# Tagged Questions

Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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### Is there an analytic solution for this Fokker-Planck equation?

The Fokker-Planck equation for a probability distribution $P(\theta,t)$: \begin{align} \frac{\partial P(\theta,t)}{\partial t}=-\frac{\partial}{\partial\theta}\Big[[\sin(k\theta)+f]P(\theta,t)-D\frac{\...
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### Examples on conceptual problems for eigenvalues in differential equations

I am currently holding a discussion class on diff eqs for engineers and I am looking for an interesting conceptual problem on eigenvalues in diff eqs. Most of the problems in 5 different books that I ...
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### Finding the inverse of a function.

Let $f:\mathbb{R}\to \mathbb{R}_+$ with $f\geq\epsilon>0$ be smooth and define $G:\mathbb{R}\to\mathbb{R}$ thus $$G(x):=\int_0^x\frac{1}{f(u)}\mathrm{d}u$$ Then it is clear that $G$ is well-...
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### Vector Laplace equation with constraint

I want to solve Laplace equation for a vector $\boldsymbol v=(v_x,v_y)$: $$\nabla^2 \boldsymbol{v}=0$$ but under the constraint that $$(1+v_x)^2+v_y^2=1$$ which becomes $v_y = -(2v_x+v_x^2)^{1/2}$. ...
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### How can one derive Stokes lines of the Stokes phenomenon of asymptotics from a differential equation?

Is there a standard technique to calculate Stokes lines and anti-Stokes lines of the Stokes phenomenon of asymptotics for a function defined as the general solution to a differential equation without ...
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### How to solve this complicated ordinary differential equation?

Consider the following non-linear ODE: $$x^2 \frac{dy}{dx} + \exp{\left(x \, \frac{d^2 y}{dx^2} \right)} = \sin \left(\frac{d^3y}{dx^3} \, \cos \left( \frac{d^4y}{dx^4} \right) \right)$$ I have no ...
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### How to solve $\frac{dy}{dx} = \frac{x^2-y^2}{x^2(y^2+1)}$

I tried to solve this using the solution of a first order differential equation but I don't think this can be reduced to that form. How to approach this problem and find $y$? Please help.
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### What differential equation might model this almost-harmonic oscillator?

I need to precisely control the motion of a damped, driven (nearly) harmonic oscillator: $$\ddot x(t) + \alpha\dot x(t) + \omega_0^2 x(t) \approx V(t)$$ I use the $\approx$ symbol because this is ...
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### Heat Kernel Asymptotics on Manifold with Boundary

On a closed Riemannian manifold $M$, the heat kernel $k_t(x, y)$ of the Laplace-Beltrami operator (or more general of any generalized symmetric Laplace-type operator acting on sections of a vector ...
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### Polar representation of conic sections $r(\theta)=\frac1{1 + e \cos\theta}$

Consider a curve given in polar coordinates by $r(\theta) = \dfrac1{1 + e \cos\theta}$, where $e\ge0$. a) Show that the distance of each point on this curve to the line $x=\frac1e$ is a constant ...
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For the equation $F(x,u,u',u'') = -au''-1 =0$ for $x\in (0,2)$ with $u(0) = 0 = u(2)$ and $a(x)$ is $1$ for $x\in (0,1)$ and $2$ for $x\in [1,2)$. Need to show that the function $$u(x) = \begin{... 0answers 116 views ### Green's function in a moving frame for a constant heat source I am looking for the Green's function of the problem in two dimensions r =(x,z), $$\nabla^2g + \frac{v}{D}\frac{\partial g}{\partial z} = -\delta (r-r_0)$$ Which ... 0answers 58 views ### Solution techniques for f'(x)=f(g(x)) I stumbled over this seemingly natural question and was surprised, that I couldn't find a satisfying answer. Differential equations of the type f'(x)=g(f(x)) are studied for all kind of classes of ... 0answers 46 views ### How to show that a leaf is topologically a cone. I am trying to understand the topological behaviour of foliations around irreducible singularities, specially in the case of singularities in the Poincaré domain. I am using the third chapter of this ... 0answers 182 views ### Coefficients of spherical solution to Laplace's equation with difficult Robin boundary conditions I'm trying to solve Laplace's equation in an (axisymmetric) external spherical domain. The controlling equation is:$$\nabla^2 f = 0$$f must dissappear at infinity, and at the surface of the ... 0answers 74 views ### Quasilinear second order ODE Consider a smooth u\colon\mathbb{R}\rightarrow\mathbb{R} satisfying$$ u^{\prime\prime}+a\left(u^{\prime}\right)^{2}+bu=0\text{ on }\mathbb{R} $$with$$ u^{\prime}\left(x\right),u^{\prime\prime}\...
Problem 1 of Section 1.2.4 of Arnold's ODE book asks, "Can the integral curves of a smooth (continuously differentiable) equation $\frac{dx}{dt} = v(x)$ approach each other faster than exponentially ...
### Solving $y^{(n)}(t)=f(t); t>0$ with initial conditions
I will use the notation $\frac{d^n y}{dt^n} \equiv y^{(n)}$. How do I solve this ODE? $$y^{(n)}(t)=f(t); t>0;\\ y(0)=y_0, y'(0)=y_1, ..., y^{(n-1)}(0)=y_{n-1}$$ What I did: The ODE is in the ...