# Questions tagged [differential-equations]

Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.

1,041 questions
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### On a system of non-linear differential equations

Consider the following system of coupled differential equations \begin{align} \dot{x}_{1}&= -b_1\sin(x_{1})+c(\sin(x_{2})-\sin(x_{3})) \\ \dot{x}_{2}&= a-2c\sin(x_{2})+b_1\sin(x_{1})-b_4\sin(...
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### Behavior of a non-linear differential equation

Let us consider the following differential equation $$\dot{x}(t)=a - b\sin(x(t)), \quad a,b\in\mathbb{R}.$$ My question. Suppose $a>|b|$ and $x(0)=x_0\in\mathbb{R}$. Can the solution to the ...
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### Unusual problem of calculus-of-variations. Attempt 2

I already tried to ask the same question here Unusual problem of calculus-of-variations, but I did a huge mistake and have no time to understand it. In the past post, I tried to simplify the problem, ...
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### ODE with Holder drift - Cauchy-Peano theorem

Consider the following ODE: $$x′(t)=b(x(t)),\quad x(0)=x_0.$$ If $b$ is bounded and Holder continuous, then the Cauchy-Peano theorem ensures that there exists a solution to the above equation (but ...
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### $N-$Green function in $\mathbb R^N$

Let $N \geq 3$. Does there exist solution of the following equation $$-\Delta_N G + G^{N-1} = \delta_0,$$ where $-\Delta_N = - \text{ div}(|\nabla \cdot |^{N-2} \nabla \cdot )$ denotes $N-$Laplace ...
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### How to find all orthogonal coordinates in a space of dimension $n$

I have been thinking for a while how to determine all the orthogonal coordinate systems in linear spaces of an arbitrary dimension $n$. The motivation for such a task comes from physics: I am ...
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### Unusual problem of calculus-of-variations

I have a domain $D:\quad$ $-1<x<1$ and $-f(x)<y<f(x)$.\ There is function $u(x,y)$ which is obey the equation $\Delta u(x,y)=0$, $\forall (x,y)\in D$ with the Dirichlet boundary condition ...
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### The canonical form of the first Painlevé equation

The first Painlevé equation is traditionally written as $$y''=6y^2+x.$$ Using scaling in both the dependent and independent variables, one can transform this equation into $$Y''=aY^2+bx$$ for ...
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### On local attractivity of a coupled non-linear differential equation

Consider a dynamical system described by the following coupled non-linear differential equation \begin{align} \dot{x}_1(t) &= v + a_{12}\sin(x_2(t)-x_1(t)) + a_{13}\sin(x_3(t)-x_1(t))\\ \dot{x}_2(...
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### Origins of the generalized shift operator exp(t*g(z)d/dz)

Charles Graves in the 1850s investigated iterated operators of the form $g(x) \frac {d}{dx}$ (see page 13 in The Theory of Linear Operators ... (Principia Press, 1936) by Harold T. Davis). Graves ...