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Consider the following integral:$I(x)=\int_0^\infty \frac{t^{x-1}e^{-t}}{t+x}dt$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. | $\boxed{I(x)\sim \frac{\Gamma(x)}{2x}}$ | $x$ | integrals | 12 |
Consider the following integral:$I(x) = \int_0^{π/4}\sqrt{sin (t)}e^{-x^2t^2}dt$ 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. | $\boxed{I(x) = \frac{1}{2}x^{-3/2}\cdot\Gamma(\frac{3}{4})}$ | $x$ | integrals | 13 |
Find the behavior of $y(y") + y' + xy = x^2$ in the limit $ x \rightarrow \infty$ to leading order. Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them with a ;. | $\boxed{y(x) = x}$ | $x$ | nonlinear_ode | 0 |
Find the behavior of $\frac{d^4y}{dx^4} = \cos(x^2 \frac{d^2y}{dx^2}) + \arctan(x^3 dy/dx) + e^x$ in the limit $ x \rightarrow \infty$ to leading order $x^4$. Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them with a ;. | $\boxed{e^x}$ | $x$ | nonlinear_ode | 1 |
Find the leading order behavior of $\frac{d^5 y}{dx^5} + x \frac{d^4 y}{dx^4}+ \frac{d^3 y}{dx^3}+ e^x\left(\frac{d^2 y}{dx^2}\right)^{2}-x^3y^3+x^4 \frac{dy}{dx}=0; [y(0) = 1, \frac{dy}{dx}(0) = 1, \frac{d^2 y}{dx^2}(0) = -1, \frac{d^3 y}{dx^3}(0)= 2, \frac{d^4 y}{dx^4}(0)= 1]$ in the limit $x \rightarrow 0$. Please p... | $\boxed{y(x)=1}$ | $x$ | nonlinear_ode | 2 |
Find the leading order behavior of $\frac{d^4y}{dx^4} + 2\frac{d^2y}{dx^2} + y^6 = 0, y(0)=1,y'(0)=0,y''(0)=-1,y'''(0)=-1$ in the limit $x \to \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them with a ;. | $\boxed{y(x) = 3.069(9.976 - x)^{-4/5} + (1 - 3.069(9.976 - x)^{-4/5})}$ | $x$ | nonlinear_ode | 3 |
Find the behavior to the second leading order of $\frac{d^4 y}{dx^4} = (\frac{d^2 y}{dx^2})^2 - \frac{d y}{dx}+ \frac{1}{x^3+1}, y(0)=0, y'(0)=1, y''(0)=0, y'''(0)=1$ in the limit $x \rightarrow 0$. Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them ... | $\boxed{y = x + \frac{1}{6}x^3}$ | $x$ | nonlinear_ode | 4 |
Find the leading order behavior of $\frac{d^4 y}{dx^4} = (\frac{d^2 y}{dx^2})^2 - \frac{d y}{dx}+ \frac{1}{x^3+1}, y(0)=0, y'(0)=1, y''(0)=0, y'''(0)=1$ in the limit $ x \rightarrow \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them with a ;... | $\boxed{y=6(4.01-x)^{-1}}$ | $x$ | nonlinear_ode | 5 |
Find the first order behavior of $y'' = \frac{2xy}{x^3 + y^3}, y(0)=1,y'(0)=1$ in the limit $x \rightarrow \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them with a ;. | $\boxed{y = 6^{1/3} x \ln(x)^{1/3}}$ | $x$ | nonlinear_ode | 6 |
Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + u^2 (1-u), \lim_{x \to \infty} u(x, t) = 0 \quad \text{and} \quad \lim_{x \to -\infty} u(x, t) = 1 $$ in the limit . Please place your final solutio... | $$ \boxed{ u(x, t) = \frac{1}{2} \left[ 1 - \tanh\left( \frac{x - t/\sqrt{2}}{2\sqrt{2}} \right) \right] } $$ | $x; t;$ | nonlinear_pde | 0 |
Given the following PDE:$$\frac{\partial y}{\partial t} = \frac{\partial^2y}{\partial x^2} - y^5, \quad y(x,t)>0$$For the $D\frac{\partial^2y}{\partial x^2}$ and $ \alpha y^5 t$ terms of the same order of magnitude, find the asymptotic behavior of $y$ at times after $y$ blows up. Please place your final solution in a $... | $\boxed{y(x,t) = (3/4)^{1/4} x^{-\frac{1}{2}}}$ | $x; t$ | nonlinear_pde | 1 |
Find a self similarity solution (soliton behavior) for the non-linear partial differential equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + 2v^2 u^2 (1 - u), \quad \lim_{x \to \infty} u(x, t) = 0, \quad \lim_{x \to -\infty} u(x, t) = 1$$ in the limit $t \rightarrow \infty$. Please place yo... | $\boxed{u(x, t) = \frac{1}{1 + e^{0.5(x - 0.5 t)}}}$ | $x; t$ | nonlinear_pde | 2 |
Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + 2 u(1-u)(u-\frac{1}{4}), \quad \lim_{x \to -\infty} u(x,t) = 1, \quad \lim_{x \to \infty} u(x,t) = 0 $$ in the limit $t \rightarrow \infty $. Please... | $$ \boxed{ u(x, t) = \frac{1}{1 + e^{x - 0.5t}} } $$ | $x; t;$ | nonlinear_pde | 3 |
Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$ \frac{\partial u}{\partial t} + 6 u^2 \frac{\partial u}{\partial x} + \frac{\partial^3 u}{\partial x^3} = 0, \quad u(x, t) > 0, \lim_{x \to \pm \infty} u(x,t) = 0 $$ in the limit $t \rightarrow \infty $. Please place... | $ \boxed{u_1(x, t) = \frac{1}{\sqrt{2}} \sech\left(\frac{x - 0.5t}{\sqrt{2}}\right)} $ | $x; t;$ | nonlinear_pde | 4 |
Find a localized self similarity solution (soliton behvaiour) for the non-linear partial differential equation $\partial_{xx} u + \tanh(u \partial_x u) \sech(u \partial_y u) + \sin^2(\partial_{xy} u) - e^{xy} = 0$, $u(0, y) = \cosh(y)$, $\partial_x(0, y) = \sinh(y)$ with $u(0, 0) = 1$ as the maximum value. Localization... | $\boxed{u(x, y) = sech(\sqrt{\frac{1}{2}} (x-\frac{y}{2}))}$ | $x; y$ | nonlinear_pde | 5 |
Find a localized self similarity solution (soliton behvaiour) for the non-linear partial differential equation $u_{tt}-u_{xx}-3\bigl(u^{2}\bigr)_{xx}-u_{xxxx}=0$ with $u(0, 0) = 1/4$ as the maximum value and $\partial_t u(-2, 1) < 0$. Localization means $u(x,t)$ and its derivatives vanish at $t= \pm \infty$. Please pla... | $\boxed{u(x, t) = \frac{1}{4\cosh^2(\frac{x-\sqrt{\frac{3}{2}}t}{2\sqrt{2}})}}$ | $x; t$ | nonlinear_pde | 6 |
Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $\partial_t u +\sqrt{u}\partial_x u + \partial_x^3 u=0, \lim_{|x|\to\infty} u(x,t)=0$ in the limit $t\rightarrow\infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. There should be a free paramet... | $\boxed{u(x,t) = \frac{225}{256} \sech^4( \frac{\sqrt{2}}{8} (x - 0.5t) )}$ | $x; t$ | nonlinear_pde | 7 |
Find a self similarity solution (soliton behaviour) for the non-linear partial differential equation $$ \partial_t u = \partial_{xx} u - \frac{(\partial_x u)^2}{u} + u \ln u \left(1 - (\ln u)^2\right) $$ that connects the stable state $u = e$ and $u = e^{-1}$ in the limit $|t| \rightarrow \infty$. Please place your fin... | $\boxed{u(x,t) = e^{\tanh\left(\frac{x}{\sqrt{2}}\right)} ; u(x,t) = e^{-\tanh\left(\frac{x}{\sqrt{2}}\right)}}$ | $x; t$ | nonlinear_pde | 8 |
Find an approximate solution describing the behavior for the non-linear partial differential equation $$ \frac{\partial u}{\partial t} = u^2 \frac{\partial^2 u}{\partial x^2} + \tan(u);\quad \frac{\partial u}{\partial x}(0, t) = 0,\quad u(x\to\pm\infty, t) = 0.$$. The initial condition is $$ u(x, 0) = \begin{cases} 1 &... | $$ \boxed{u(x,t) \approx \frac{\pi}{2} - \sqrt{2(0.1-t)} - 5 \frac{x^2}{\sqrt{0.1-t}}} $$ | $x; t$ | nonlinear_pde | 9 |
Find an approximate solution describing the behavior for the non-linear partial differential equation $$ \frac{\partial u}{\partial t} = u^4 \frac{\partial^2 u}{\partial x^2} + \tan(u);\quad \frac{\partial u}{\partial x}(0, t) = 0,\quad u(x\to\pm\infty, t) = 0.$$. The initial condition is $$ u(x, 0) = \begin{cases} 1 &... | $$ \boxed{u(x,t) \approx \frac{\pi}{2} - \sqrt{2(0.1-t)} - 5 \frac{x^2}{\sqrt{0.1-t}}} $$ | $x; t$ | nonlinear_pde | 10 |
Find an approximate analytical solution $u(x, t)$ to the following nonlinear partial differential equation $$\frac{\partial u}{\partial t} + 3u^2 \frac{\partial u}{\partial x} = 0.3 \frac{\partial^2 u}{\partial x^2} - 1.5 u; \quad u(0, t) = 1 \quad \text{for } t > 0.$$ The solution is sought for $x \ge 0$ and $t \ge 0$... | $$\boxed{u(x, t \rightarrow \infty) \approx \left(1 + \frac{\sqrt{5}}{4}\right) e^{-\sqrt{5}x} - \frac{\sqrt{5}}{4} e^{-3\sqrt{5}x}}$$ | $x; t$ | nonlinear_pde | 11 |
Find a self-similarity solution (soliton behavior) for $\partial_t u = \partial_{xx} u + u (4 - u^2)$ that connects the $u = 0$ solution in the $t \rightarrow -\infty$ limit to $u = 2$ in the $t \rightarrow \infty$ limit, subject to the boundary condition of $u(0, 0) = 1$. Please place your final solution in a $\boxed{... | $$\boxed{u(x, t) = 2(1 + e^{-\sqrt{2}(x + 3\sqrt{2}t)})^{-1}}$$ | $x; t$ | nonlinear_pde | 12 |
Find a self-similarity solution (soliton behavior) for $\partial_{tt} u - \partial_{xx} u + 2u ((\partial_t u)^2 - (\partial_x u)^2) = 2u^5 - u$ that travels at velocity $v = 1/\sqrt{2}$, subject to the boundary condition of $u(0, 0) = 1$. Please place your final solution in a $\boxed{}$ LaTeX Environment. The only rem... | $$\boxed{u(x, t) = 2(e^{\sqrt{2}(x - (1/\sqrt{2})t)} + e^{-\sqrt{2}(x - (1/\sqrt{2})t)})^{-1}}$$ | $x; t$ | nonlinear_pde | 13 |
Find a solution for the non-linear partial differential equation $\frac{\partial u}{\partial t} = -5u\frac{\partial u}{\partial x} -2.5u^2\frac{\partial u}{\partial x} - 0.5\frac{\partial^3 u}{\partial x^3},\lim_{x \to \pm \infty} u(x,t) = 0,u'(x,t) = 0,u''(x,t) = 0$. Please place your final solution in a $\boxed{}$ La... | $\boxed{u(x, t) = \tanh\left( \frac{x }{\sqrt{2}} \right)}$ | $x; t$ | nonlinear_pde | 14 |
Find a self-similarity solution for the non-linear partial differential equation $ \partial_t u = \partial_{xx} u - \frac{(\partial_x u)^2}{u} + u \ln u \left(1 - (\ln u)^2\right) - \delta \partial_x u $ where $\delta$ is a real constant in the limit $|t| \rightarrow \infty$ Please place your final solution in a $\boxe... | $\boxed{u(x,t) = e^{\tanh\left(\frac{x-\delta t}{\sqrt{2}}\right)}; u(x,t) = e^{-\tanh\left(\frac{x-\delta t}{\sqrt{2}}\right)}}$ | $x; t; \delta$ | nonlinear_pde | 15 |
Find a self-similar solution (soliton behaviour) for the non-linear partial differential equation $\partial_t u - \frac{10}{\sqrt{30}} \, \partial_x u = \frac{2}{5} \, \partial_x^2 u + 2 u (1 - u), \quad \lim_{x \to -\infty} u(x,t) = 1, \quad \lim_{x \to \infty} u(x,t) = 0$ in the limit $t \rightarrow \infty$. Please w... | $\boxed{u(x,t) = \frac{1}{\left(1 + \exp\left[\sqrt{\frac{5}{6}}\left(x + \frac{10}{\sqrt{30}} t \right)\right] \right)^2}}$ | $x; t$ | nonlinear_pde | 16 |
Find a self similarity solution for the non-linear partial differential equation $ \partial_t u = \partial_{xx} u + (2u-\sqrt{5})(1-u^2), \lim_{x \to -\infty} u(x,t) = -1, \lim_{x \to \infty} u(x,t) = 1 $ in the limit $t \rightarrow \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. | $ \boxed{ u(x,t)=\tanh(x-\sqrt{5}t) }$ | $x; t$ | nonlinear_pde | 17 |
Please solve the non-linear partial differential equation $\frac{\partial^2 u}{\partial t^2}+ \frac{\partial^2 u}{\partial x \partial t}= \left( \frac{\partial u}{\partial t} \right)^2+ \frac{\partial u}{\partial x} \cdot \frac{\partial u}{\partial t}$ with initial conditions $u(x, 0) = 0, \quad u_t(x, 0) = x^2$ . Plea... | $\boxed{u(x, t) = -\ln\left(1 - \frac{t^3}{3} + x^2 t - x t^2\right)}$ | $x; t$ | nonlinear_pde | 18 |
Find a self-similarity solution (soliton behaviour) for the non-linear partial differential equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + 5u(1-u)$$ with boundary condition: $\lim_{x \to -\infty} u(x,t) = 1,\lim_{x \to \infty} u(x,t) = 0$. Please place your final solution in a $\boxed{}$... | $\boxed{u(x,t) =\frac{1}{1 + e^{\frac{\sqrt{5}}{2}(x - 2\sqrt{5}\, t)}}}$ | $x; t$ | nonlinear_pde | 19 |
Consider the PDE $\u_t + \frac{2x}{t} u_x = u_{xx} + (1-u^2) \sinh(x), \quad u(x,1) = \frac{1}{4}e^{-x^2} -1, \lim_{|x| \to \infty} u(x, t) = -1$. Find the solution in the limit $t \to \infty$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a ... | $\boxed{ u(x,t) = \tanh(-\cosh(x) + x^2/t)}$ | $x; t$ | nonlinear_pde | 20 |
For $0 < x < 1$, $t > 0$ and a small parameter $0 < \epsilon \ll 1$ consider the PDE $u_t = \epsilon u_{xx} + u(1-u), \quad u(0, t) = 1, u(1, t) = 1/2, u(x, 0) =1.$. Find the leading order solution as $t \to \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment | $\boxed{u(x,t \to \infty) \approx \frac{3}{2} \left(\frac{(5+2\sqrt{6}) e^{(1-x)/\sqrt{\epsilon}} - 1}{(5+2\sqrt{6}) e^{(1-x)/\sqrt{\epsilon}} + 1}\right)^2 - \frac{1}{2}}$ | $x; t; \epsilon$ | nonlinear_pde | 21 |
Find a self similarity solution (traveling wave front) for the reaction-diffusion equation $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + 4u(1 - u^2) $$ that connects the stable state $u=-1$ as $x \to -\infty$ to the stable state $u=+1$ as $x \to +\infty$. The solution should satisfy $u(x,t) = 0... | $\boxed{u(x,t) = \tanh(\sqrt{2} x)}$ | $x; t$ | nonlinear_pde | 22 |
Find the solution behavior of $\frac{\partial u}{\partial t} = \frac{\partial}{\partial x} \left( u^2 \frac{\partial u}{\partial x} \right)$ in the limit $t \to \infty$ (similarity solution) with boundary conditions $\lim_{|x| \to \infty} u(x,t) = 0 \text{ and initial mass } \int_{-\infty}^{\infty} u(x,0) dx = M$. Plea... | $\boxed{u(x,t) = \frac{1}{(4t)^{1/4}} ( \frac{12}{\pi} - (\frac{x}{(4t)^{1/4}})^2 )^{1/2}}$ | $x; t$ | nonlinear_pde | 23 |
Solve the following nonlinear partial differential equation $$\partial_t u + \partial_x u = u \partial_{xx}u + (x-t)^{-2}$$by finding a travelling wave solution, and determine its leading-order behavior as $|x-t| \to \infty$. Your final answer should contain only the variables $x$ and $t$. In particular, do not define ... | $\boxed{(\ln((x-t)^2))^{1/2}}$ | $x; t$ | nonlinear_pde | 24 |
Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $u_{tt} - u_{xx} - 6 (u^2)_{xx} - u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = \frac{1}{2}$. | $\boxed{u(x,t) = \frac{1}{2} \text{ sech}^2(\frac{1}{\sqrt{2}} (x - \sqrt{3}t)); u(x,t) = \frac{1}{2} \text{ sech}^2(\frac{1}{\sqrt{2}} (x + \sqrt{3}t))}$ | $x; t$ | nonlinear_pde | 25 |
Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $u_{tt} - u_{xx} - 2 (u^3)_{xx} - u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = \frac{\sqrt{3}}{3}$. | $\boxed{u(x,t) = \frac{\sqrt{3}}{3} \text{ sech}(\frac{1}{\sqrt{3}} (x - \frac{2}{\sqrt{3}}t)); u(x,t) = \frac{\sqrt{3}}{3} \text{ sech}(\frac{1}{\sqrt{3}} (x + \frac{2}{\sqrt{3}}t))}$ | $x; t$ | nonlinear_pde | 26 |
Find a self-similarity solution (soliton behaviour) for the non-linear partial differential equation $$\frac{\partial u}{\partial t} + u^2 \frac{\partial u}{\partial x} + \frac{\partial^3 u}{\partial x^2 \partial t} = 0$$ with boundary condition: $\lim_{x \to -\infty} u(x,t) = 0,\lim_{x \to \infty} u(x,t) = 0$. Please ... | $\boxed{u(x,t)= \sqrt{6} \, \mathrm{sech}(x-t)}$ | $x; t$ | nonlinear_pde | 27 |
Suppose we have the following Reaction-Diffusion type Partial Differential Equation, $ \partial_t u = \partial_{xx} u - \alpha (u - \frac{1}{5}) + \beta (u - \frac{1}{5})^3 $, for some function $u(x,t)$ where $\alpha,\beta > 0$. Please find a self similarity solution (solition behavior) that connects the state $u = \fr... | $\boxed{u(x,t) = \sqrt{\frac{2 \alpha}{\beta}}\text{sech}(\sqrt{\alpha} z) + \frac{1}{5}; u(x,t) = -\sqrt{\frac{2\alpha}{\beta}}\text{sech}(\sqrt{\alpha} z) + \frac{1}{5}}$ | $x; t;\alpha;\beta; z$ | nonlinear_pde | 28 |
Suppose we have the following Partial Differential Equation, $\partial_{xxx}u + \partial_xu(1+\partial_xu) = \partial_tu$, for some function $u(x,t)$. Please find a traveling wave solution that connects the steady states $u = 1$ in the limit $t \rightarrow -\infty $ and $u = 3$ in the limit $t \rightarrow +\infty$. Pla... | $\boxed{u(x,t) = 2+\tanh(\frac{1}{6}(x+\frac{10}{9}t))}$ | $x; t$ | nonlinear_pde | 29 |
Find a traveling-wave solution to the nonlinear partial differential equation $$\partial_t u + \partial_x u = -u^3 \partial_{xx}u + (x-t)^2 + 1$$ and determine its leading-order behavior as $|x-t| \to \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. The only remaining parameters in your fina... | $$\boxed{u(x,t) = \sqrt{2} (x-t) (\ln(|x-t|))^{\frac{1}{4}}}$$ | $x;t$ | nonlinear_pde | 30 |
Find the solution that contains a logarithmic term to the nonlinear PDE for $x \in (0, \pi)$, $t > 0$, $ t u_t - x u_x - (3 - 2x \cot(x) ) u \log u = 0, \quad u(0, t) = 1, u(\pi, t) = 1, u(x, 0) = 1 $. Please place your final solution in a $\boxed{}$ LaTeX Environment. The only remaining parameters in your final soluti... | $\boxed{ u(x,t) = \exp(\sin^2(x) t^3 \log(1 + xt)) }$ | $x; t$ | nonlinear_pde | 31 |
Find the solution behavior to$u \partial_{xt} u - \partial_x u \partial_t u + t u \partial_x u + (1 + t^2) \sin(2x) u^2 = 0, u(0, t) = e^{-t}, u_t(x, 0) = \cos^2(x)$ in the limit $t \to \infty$Please place your final solution in a $\boxed{}$ LaTeX Environment. | $\boxed{e^{\cos(x)^2 t - \log(1 + t^2)}}$ | $x; t$ | nonlinear_pde | 32 |
Approximate a self similar traveling wave solution $ \partial_t u = \partial_{xx} u + u(1-u)(u-\frac{1}{4}) + x \left(\partial_t u + \frac{\sqrt{2}}{4} \partial_x u\right)$ in the limit $t \to \infty$. Return one exact expression for u(x,t). Please place your final solution in a $\boxed{}$ LaTeX Environment.If there ar... | $\boxed{u(x,t) = \frac{1}{1+\exp{\left(\frac{x}{\sqrt{2}}-\frac{t}{4}\right)}}}$ | $x;t$ | nonlinear_pde | 33 |
Find an approximate analytical solution $u(x, t)$ to the following nonlinear partial differential equation $$u_t + 0.1 (u_x)^2 = u_{xxxx} - 16 u; \quad u(0, t) = 1, u(x, 0) = 0 $$ The solution is sought for $x \ge 0$ and $t \ge 0$. Derive an approximate analytical solution $u(x, t)$ that captures the dominant behavior ... | $\boxed{u(x, t) \sim \frac{599}{600}e^{-2x} + \frac{1}{600}e^{-4x}}$ | $x; t$ | nonlinear_pde | 34 |
Find a solution to the nonlinear partial differential equation $ \partial_t u + \partial_x u = -\left(u + \frac{1}{u}\right) \partial_{xx}u + (x-t)^2 + \frac{1}{x-t}$ for its leading-order behavior as $|x-t| \to \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment.If there are multiple solutions,... | $\boxed{u(x,t) \sim \frac{1}{\sqrt{2}} (x-t)^2}$ | $x;t$ | nonlinear_pde | 35 |
Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $u_{tt} - u_{xx} - 8(u^2)_{xx} - u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = 2$. | $\boxed{\frac{2}{\cosh^2(2\sqrt{\frac{2}{3}}(x-\sqrt{\frac{35}{3}}t))}}$ | $x; t$ | nonlinear_pde | 36 |
Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $u_{tt} - u_{xx} - 9(u^3)_{xx} - u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = 1$. | $\boxed{u(x,t)=\frac{1}{\cosh(\frac{3}{\sqrt{2}}(x+\sqrt{\frac{11}{2}}t))}}$ | $x; t$ | nonlinear_pde | 37 |
Solve the following nonlinear partial differential equation $$ \partial_t u + u^2 \partial_x u = \partial_{xx}u - (x-t)^{-2} $$ by finding a travelling wave solution of the form $u(x,t) = U(x-t)$, and determine its leading-order behavior as $x-t \to \infty$. Your final answer should contain only the variables $x$ and $... | $\boxed{u(x,t) = 1 + (x-t)^{-1/2}}$ | $x;t$ | nonlinear_pde | 38 |
Solve the following nonlinear partial differential equation $$\partial_t u + \frac{3}{2} \partial_x u = u^2 \partial_{xx}u + \frac{2}{3}\left(x-\frac{3}{2}t\right)^{-2}$$ by finding a travelling wave solution, and determine its leading-order behavior as $|x-\frac{3}{2}t| \to \infty$. Your final answer should contain on... | $\boxed{\left(2\ln\left(x-\frac{3}{2}t\right)\right)^{1/3}}$ | $x;t$ | nonlinear_pde | 39 |
solve$ \partial_t u = (u^3-2(x-2t)) \partial_x u - \partial_x \left( u \partial_x u \right) - (x-2t)^{-3} $as $t\to\infty, u(\pm infty,t)\to 5$ away from any divergences to a nonconstant function. Place one final solution in a $\boxed{}$ LaTeX Environment | $\boxed{u(x,t)=-\frac{1}{254 (x-2t)^2}-\frac{2}{16129 (x-2t)}+\frac{4 \log (x-2t)}{2048383}-\frac{4 \log (127-2 (x-2t))}{2048383}+5}$ | $x;t$ | nonlinear_pde | 40 |
solve$ \partial_t u = u^2 \partial_x u - \partial_x \left( u \partial_x u \right) - (x-2t)^{-3/2} $as $t\to\infty, u(\pm infty,t)\to 5$ away from any divergences to a nonconstant function. Place one final solution in a $\boxed{}$ LaTeX Environment | $\boxed{u(x,t)=-2(x-2t)^{-1/2}/(2+5^3)+5}$ | $x;t$ | nonlinear_pde | 41 |
Suppose we have the following Partial Differential Equation, $\partial_{xxx}u + \partial_{t}u(1-\partial_xu)= 0$, for some function $u(x,t)$. Please find a travelling wave solution that connects the steady states $u = 1$ in the limit $t \rightarrow -\infty $ and $u = 3$ in the limit $t \rightarrow +\infty$. Place your ... | $\boxed{u(x,t) = 2+\tanh(\frac{3}{2}(x-9t))}$ | $x;t$ | nonlinear_pde | 42 |
solve$ \partial_t u = (u^3-2(x-2t)) \partial_x u - \partial_x \left( (u^2+3) \partial_x u \right) - (x-2t)^{-3} $as $t\to\infty, u(\pm infty,t)\to 5$ away from any divergences to a nonconstant function. Place ONE final solution in a $\boxed{}$ LaTeX Environment | $\boxed{u(x,t)=-\frac{1}{254 (x-2t)^2}-\frac{2}{16129 (x-2t)}+\frac{4 \log (x-2t)}{2048383}-\frac{4 \log (127-2 (x-2t))}{2048383}+5}$ | $x;t$ | nonlinear_pde | 43 |
Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \frac{\partial^2 u}{\partial x^2} - u^2; x \geq 0, t > 0; u(0, t) = \frac{1}{\sqrt{t}}, u(x,0) = 0$ in the limit $t \rightarrow \infty$. Please place yo... | $\boxed{u(x,t) = \frac{1}{\sqrt{t}} e^{-x/\sqrt{t}}}$ | $x;t$ | nonlinear_pde | 44 |
Find the asymptotic solution with first two leading terms for the non-linear partial differential equation $\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \frac{\partial^2 u}{\partial x^2} - u^2; x \geq 0, t > 0; u(0, t) = 1, u(x,0) = 0$ in the limit $t \rightarrow \infty$. Please place your final so... | $\boxed{u(x,t) = \frac{1}{1 + 2.5x} + 0.611 e^{-0.5 x} - \frac{0.509}{1 + 2.5x} e^{-0.5 x}}$ | $x;t$ | nonlinear_pde | 45 |
Find an approximate analytical solution $u(x, t)$ to the following nonlinear partial differential equation $u_t +0.1(u_x)^2+0.05uu_{xx} = u_{xxxx}-16u+0.2e^{-3x}, \quad u(0,t)=1, u(x,0)=0 $ The solution is sought for $x \ge 0$ and $t \ge 0$. Derive an approximate analytical solution $u(x, t)$ that captures the dominant... | $\boxed{$u(x,t)=0.9994170441764800e^{-2x}+0.003076923076923077e^{-3x}-0.002484639053254438e^{-4x}-0.000009318214941557116e^{-5x}-0.000000009985207100591716e^{-6x}}$ | $x; t$ | nonlinear_pde | 46 |
Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $u_t - u_{xx} - u^3 = (1-2t)\cos(x)e^{-t^2} - \cos^3(x)e^{-3t^2}, u(x,0) = \cos(x)$ in the limit N/A. Please place your final solution in a $\boxed{}$ LaTeX Environment.DONT USE \operatorname in your boxed solution. The... | $\boxed{u(x,t) = \cos(x) e^{-t^2}}$ | $x; t$ | nonlinear_pde | 47 |
Find a nontrivial self similarity solution (soliton behaviour) for the non-linear partial differential equation $$ \partial_t u + \left(\frac{45}{16} u^{1/2} - \frac{3}{2} u\right)\partial_x u + \partial_x^3 u=0 $$ The wave is assumed to travel with velocity $v=0.5$. The boundary condition is $\lim_{|x|\to\infty} u(x,t... | $\boxed{u(x,t) = \frac{16}{\left(6 + e^{\frac{x-0.5t}{2\sqrt{2}}} + e^{-\left(\frac{x-0.5t}{2\sqrt{2}}\right)}\right)^2}}$ | $x; t; t_0$ | nonlinear_pde | 48 |
Find a self-similar solution to $$u_t=(u^2)_{xx}$$ for $$x\in \mathbb{R}, t>0$$. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants. Box your final answer. | $$\boxed{u(x,t)=t^{-1/3}\cdot\max\{1-\frac{x^2}{12t^{2/3}},0\}}$$ | $x; t$ | nonlinear_pde | 49 |
For the nonlinear PDE $$u_t + 6u\cdot u_x + u_{xxx} = 0$$ with $x\in [-L,L] \ \text{(periodic)}, \ t>0$, seek a traveling‐wave solution. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants. Box your final answer. | $$\boxed{u(x,t) = \frac{1}{2}\cdot\text{sech}^2\left(\frac{1}{2}\cdot(x - t)\right)}$$ | $x; t$ | nonlinear_pde | 50 |
Find a self similarity solution (soliton behavior) for the non-linear partial differential equation $ \partial_t u + u^2 \partial_x u + \partial_x^3 u = 0 $ in the limit $t\to\infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=6$. If there are multip... | $$\boxed{u(x,t) = 6 \sech{\left( \sqrt{6} (x - 6t) \right)}}$$ | $x; t; v$ | nonlinear_pde | 51 |
Find a self similarity solution (soliton behavior) for the non-linear partial differential equation $ \partial_t u + u^{1/2} \partial_x u + 1.5\partial_x^3 u = 0 $ in the limit $t\to\infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=0.5$. If there a... | $$\boxed{u(x,t) = \frac{225}{256} \sech^4\left( \frac{\sqrt{3}}{12}(x - 0.5t) \right)}$$ | $x; t; v$ | nonlinear_pde | 52 |
Find a self similarity solution (soliton behavior) for the non-linear partial differential equation $ \partial_t u + u^{3/2} \partial_x u + \partial_x^3 u = 0 $ in the limit $t\to\infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=0.5$. If there are ... | $$\boxed{u(x,t) = ( \frac{35}{16} )^{2/3} \sech^{4/3}( \frac{3\sqrt{2}}{8}(x - 0.5t) )}$$ | $x; t; v$ | nonlinear_pde | 53 |
Find a self similarity solution (soliton behavior) for the non-linear partial differential equation $ \partial_t u + 2u^{1/2} \partial_x u + \partial_x^3 u = 0 $ in the limit $t\to\infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=0.5$. If there are... | $$\boxed{u(x,t) = \frac{225}{1024} \sech^4\left( \frac{\sqrt{2}}{8}(x - 0.5t) \right) }$$ | $x; t; v$ | nonlinear_pde | 54 |
Find the self-similar (soliton-like) solution $u(x,t)$ for the nonlinear partial-differential equation $$ \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \frac{\partial^2 u}{\partial x^2} - u^3, u(0,t)=1, u(x,0)=0, x\ge 0, t>0 $$ in the limit $t\to\infty$ | $$ \boxed{U(x)=\frac{1}{1+\frac{1}{2}x}} $$ | $x$ | nonlinear_pde | 55 |
Find the self-similar (soliton-like) solution $u(x,t)$ for the nonlinear partial-differential equation $$\partial_t u + u^{1/2}\,\partial_x u = \partial_{xx}u - 2.5\,u^2, u(0,t)=1, u(x,0)=0,\;x\ge0,\;t>0$$ in the limit $t\to\infty$ | $$ \boxed{U(x)=\frac{1}{(1+0.5\,x)^{2}}} $$ | $x$ | nonlinear_pde | 56 |
Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$\partial_t u = partial_{xx} u + u^2 - u; u(x;0) = 0.1 \quad (|x| \leq 3), u(x;0) = 0 \quad (|x| \gt 3); \lim_{x \to -\infty} u(x,t) = 0, \quad \lim_{x \to \infty} u(x,t) = 0$$ in the limit $t\to\infty$. Please place y... | $$\boxed{\frac{0.6}{\sqrt{4 \pi t}} e^{-t} e^{-\frac{x^2}{4t}}}$$ | $x; t$ | nonlinear_pde | 57 |
Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$\partial_{t} u = \partial_{xx} u - 1.5 \, \partial_{x} u + 0.2 \, u, \quad u(x,0) = \frac{1}{\sqrt{0.1}} e^{-\frac{x^2}{0.4}}, \quad u(+\infty,t) = u(-\infty,t) = 0$$. | $$\boxed{\frac{1}{\sqrt{t+0.1}} e^{-\frac{(x-1.5t)^2}{4(t+0.1)} + 0.2t}}$$ | $x; t$ | nonlinear_pde | 58 |
Find a nontrivial self similarity solution (soliton behaviour) for the non-linear partial differential equation $$ \partial_t u + \left(\frac{15}{8} u^{1/2} + \frac{9}{4} u\right)\partial_x u + \partial_x^3 u=0 $$ The wave is assumed to travel with velocity $v=0.5$. The boundary condition is $\lim_{|x|\to\infty} u(x,t)... | $$ \boxed{u(x,t) = \frac{40}{\left(2\sqrt{10}+5e^{\frac{x-0.5t}{2\sqrt{2}}} + 5e^{-\left(\frac{x-0.5t}{2\sqrt{2}}\right)}\right)^2} ; u(x,t)=0} $$ | $x; t; v$ | nonlinear_pde | 59 |
Find the self similarity solution for the nonlinear partial differential equation $$\partial_t = \partial_x (u^3 \partial_xu) $$ Where the initial shape of the solution is $u(x, t)=u(x, 0)= 0.9 * \exp(-x^2 /2)$ There should be a free parameter A in the solution which you should set to $A =1.1$. Please put your final so... | $$\boxed{u(x,t) = \frac{1}{(5t)^{\frac{1}{5}}} \sqrt[3]{-\frac{3}{2}\left(\frac{x}{(5t)^{\frac{1}{5}}}\right)^2 + A}}$$ | $x; t; A$ | nonlinear_pde | 60 |
Find the solution for the nonlinear partial differential equation $$\partial_t u = \partial_{xx}u + u^2 (1 - u^2)$$ | $$\boxed{u(x,t)=\tanh( \frac{1}{\sqrt{2}}(x - \sqrt{2}t))}$$ | $x; t$ | nonlinear_pde | 61 |
Find a traveling-wave solution to $u_t = -u_{xx} - 2 u u_x^2/(1 - u^2)$, such that $u \rightarrow 0$ as $t \rightarrow \infty$, $u \rightarrow 1$ as $t \rightarrow -\infty$, $u(0,0) - 0.76159 < 0.01$, and $u_x(0,0) - 0.41997 < 0.01$. Please place your final solution in a $\boxed{}$ LaTeX Environment. The only remaining... | $$\boxed{u(x,t) = \tanh(e^{x - t})}$$ | $x; t$ | nonlinear_pde | 62 |
Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $2u_{tt} - 3u_{xx} - 6(u^2)_{xx} - 5u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = 13$. Please place your final solution in a $\boxed{}$ LaTeX Environment. | $$ \boxed{ u(x,t) = 13\,\mathrm{sech}^2\left(\sqrt{\frac{13}{5}}\left(x-\sqrt{\frac{55}{2}}t\right)\right) } $$ | $x; t$ | nonlinear_pde | 63 |
Find a self-similar solution to $$u_t=(u^5)_{xx}$$ for $$x\in \mathbb{R}, t>0$$. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants. Please place your final solution in a $\boxed{}$ LaTeX Environment. | $$\boxed{u(x,t)=t^{-1/6}\cdot\max\{1-\frac{x^2}{15t^{1/3}},0\}}$$ | $x; t$ | nonlinear_pde | 64 |
Find a self-similarity solution for the non-linear partial differential equation: $d_t u = d_{xx}u + u(u - \alpha)(1 - u)$ where $\alpha$ is a constant such that $0 < \alpha < 1/2$. The solution connects $u=1$ ($t \rightarrow -\infty$) to $u=0$ ($t \rightarrow +\infty$). Determine the wave speed $c$ in terms of $\alpha... | $$ \boxed{c(\alpha) = \frac{1-2\alpha}{\sqrt{2}}} $$;$$ \boxed{u(x,t) = \frac{1}{1+e^{\frac{1}{\sqrt{2}}(-x + \frac{\sqrt{2}}{4}t + 1)}}} $$ | $x; t; \alpha$ | nonlinear_pde | 65 |
Find a self-similarity solution for $$ \partial_t u = \partial_{xx}(u^2) - \partial_{xxxx}u + (\partial_x u)^2 $$ with maximum value at $u(0,0) = 1$. | $$ \boxed{ u(x,t) = (t+K_0^2)^{-1/2} f_s\left( \frac{x}{(t+K_0^2)^{1/4}} \right) } $$ | $x; t;K_0;f_s$ | nonlinear_pde | 66 |
Approximate the expectation of the solution of SDE $$dX_t = \Bigl(-\frac{\sin(t^{-1/2})}{t^2}-\frac{1+\tfrac1t}{2t^{3/2}}\cos(t^{-1/2})-\frac{B_t}{2t^{3/2}}+\frac1{t+1}+B_t^2\bigl(6t^{-3}-\frac1{(t+1)^2}\bigr)-3\frac{B_t^4}{t^4}\Bigr)\,dt + \Bigl(\frac1{\sqrt t}+\frac{2B_t}{t+1}+\frac{4B_t^3}{t^3}\Bigr)\,dB_t.$$ in the... | $\boxed{\frac{1}{1 + \sqrt{t}}}$ | $t$ | other | 0 |
Approximate the expectation function of the solution of SDE $dX_t = \Bigl[\bigl(\frac1t-\frac1{t^2}+\frac2{t^3}\bigr)+\ln(t)\bigl(\frac1{t^2}-\frac4{t^3}\bigr)-\tfrac12\ln(t)\bigl(1-\frac1t+\frac2{t^2}\bigr)\Bigr]\cos(B_t)\,dt - \ln(t)\bigl(1-\frac1t+\frac2{t^2}\bigr)\sin(B_t)\,dB_t$ in the limit $ t > 0$ Please provid... | $\boxed{e^{-\frac{t}{2}} \ln(t) (1 - \frac{1}{t} + \frac{2}{t^2})}$ | $t$ | other | 1 |
Approximate the expectation function of the solution of SDE $$dX_t=\exp\bigl(\sqrt{t}\sin(2t)\,B_t\bigr)\Bigl(-\frac{2t}{(1+t^2)^2}+\frac{B_t}{1+t^2}\Bigl(\frac{\sin(2t)}{2\sqrt{t}}+2\sqrt{t}\cos(2t)\Bigr)+\frac{(\sqrt{t}\sin(2t))^2}{2(1+t^2)}\Bigr)\,dt+\frac{\sqrt{t}\sin(2t)}{1+t^2}\exp\bigl(\sqrt{t}\sin(2t)\,B_t\bigr... | $\boxed{u(x,t) = \frac{1}{1 + t^2} e^{t^2 \sin^2(2t)}}$ | $t$ | other | 2 |
Approximate the expectation function of the solution of SDE $$dX_t=\exp\bigl(\cos(t)B_t\bigr)\Bigl(-\frac{2t}{(1+t^2)^2}-\frac{B_t\sin(t)}{1+t^2}+\frac{\cos^2(t)}{2(1+t^2)}\Bigr)\,dt+\frac{\cos(t)}{1+t^2}\exp\bigl(\cos(t)B_t\bigr)\,dB_t.$$ in the limit $t \to \infty$ Please provide your answer in LaTeX \boxed{} environ... | $\boxed{\frac{1}{1 + t^2} e^{\frac{t}{2}\cos^2(t)}}$ | $t$ | other | 3 |
Approximate the expectation function of the solution of SDE $$dX_t =\exp\!\Bigl(\tfrac{\sqrt2\,\sin(t)}{t^{1/4}}\,B_t\Bigr)\Bigl(-\frac{2t}{(1+t^2)^2}+\frac{B_t}{1+t^2}\Bigl(\frac{\sqrt2\,\cos(t)}{t^{1/4}}-\frac{\sqrt2\,\sin(t)}{4\,t^{5/4}}\Bigr)+\frac{\sin^2(t)}{t^{1/2}(1+t^2)}\Bigr)\,dt+\frac{\sqrt2\,\sin(t)}{t^{1/4}... | $\boxed{\frac{1}{1 + t^2} e^{\sqrt{t} \sin^2(t)}}$ | $t$ | other | 4 |
Approximate the expectation function of the solution of SDE $$ dX_t=\exp\Bigl(\tfrac{t}{\sqrt3}\sin\bigl(\tfrac{t}{3}\bigr)B_t\Bigr)\Bigl(-\tfrac{2B_t^2}{t^3(1+t^2)}-\tfrac{2t(1+\tfrac{B_t^2}{t^2})}{(1+t^2)^2}+\tfrac{(1+\tfrac{B_t^2}{t^2})B_t}{1+t^2}(\tfrac{\sin\tfrac{t}{3}}{\sqrt3}+\tfrac{t\cos\tfrac{t}{3}}{3\sqrt3})+... | $\boxed{\frac{1}{1 + t^2} e^{\frac{t^3}{6} \sin^2(t/3)}}$ | $t$ | other | 5 |
Find the leading order WKB approximation for the specific differential equation: $$ \epsilon^2 y''(x) + (1+x^2) y(x) = 0 $$ with initial conditions at $y(0) = 1, y'(0) = 0$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1$). Use only the variables and constants given in the problem; do not define add... | $\boxed{ y_{WKB}(x) \approx (1+x^2)^{-1/4} \cos\left( \frac{1}{\epsilon} \left[ \frac{1}{2} x \sqrt{1+x^2} + \frac{1}{2} arcsinh(x) \right] \right)} $ | $x; \epsilon$ | wkb | 0 |
Find the leading order WKB approximation for the specific differential equation: $\epsilon^2 y'' = (x+1)y$ with initial conditions at $y(0)=0, y'(0)=1$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1$). Use only the variables and constants given in the problem; do not define additional constants. Pl... | $\boxed{y(x) \sim \epsilon (x+1)^{-1/4}\sinh\left[\frac{2\left((x+1)^{3/2}-1\right)}{3\epsilon}\right]}$ | $x; \epsilon$ | wkb | 1 |
Find the leading order WKB approximation for the specific differential equation: $$ \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 g... | $\boxed{y(x) \sim 2\cosh(1) (1+\cos x)^{-1/4}\sinh\left[\frac{2\sqrt{2}\sin(x/2)}{\epsilon}\right]}$ | $x; \epsilon$ | wkb | 2 |
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 additiona... | $\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)]}$ | $x; \epsilon$ | wkb | 3 |
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$ | $\boxed{y(x) = \frac{\epsilon}{\sqrt{2(2+x+3x^2)}} \sinh[\frac{2x+\frac{1}{2}x^2+x^3}{\epsilon}]}$ | $x; \epsilon$ | wkb | 4 |
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$ | $\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}}}$ | $x; \epsilon$ | wkb | 5 |
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$ | $\boxed{y(x) =\epsilon e^{-x/4}\sinh\left[\frac{2\left(e^{x/2}-1\right)}{\epsilon}\right]}$ | $x; \epsilon$ | wkb | 6 |
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 \boxe... | $\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}) } $ | $x; E_1$ | wkb | 7 |
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. | $\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)}$ | $x; \epsilon$ | wkb | 8 |
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 additiona... | $\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)}$ | $x; \epsilon$ | wkb | 9 |
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. | $\boxed{y(x) = \frac{1+\epsilon}{2} (1 + sin^2(x))^{-\frac{1}{4}} e^{1.2160 * x / \epsilon}}$ | $x; \epsilon$ | wkb | 10 |
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. | $\boxed{y(x) \sim \epsilon (1+x)^{-1/2}\sinh[\frac{2x+x^2}{2\epsilon}]}$ | $x; \epsilon$ | wkb | 11 |
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. | $$\boxed{y(x) \sim \epsilon (1+x^2)^{-1/2} \sinh\left(\frac{x+x^3/3}{\epsilon}\right)}$$ | $x; \epsilon$ | wkb | 12 |
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. | $\boxed{y(x) \sim x^{1/2}[(\frac{1}{2}-\frac{\epsilon}{4})x^{1/\epsilon}+(\frac{1}{2}+\frac{\epsilon}{4})x^{-1/\epsilon}]}$ | $x; \epsilon$ | wkb | 13 |
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. | $\boxed{y(x) = \frac{sin(\frac{x}{\sqrt{\epsilon}})}{sin(\frac{1}{\sqrt{\epsilon}})}}$ | $x; \epsilon$ | wkb | 14 |
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 $\boxed{}$ La... | $\boxed{y = (1+\frac{1}{\cosh(1)}) \sec(\frac{2.1633}{2\sqrt{\epsilon}})\cos(\frac{0.00128074x^4 + 0.07861894x^3 - 0.00834220x^2 + 0.99478970x - 0.00117990}{\sqrt{\epsilon}})- \frac{1}{\cosh(x)}}$ | $x; \epsilon$ | wkb | 15 |
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 $x\to\0^+$. | $\boxed{y(x) =-0.47991x^{4/3}\exp{\left( \frac{3}{2} x^{-1/3} \right)}\cos\left( \frac{3\sqrt{3}}{2} x^{-1/3} -1.0750 \right) }$ | $x$ | wkb | 16 |
Find the leading order behavior of $\epsilon y'' = e^xy$ subject to boundary conditions $[y(0) = 0, y'(0) = 1]$ in the limit of $x\to\0^+$. | $\boxed{y(x) = \sqrt{\epsilon} e^{-\frac{x}{4}}\sinh(\frac{2e^{\frac{x}{2}}-2}{\sqrt{\epsilon}})}$ | $x; \epsilon$ | wkb | 17 |
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