# Questions tagged [differential-equations]

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

1,039 questions
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### Stability of an equilibrium for a third order control system with an integral regulator limited by the stop-type element

In what publication is the following theorem prove carried out? The method of the Lyapunov functions is preferable. Thanks. Consider the system consisting of the controlled object and regulator. The ...
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### Does stability of equilibrium point preserved by permutation matrix (symmetry)?

Given the following differential equations: \begin{aligned} \dot{x}_1 &= f_1(x_1,\ldots,x_n) \\ \vdots \\ \dot{x}_n &= f_n(x_1,\ldots,x_n) \end{aligned} In ...
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### Galerkin Finite element for solving third order time dependent partial differential equation inti weak form

How to solve the third order time dependent partial differential equation (i.e. u_t + 6u_x + u_xxx = 0) into weak form using galerkin finite different method?
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### Differential equation changing sign almost everywhere

Conjecture: Let $f:\mathbb{R}→\mathbb{R}$ be an everywhere differentiable function and assume that $f(x)+f′(x)∈ \{-1,1\}$ almost everywhere and $f'(0)=0$. Then is $f$ necessarily a constant function? ...
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### Spectral decomposition of a specific operator

To understand a crucial example in representation theory, I need the explicit spectral decomposition of the differential operator $$Df(x)=(1+x^2)f''(x)+2xf'(x)$$ on $L^2({\mathbb R})$. I'm not an ...
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### Is the $\hat{A}$-genus invariant under crepant birational maps between smooth algebraic varieties?

The degree of the Todd class of the tangent bundle of a variety $Y$ gives the holomorphic genus of $Y$, which is a birational invariant. We also know that the $\hat{A}$-roof of a smooth variety $Y$ ...
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### A nodal theorem in 1D

Consider a 1D zero-energy Schrödinger equation on the half-line, $(-\partial_x^2 + V(x))\psi(x)=0, \quad x \in (0, \infty)$ with a zero boundary condition $\psi(0) = 0$. Is it true that if the zero-...
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### Convolution of viscosity solutions and subharmonic functions

Suppose that $u : \mathbb{C}^n \rightarrow \mathbb{R}$ is continuous. We say that $u$ is a viscosity subsolution (resp. viscosity supersolution) for the Laplace's equation if for all $\varphi \in C^2$ ...
I'm struggling to solve the following problem: Find the image of $(1,0)$ vector in $(0,0)$ point under the action of system's flow: \begin{cases} x'= e^{3x}-e^{4y} \\ y'= e^{x}-e^{y} \end{cases} I ...