# 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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### Distributional PDE solutions as topological linear duals of PDE solutions

Let $$P \;\colon\; \Gamma_\Sigma(E) \to \Gamma_\Sigma(\tilde E^\ast)$$ be a formally self-adjoint hyperbolic linear differential operator ($\tilde E^\ast$ denotes the densitized dual of a ...
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### An elementary question about integration by parts! [closed]

Let $f,g: R \rightarrow R$ be two positive increasing functions. Under what (non-trivial) conditions one can guarantee that $\int_{0}^{\infty}f'g dx\geq \int_{0}^{\infty}g'fdx$.
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### diffusion coefficient derived from simple random walk in a 1D semi-infinite domain

Suppose we have a 1D domain $x\in[0,\infty)$ and particles released at $x=0$ are doing simple random walks along the domain with reflecting boundary conditions at x=0. Then we can write down the ...
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### Holonomic modules and Holonomic functions

Let $$f_{d}(h):=\sum_{k=1}^{d}(-1)^k\binom{d-1}{k-1}\prod_{i=1}^{d}G((i-k)h) .$$ I have proved that $F(x):=\sum_{d=1}f_{d}\frac{x^{d}}{d!}\in \mathbb{C}(h)[[x]]$ is holonomic and arrive at a ...
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### Differential equations with infinite-dimensional Lie groups

I am no expert in solving DEs by symmetry methods, but from pure interest - is it possible for a differential equation to have an infinite-dimesional Lie group as a symmetry group?
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### Continuous Dependence of ode solution on parameters [closed]

Let $f:V\rightarrow \mathbb{R}^n$ be locally Lipschitz ($V$ is a subset of $\mathbb{R}\times\mathbb{R}^m\times \mathbb{R}^n$). Suppose we have a function $x:[t_0,\beta[\times W\rightarrow \mathbb{R}^n$...
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### The entire parametrization of leaves of singular holomorphic foliation of $\mathbb{C}P^2$

What is an example of an entire non constant holomorphic function $\gamma: \mathbb{C} \to \mathbb{C}P^2$ such that the image of $\gamma$ is a leaf of a singular holomorphic foliation of ...
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### Closed orbit for vector field $f(\bar{z})$ where $f$ is holomorphic function

Edit : According to the comments of Michael Renardy and Christian Remling I revise the question as follows: Is there a vector field $X$ on an open set $U\subseteq \mathbb{R}^2$ such ...
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### ODEs whose finite-time solutions are not L^2 on their interval of definition

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be analytic and consider the ODE $$x'(t)=f(x(t)).$$ It is well-known that if $(t_{min},t_{max})$ is the maximal domain of a solution $x$ and $t_{max}<\infty$, ...
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### How to find maximize the variable in the given scenario?

Consider an aloha like wireless communication algorithm with $n$ nodes, with each node transmitting with an exponential distribution with rate $\lambda_o$, with $\tau$ be the packet transmission time. ...
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### Ratio dependent predator prey model

In the article on Global qualitative analysis of a ratio-dependent predator–prey system- Kuang, 1998 The system is where a, K, c, m, f, d are positive constants that stand for prey intrinsic ...
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### The number of limit cycles of a quadratic vector field with a unique singularity

Is there a uniform upper bound for the number of limit cycles of a quadratic vector field which has a unique singular point in the plane?
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### Solving ODE via contact geometry

I have been reading H. Geiges' "A Brief History of Contact Geometry and Topology". According to him contact transformations were introduced as a geometric tool to study systems of differential ...
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### Two semi stable limit cycles with disjoint interior

What is a precise example of a quadratic vector field on the plane with at least $1$ semi stable limit cycles? Furthermore, is there a quadratic polynomial vector field on the plane with two ...
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### A cubic system with two nested limit cycles with opposite orientations(2)

The second part of Hilbert's 16th problem not only concerns "The number of limit cycles of a polynomial vector field", but also the position and configuration of of those limit cycles with respect to ...
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### Error bounds for non-autonomous systems with respect to input

The error bounds for an ordinary differential equation: $$\dot{x}(t) = f(x(t))$$ with respect to initial conditions $x(t_0) = x_0$, $\hat{x}(t_0)=\hat{x}_0$ \begin{...
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Assume that $P(x,y),Q(x,y)\in \mathbb{R}[x,y]$ are two polynomials of degree $2$ with $P(0,0)=Q(0,0)=0.$ Suppose that the vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ has a center ...
The general Volterra Equation of the second kind in convolution form can be described by: $$\phi(x) = \int_a^x K(x-t)\phi(t)\, \mathrm{d}t + f(x), \text{ for } x\geq a$$ Suppose we wish to ...