# Questions tagged [differential-equations]

Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related 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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194 views

### 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 ...

**2**

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**1**answer

360 views

### 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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**2**answers

105 views

### 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 ...

**0**

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**0**answers

57 views

### $N-$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 ...

**0**

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**1**answer

79 views

### 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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186 views

### 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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14 views

### Structural properties (higher order monotonicity) preserved by linear differential operators

Consider the following linear differential equation:
$(\rho+x)\ f_i(x,y)+k(x-b)\ \partial_x\ f_i(x,y)=x\ f_{i-1}(x,y),\quad x\geq x_0>b>0$
where $i\geq1$, $f_0(x,y)=0$ for any $(x,y)\in\...

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**2**answers

166 views

### Convexity of a solution of a first order linear ODE

I have a very simple linear first order ODE.
$$v(x) = c x + A - B x(1-x) v'(x)$$
$c, A ,B \in(0,1)$. The domain is $(\underline{x}, 1)$. where $\underline x > 0$.
I am guessing that for any ...

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**1**answer

430 views

### Differential algebraic geometry vs Diffiety theory

Algebraic geometry is said to be useful to study not only specific solutions of polynomial equations but to understand the intrinsic properties of the totality of solutions of a system of equations.
...

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80 views

### Proving that system is Hamiltonian

This question is moved from math stackexchange, seems like it is a more advanced question. Here the link from the original question: https://math.stackexchange.com/questions/2666194/proving-that-...

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**1**answer

145 views

### Dirichlet fractional Laplacian and zero boundary conditions

Does there exists a non-zero function $$f\in C_0([0,1]):=\{f:[0,1]\to \mathbb R:\ f\text{ is continuous and } f(0)=f(1)=0\},$$ such that $(-\Delta)^{\frac\alpha 2}f\in C_0([0,1]) $, where $(-\Delta)^{\...

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47 views

### 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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**1**answer

111 views

### 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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**2**answers

170 views

### Reference request: a singular differential equation

I need the following result (which I believe to be true though I was too lazy to write down a complete proof).
Let $f$ be a function of two complex variables analytic at the origin and $a\not\in\...

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76 views

### A singular foliation analogy of the Riemann Hilbert problem

Note:
In this question by $\mathbb{C}P^1 \subset \mathbb{C}P^2$ we mean that we choose the line at infinity in the form $\{[0,y,z]\in \mathbb{C}P^2\} $ which is identified by $\mathbb{C}P^1$.
...

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34 views

### Is there a way to set a system of logistic differential equations to sum to 0

so I am working on this personal project in which I have a set of differential equations that are all of logistic growth form
$$
\dfrac{dS_k}{dt}=S_k\left(1-\dfrac{S_k}{\sigma_k(S_1,S_2,...,S_n)}\...

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41 views

### Redundant boundary condition of a $1$st order ODE?

Consider the following $1$st order eigen ODE system of 2 components $(\alpha,\beta)(x)$ defined for $x\in[0,L]$
$$
F_{l+1}\beta+m\alpha=\lambda\alpha\\
F_{-l}\alpha-m\beta=\lambda\beta
$$
where $\...

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22 views

### Two variable delta-finite function

Let ore algebra $\mathbb{O}:=\mathbb{C}(x,y)[D_x , 1,D_x][D_y,1,D_y]$
Let $F(x,y):=\sum_{m,n}a_{m,n}x^m y^n$ is a $\partial$ finite function in two variable over the field of rational function $k:=\...

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46 views

### Region of attraction of simple ODE with perturbation

Consider the following simplest example:
$$\dot{x} = x(x-1)(x+1)$$ $[-1,1]$ is the ROA.
Now consider the two dimensional case:
\begin{equation}
\begin{aligned}
&\dot{x} = x(x-1)(x+1)\\
&...

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103 views

### Stability when linearization fails

The dynamics of the $j$th system:
\begin{equation}
\begin{split}
\dot{\overline r}_j &= h (\overline r_j)
\,\, - \varepsilon \omega_{\mathrm{sw}}R_{\mathrm{Th}}\sqrt{\xi^2+\chi^2}\sum^N_{k=1} \...

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107 views

### Does a smooth dynamical system always come with a metric

Warning: My education in formal mathematics is very weak so I apologize for any confusions/errors in the following, please don't hesitate to correct me.
Question: Consider a smooth dynamical system $...

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**1**answer

151 views

### Does asymptotic behavior guarantee uniqueness?

Suppose $w$ is a solution of
$$\frac{d^2}{dx^2}w+\{u(x)+k^2\}w=0$$
with asymptotic condition
$$\lim_{x\rightarrow \infty}w(x)e^{ikx}=1$$
and $u\in L^1_1(\mathbb{R})=\{f:\int_\mathbb{R}(1+|x|)|f|dx<...

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**2**answers

146 views

### Rotation number of composition

Let $f,g:S^1 \to S^1$ be orientation-preserving homeomorphisms. Consider the lift $F,G:\mathbb R \to \mathbb R$. Let $\rho(G)$ and $\rho(F)$ be a rotation numbers. What we can say about rotation ...

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184 views

### Canard limit cycle for certain singularly perturbed system(Is there a contradictory situation?)

From the figures of page 478 and 479 of this paper one find that the author probably means that we have a (canard) limit cycle for the system
$$\begin{cases} x'=y-x^2\\ y'=\epsilon(a-x) ...

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34 views

### Harnack type Estimates for a p-Poisson equation with constant source term

Let $B=B_1(0)\subset \mathbb R^N$ and let $u\geq 0$ solve the PDE
$$
-\Delta_p u=1\,\,\mbox{in $B$}
$$
Let another function $f$ be such that
$$
\begin{cases}
-\Delta_p f =1 \;\;\mbox{in $B$}\\
f=0 \...

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34 views

### Examples of Differential Equations with Strongly Monotone Operator

Let $E \subset H \subset E^{*}$, where $E$ is a real Banach space, $E^{*}$ its dual and $H$ is a real Hilbert space. Embendings are continuous and dense. Let $\langle v_1,v_2\rangle$ be a dual pair ...

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**1**answer

50 views

### Relation between symmetries and asymptotics for Painlevé equations

The first Painlevé equation
$$P_I:y''=6y^2-x $$
has the symmetries $$x \mapsto \omega x \\y \mapsto \omega^3 y$$
for any fifth root of unity $\omega$. At the same time, the near-infinity asymptotics ...

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28 views

### Implementing the Fair-Luke algorithm

The Fair-Luke algorithm, as appears in Rational Approximations to the Solution of the Second Order Riccati Equation, constructs rational functions which approximate solutions of the so-called "...

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144 views

### Controlling solutions of a second order linear differential inequality

A slightly less general version of this question was asked, in a subsequent comment, by the OP of the question at
Controlling subsolutions of a second order linear ODE
Let $f:[0,\infty) \to \mathbb{...

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20 views

### equivalence of covariance matrix construction using convolution method and diffusion operator method?

One can define a correlation by making convolutions of two radially symmetric functions.
\begin{equation*}
C_0(r)=B_1*B_2(r) [B_1*B_2(0)]^{-1}
\end{equation*}
On the other hand, one can also ...

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**1**answer

151 views

### the asymptotic behaviour of function as $\lambda \to -\infty$

Let's consider the following differential equation on $\mathbb{R}$:
$$-u''(x)+u(x)-V(x)u(x)=\lambda u(x),$$ where $\lambda<1$ and $V$ is a bounded.
We consider only that solution $u(x) \in C^1$ ...

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**1**answer

82 views

### Reference request - existence of movable essential singularities

On the Wikipedia page regarding the Painlevé transcendents it says:
Émile Picard pointed out that for orders greater than 1, movable essential singularities can occur, and found a special case of ...

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**2**answers

199 views

### Can one obtain this ODE as an Euler-Lagrange equation?

Some of the second order ODE can be considered as Euler-Lagrange equations for an appropriate Lagrangian. However this is true not for arbitrary second order equation. But some of important equations ...

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**1**answer

207 views

### Controlling subsolutions of a second order linear ODE

Let $f:[0,\infty) \to \mathbb{R}$ obey the differential inequality
$$f'' - 2\alpha f' + 2\alpha f \leq 0$$
where $0 < \alpha < 2$ is some constant. If $f(0) = 0$ and $f'(0) = 1$, can I say that $...

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293 views

### Non-linear first order ODE $ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{Axy \ + \ By^2 \ + \ Cy}{Dxy \ + Ey \ +\ Fx \ + G}$

I am trying to solve an ODE which has the following form:
$$ \dfrac{\mathrm{d}y}{\mathrm{d}x} = \frac{Axy \ + \ By^2 \ + \ Cy}{Dxy \ + Ey \ +\ Fx \ + G}$$ with an initial condition $y(x_0) = y_0 \\ $....

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145 views

### flow of holomorphic vector field [closed]

Let $(M,J)$ be a complex manifold, where $J$ is the integrable complex structure. Let $X$ be a holomorphic vector field on $M$ and let $\varphi_{t} : M\rightarrow M $ be its flow. Question: Is $\...

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92 views

### Stochastic Approximation in Reproducing Kernel Hilbert Space

Consider an iterative algorithm with incremental updates
\begin{align}
x_{t+1} = x_t + \alpha_t \cdot [ h(x_t) + M_{t+1}],
\end{align}
where $\{x_t \}_{t \geq 0}$ is in a reproducing kernel Hilbert ...

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**0**answers

171 views

### Location of the endpoints of two parametric curves

I have two curves, $C_1$ and $C_2$ parametrized by $\theta$, the angle of the outward normal with the X-axis.
$C_1$ is given by the following equations (say $r = 0.2$):
\begin{align*}
\frac{dx}{d\...

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52 views

### Existence of a couple of functions solution of a differential equation (with additional constraint)

I would like to know if we can find a real function $v(x)$ and a complex function $f(x)$, such that they solve the following differential equation (with $\alpha$ a complex, $0<Re(\alpha)<1$):
$$...

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85 views

### How do I obtain the first fundamental form in terms of the arclength and unit normal of an arbitrary ray in the real plane?

My question is based on the 1981 paper "Computation of wave fields in inhomogenous media" by Cerveny et.al. (I will add any appropriately needed information to this posted question, so there is no ...

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54 views

### When are Green's functions causal convolution kernels

Let $L$ be a linear differential operarator acting on distributions over $\mathbb{R}$ and $G(t, s)$ be a Green's function, i.e., a solution to $LG(t, s) =\delta(t-s)$.
$G$ is said to be causal if $G(...

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83 views

### 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 ...

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**1**answer

98 views

### Basis for solutions hypergeometric differential equation

In the book "Theorie der gewöhnlichen Differentialgleichungen" by Bieberbach, page 240, there is a solution to the hypergeometric differential equation
$z(z-1)w^{\prime \prime}+(2z-1)w^{\prime}+\frac{...

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**2**answers

198 views

### Is this Riccati equation (“Josephson junction”) always phase-locked at integer rotation numbers?

Given parameters $(a,k,A) \in \mathbb{R}^3$, we consider on $\mathbb{S}^1$ the $2\pi$-periodic ODE
$$ \dot{\theta} \ = \ - a\sin(\theta) + k + A\cos(t) \hspace{4mm} \mathrm{mod} \ 2\pi. $$
Identifying ...

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**1**answer

262 views

### Gronwall's inequality for higher order derivatives

Gronwall's inequality says that solutions to the initial value problem $u'(t) \leq \beta(t)u(t)$ with $u(0)=u_0$ are bounded by solutions to the problem with inequality replaced with equality for $t\...

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78 views

### Some quasi differential equations

This question is inspired by the concept of "Differential Inclusion".
The ring of entire holomorphic functions is denoted by $Hol(\mathbb{C})$.
Is there a complete classification of all $f\in Hol(\...

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**2**answers

119 views

### Planar polynomial vector field for a harmonic pair of polynomials

Has the system of ODEs
$$\frac{dx}{dt}=P(x,y)\\
\frac{dy}{dt}=Q(x,y)
$$
been studied for the special case of the polynomials $P$ and $Q$ being a harmonic pair, i.e. the real and imaginary part of ...

**1**

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**0**answers

80 views

### Relating to the work of Václav Hlavatý

Upon looking for a topic for my phd thesis, I came upon the work of Václav Hlavatý, whose work concerning unified field theory seemed very interesting to me. My question is this; what are some similar ...

**6**

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**1**answer

196 views

### Pursuit solutions to the Rock-paper-scissors flow and delay differential equations

The Rock-paper-scissors flow is the following reaction-diffusion system
$$r_t = \Delta r + rs-rp,$$
$$p_t = \Delta p + pr-ps,$$
$$s_t = \Delta s + sp-sr.$$
We can assume $r,p,s\geq 0$, $r+p+s$ is ...