initial conditions at $y(0) = 1, y(1) = e^{-1}$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1$). Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x)\sim \frac{(1+x)^{-1/2}}{e^{3/(2\epsilon)}-e^{-3/(2\epsilon)}} [(\sqrt{2}e^{-1}-e^{-3/(2\epsilon)})\exp((x + x^2/2)/\epsilon) +(e^{3/(2\epsilon)}-\sqrt{2}e^{-1})\exp(-(x + x^2/2)/\epsilon)]}$

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order WKB approximation for the specific differential equation: $\epsilon^2 y'' = (2+x+3x^2)^2y$ with initial conditions at $y(0) = 0, y'(0)=1$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1$

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

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order WKB approximation for the specific differential equation: $\epsilon^2 y'' = x y$ with initial conditions at $y(1)=1,y'(1)=1$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1$

solution: $\boxed{\frac{\left(\left(4 - 5 \epsilon\right) e^{\frac{4}{3 \epsilon}} + \left(5 \epsilon + 4\right) e^{\frac{4 x^{\frac{3}{2}}}{3 \epsilon}}\right) e^{- \frac{2 \left(x^{\frac{3}{2}} + 1\right)}{3 \epsilon}}}{8 \sqrt[4]{x}}}$

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order WKB approximation for the specific differential equation: $\epsilon^2 y'' = e^x y$ with initial conditions at $y(0)=0, y'(0)=1$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1$

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

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order WKB approximation for the lowest order normalized eigenfunction of the differential equation: $y'' = -E_1 (x+\pi)^4 y$ with boundary conditions at $y(0)=0, y(\pi)=0$ where $E$ is a positive real value. Normalization is: $\int_0^\pi [y(x)^2 (x+\pi)^4] dx = 1$.Put your final answer in a LaTeX \boxed{} environment.

solution: $\boxed{y(x)=\sqrt{\frac{6}{7\pi^3 }}\frac{1}{(x +\pi)} \sin(\frac{x^3 + 3 \pi x^2 + 3x \pi^2}{7\pi^2}) } $

type: wkb


prompt: Find the leading order WKB approximation of $\epsilon^2 y'' = (1 + x \sin(x)) y$ subject to boundary conditions $y(0) = 0$, $y(1)= 1$ in the limit $\epsilon \rightarrow 0^+$. Approximate any integrals with a polynomial in $x$, up to third order in $x$. Put your final answer in a LaTeX \boxed{} environment.

solution: $\boxed{y = (1 + \sin(1))^{1/4} \left(\sinh\left(\frac{7}{6\epsilon}\right)\right)^{-1}(1 + x\sin(x))^{-1/4} \sinh\left(\frac{(x + x^3/6)}{\epsilon}\right)}$

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order WKB approximation for the specific differential equation: $\epsilon y''(x) + (1+x)^2 y(x) = 0$ with initial conditions at $ y(0) = 0, y(1) = 1$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1$). Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.

solution: $\boxed{y(x) \approx \frac{\sqrt{2}}{\sin\left(\frac{3}{2\sqrt{\epsilon}}\right)} \frac{1}{\sqrt{1+x}} \sin\left(\frac{x + x^2/2}{\sqrt{\epsilon}}\right)}$

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order behavior of $\epsilon^2 y''(x) = [1 + sin(x)^2]y$ subject to boundary conditions $y(0) = 1, y'(0)=1$ in the limit of large x.

solution: $\boxed{y(x) = \frac{1+\epsilon}{2} (1 + sin^2(x))^{-\frac{1}{4}} e^{1.2160 * x / \epsilon}}$

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order behavior of $\epsilon^2 y'' = (1+x)^2 y$ subject to boundary conditions $y(0) = 0$, $y'(0)= 1$ in the limit of large x.

solution: $\boxed{y(x) \sim \epsilon (1+x)^{-1/2}\sinh[\frac{2x+x^2}{2\epsilon}]}$

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order behavior of $\epsilon^2 y'' = (1+x^2)^2 y$ subject to boundary conditions $y(0) = 0$, $y'(0)= 1$ in the limit of large x.

solution: $$\boxed{y(x) \sim \epsilon (1+x^2)^{-1/2} \sinh\left(\frac{x+x^3/3}{\epsilon}\right)}$$

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order behavior of $\epsilon^2 y'' = (1/x^2) y$ subject to boundary conditions $y(1) = 1$, $y'(1)= 0$ in the limit of large x.

solution: $\boxed{y(x) \sim x^{1/2}[(\frac{1}{2}-\frac{\epsilon}{4})x^{1/\epsilon}+(\frac{1}{2}+\frac{\epsilon}{4})x^{-1/\epsilon}]}$

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order behavior of $\epsilon y'' + y = 0$ subject to boundary conditions $y(0) = 0$, $y(1) = 1$ in the limit of large x.

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

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order behavior of $\epsilon y'' + cosh(x)y = -1$ subject to boundary conditions $y(-1) = 1$, $y(1) = 1$ in the limit of large x. Write your answer as a single expression for y(x) without defining any functions or variables beyond what is given in the problem. Place this answer in a single $\boxed{}$ LaTeX environment.

solution: $\boxed{y = (1+\frac{1}{\cosh(1)}) \sec(\frac{2.1633}{2\sqrt{\epsilon}})\cos(\frac{0.00128074x^4 + 0.07861894x^3 - 0.00834220x^2 + 0.99478970x - 0.00117990}{\sqrt{\epsilon}})- \frac{1}{\cosh(x)}}$

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order behavior of $x^4y'''=y$ subject to boundary conditions $y(1)=1,y'(1)=0,y''(1)=0$ in the limit of $x\to\0^+$.

solution: $\boxed{y(x) =-0.47991x^{4/3}\exp{\left( \frac{3}{2} x^{-1/3} \right)}\cos\left( \frac{3\sqrt{3}}{2} x^{-1/3} -1.0750 \right) }$

type: wkb


prompt: Find the leading order behavior of $\epsilon y'' = e^xy$ subject to boundary conditions $[y(0) = 0, y'(0) = 1]$ in the limit of $x\to\0^+$.

solution: $\boxed{y(x) = \sqrt{\epsilon} e^{-\frac{x}{4}}\sinh(\frac{2e^{\frac{x}{2}}-2}{\sqrt{\epsilon}})}$

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order WKB approximation for the specific differential equation: $\epsilon^2 y'' = \frac{e^{2x}}{(1+e^x)^4} y$ with initial conditions at $y(0)=0, y'(0)=1$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1)$. Give your final answer in a \boxed latex environment.

solution: $\boxed{y(x) \sim 4\epsilon \cosh(x/2)\sinh\left[\frac{e^x-1}{2\epsilon(1+e^x)}\right]}$

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order WKB approximation for the specific differential equation: $\epsilon^2y''=x^2y$ with initial conditions at $y(1)=0; y(2)=1$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1)$. Give your final answer in a \boxed latex environment.

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

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order WKB approximation for the specific differential equation: $\epsilon y'' + y = 0$ with initial conditions at $y(0) = 0, y(1) = 1$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1)$. Give your final answer in a \boxed latex environment.

solution: $\boxed{ \frac{\sin \left(\frac{x}{\sqrt{\epsilon}}\right)}{\sin \left(\frac{1}{\sqrt{\epsilon}}\right)}}$

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order WKB approximation for the specific differential equation: $\epsilon y''' + y = 0$ with initial conditions at $y(0) = 1, y(\epsilon^{1/3}) = e^{-1}, y(-\epsilon^{1/3}) = e^{1} $ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1)$. Give your final answer in a \boxed latex environment.

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order behavior of $\epsilon^2 y'' + y = 0$ subject to boundary conditions $y(0)=0, y(1)=1$ in the limit of small $\epsilon$. Give your final answer in a \boxed latex environment.

solution: $\boxed{sin(x/\epsilon)/sin(1/\epsilon)}$

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order approximation of $y(x)$ from the differential equation $\epsilon^2 y''(x)=(\sin x) y$ subject to boundary conditions $y(\frac{\pi}{2}) = 1, y'(\frac{\pi}{2}) = 0$ in the limit of $\epsilon \to 0$. The answer should be in terms of the incomplete beta function $B_z(a, b)$ where $B_z(a, b)=\int_0^z t^{a-1}(1-t)^{b-1} d t, 0 \leq z \leq 1$

solution: $\boxed{\frac{1}{(\sin x)^{\frac{1}{4}}} \cosh \left(\frac{1}{2 \epsilon}\left[B_{(\sin(x))^2}\left(\frac{3}{4}, \frac{1}{2}\right)-B\left(\frac{3}{4}, \frac{1}{2}\right)\right]\right)}$.$

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order behavior of $y''=(\cot x)^4 y$ in the limit of small x that satisfies the conditions $y(1) = 1; y'(1) = 1$. Leading order does not necessarily mean only one term. Use only the variables and constants given in the problem; do not define additional constants. Give your final answer in a \boxed latex environment.

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

type: wkb


prompt: Find the leading order WKB approximation for the specific differential equation: $\epsilon^2 y'' = y \cosh^2 x$ with initial conditions at $y(0) = 0; y'(0) = 1$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1)$. Give your final answer in a \boxed latex environment.

solution: $\boxed{y(x) = \epsilon (\cosh x)^{-1/2} \sinh\left( \frac{1}{\epsilon} \sinh x \right)}$

parameters: $x; \epsilon$

type: wkb


prompt: Find the leading order WKB approximation for the specific differential equation: $\epsilon^2 y'' + \cosh(x) y' + \sinh(x) y = 0$ with initial conditions at $y(0) = 0$, $y(1) = 1$. where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1)$. Give your final answer in a \boxed latex environment.

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

parameters: $x; \epsilon$

type: wkb


prompt: Find the WKB approximation up to the second leading order for the specific differential equation as $x\to 0$ $3x^5y'''=y$ with initial conditions at $y(1)=1, y'(1)=0, y''(1)=0$ as $x\to 0$. Give ONE final answer in terms of y and x and numbers (not arbitrary constants) in a $\boxed{}$ latex environment.

solution: $\boxed{y(x) \approx x^{5/3} \left( 1.9461391296842614 e^{-\frac{3^{2/3}}{2}x^{-2/3}} + e^{\frac{3^{2/3}}{4}x^{-2/3}}\left(-1.4641592737823024 \cos\left(\frac{3^{7/6}}{4}x^{-2/3}\right) + 1.3969897071670665 \sin\left(\frac{3^{7/6}}{4}x^{-2/3}\right) \right) \right)}$

type: wkb


prompt: Find the WKB approximation up to the second leading order for the specific differential equation as $x\to 0$ $x^6 y''' + y = 0$ with initial conditions at $y(1)=1, y'(1)=0, y''(1)=0$ as $x\to 0$. Give ONE final answer in terms of y and x and numbers in a $\boxed{}$ latex environment.

solution: $\boxed{y(x) \approx x^2 \left( 0.6131324019838598 e^{x^{-1}} + e^{-x^{-1}/2}\left(-0.712093006628941 \cos\left(\frac{\sqrt{3}}{2}x^{-1}\right) -0.8372865406850002 \sin\left(\frac{\sqrt{3}}{2}x^{-1}\right) \right) \right)}$

type: wkb