# Questions tagged [differential-equations]

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

**3**

votes

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59 views

### Limits of a simple damped system

Definition: Let $F_n(s) = \frac{1}{s^{n+1}(1+s)^n}$ be the Laplace transform of $f_n(t)$.
Required Result: To show $\lim_{n\rightarrow\infty}f_n(n+n/e) < o(n)$.
Ideas:
Let $G_n(s)=\frac{1}{s^{n+...

**12**

votes

**4**answers

723 views

### History of ODE and PDE reference request

Is there any reference (book or articles) which made the history (up to the modern times) and the conceptual development of Ordinary Differential Equations and Partial Differential Equations? It will ...

**1**

vote

**0**answers

114 views

### Locally Lipschitz sufficiently implies a Gronwall inequality?

In the paper [1], it seems to me the authors implicitly use a local Lipschitz property to deduce a Gronwall's inequality. I am not able to justify/show that this is indeed the case and perhaps someone ...

**2**

votes

**4**answers

217 views

### Complex differential equations

I'm looking for a gentle an concise introduction to complex-variable differential equations. Eventually, I need to look at complex PDEs, but I assume one starts with complex ODEs.
Mostly, I'm just ...

**1**

vote

**1**answer

73 views

### Solve nonlinear equation

Suppose that $f:E\to F$(between Banach spaces), is of the form
$$f(x)=f(0)+D(x)+N(x).$$
Here $D$ is a linear term, whose kernel is of finite dimension, and admits a right inverse $G$, i.e. $D(G)(\...

**0**

votes

**0**answers

54 views

### 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 ...

**1**

vote

**1**answer

28 views

### Does stability of equilibrium point preserved by permutation matrix (symmetry)?

Given the following differential equations:
\begin{equation}
\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}
\end{equation}
In ...

**0**

votes

**1**answer

60 views

### 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?

**1**

vote

**2**answers

90 views

### One inequality connected with the linear second order ODE

Is the following statement true?
Let $ a>0, b>0, h>0 $, $x(t)$ be the solution of the differential equation
$ \ddot{x}+a \dot{x}+bx=h$
with initial conditions $x(0)=u<0 , \dot{x}(0)...

**5**

votes

**0**answers

102 views

### Algebraic independence of limit cycles of Lienard equation

It is well known that the Van der Pol equation $$\begin{cases} \dot x=y-(x^3-x)\\\dot y=-x \end{cases}$$ has no an algebraic limit cycle.
According to this fact, we search for a related ...

**0**

votes

**0**answers

67 views

### What is the unitary $1$-parameter group generated by a vector field on a manifold?

If $M$ is a (compact, Riemannian) manifold, and one complexifies its tangent bundle, is it possible to give a meaning to the "unitary group"-like expression $t \mapsto \exp (\mathrm i t X)$ when $X$ ...

**6**

votes

**1**answer

125 views

### Ordinary differential operators satisfying braid relation?

Let $W$ be the algebra of linear ordinary differential operators with analytic coefficients $C^{\omega}(\mathbb{R})[\partial_x]$ (with multiplication given by composition). Do there exist two elements ...

**1**

vote

**0**answers

26 views

### Mean of a periodic velocity field and trajectory displacement bound

Suppose $u(t,x)$ is a smooth velocity field on $[0,\infty)\times \mathbb{R}$ and periodic in space, i.e., $u(t,0)=u(t,1)$ $\forall t$. Assume that $\int_0^1 u(t,x) \,dx = c$, independent of time. Let $...

**18**

votes

**1**answer

2k views

### 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?
...

**3**

votes

**1**answer

115 views

### 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 ...

**10**

votes

**0**answers

271 views

### 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 $Y$ ...

**7**

votes

**1**answer

610 views

### (In)stability of a two-dimensional dynamical system

Consider the following system of coupled differential equations
$$
\dot{x}_1(t) = -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\
\dot{x}_2(t) = -\gamma x_2(t) - \cos(\...

**2**

votes

**0**answers

152 views

### A question about whether an operator can be lipschitz or not

Let assume $ X= \Omega \times (0,T) $ where $\Omega \subset \mathbb{R} $ is a bounded open set and $T>0$.
Now define the operator $ \mathcal{A} : C^{\sigma, \sigma/2}(X) \to C^{\sigma, \...

**1**

vote

**0**answers

82 views

### 2nd oder evolution equations and regularity results of their solution

I am interested in regularity results for solutions to 2nd order evolution equations in the shape of
$$
u''(t) + A(t)u(t) = f(t) \text{ in } V^*\text{ a.e. in }S, \\
u(0) = u_0 \text{ in } H, u'(0)...

**1**

vote

**1**answer

82 views

### Behaviour of solutions to $(A-r)f=0$ in the limit $r \to \infty$

Define the second order linear differential operator associated with $X$ (Here $X$ is the unique strong solution to appropriate Ito SDE) by $$A = \frac{1}{2} \sigma^2(x) \frac{d^2}{dx^2} + \mu(x) \...

**1**

vote

**1**answer

178 views

### Classification of a system of two second order PDEs with two dependent and two independent variables

If we have a second order quasilinear PDE of the form
$A\frac{\partial^2 u}{\partial x^2}+B\frac{\partial^2 u}{\partial y^2}+2C\frac{\partial^2 u}{\partial x\partial y}+ lower\,\, order \,\, terms=0$...

**1**

vote

**1**answer

127 views

### Positive Solutions of second-order ODE

Consider second-order ordinary differential equations of the form
$u''(t)=a(t)u(t)-2$
I'm interested in general criteria on the function $a(t)$, which guarantee respectively rule out the existence ...

**1**

vote

**0**answers

52 views

### Well-posedness for a general linear grad-div PDE

Let $U \subset \mathbb{R}^n$ be an open bounded domain with smooth boundary ($\partial U$ is a closed manifold) and let $A(x)\in C^1(U, M_{n \times n}(\mathbb{R}))$, $b(x) \in C^0(U, \mathbb{R}^n)$ ...

**0**

votes

**1**answer

57 views

### Well-posedness for equations of the form $u_t = grad[V(u)]$ and $u_{tt}=grad[V(u)]$?

Let $V \in C^{1}(\mathbb{R}^n, \mathbb{R})$ consider the following PDE:
$$u_t = grad[V(u)]$$
For $u \in C^{1}([0,1]^n\times [0,T),\mathbb{R}^n)$, with boundary conditions specified on the $n$-...

**6**

votes

**0**answers

119 views

### Conditions to the existence of periodic orbits of non vanishing vector fields on $\mathbb{T}^2$

I'm doing a research about Filippov systems on $\mathbb{S}^3$ with discontinuities on $\displaystyle\frac{1}{\sqrt{2}}\cdot\mathbb{T^2} =\left\{\displaystyle\frac{1}{\sqrt{2}}x ; \ x \in \mathbb{S}^1\...

**3**

votes

**0**answers

67 views

### Parametrix of external product of elliptic operators

Let $S$ and $T$ be Dirac-type operators on vector bundles $E\rightarrow M$ and $F\rightarrow N$ respectively, where $M$ and $N$ are manifolds. Suppose $Q_S$ and $Q_T$ are parametrices for $S$ and $T$. ...

**10**

votes

**0**answers

412 views

### Witt's proof of Gelfand-Mazur / Ostrowski's Theorem

Previously asked on Math Stackexchange without answers.
Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...

**1**

vote

**0**answers

74 views

### Reference for numerical solutions for differential equations like $f'(x)=f(x+1)+f(x-1)$

One can solve a delay differential equation (like for example $f'(x)=f(x-1)$) if we have a function as a bounded condition (in my example we need to know $f$ on $[0,1)$) and then use a simple forward ...

**7**

votes

**2**answers

391 views

### Generating function of $SO(N)$ random matrix

I am interested in the generating function of $SO(N)$ random matrix, that is, I want to compute
$$
Z_N[J]=\int dM e^{{\rm Tr} (J^T M)},
$$
where $dM$ is the $SO(N)$ Haar measure, and $J$ is an ...

**3**

votes

**0**answers

99 views

### If the sum of everywhere linearly independent vector fields are periodic, are the component vector fields periodic?

I feel like the above must be true but embarrassingly cannot seem to prove it. Take linearly independent, commuting vector fields $X$ and $Y$ on a manifold and corresponding flows $\Phi^t_X$, $\Phi^...

**3**

votes

**2**answers

253 views

### Are there vector fields which are gradients with respect to one metric but not another? [closed]

Is it possible for a vector field on a smooth manifold $M$ to be a gradient with respect to a Riemannian metric $g$, but not a gradient with respect to a different Riemannian metric $h$?
For ...

**-1**

votes

**1**answer

70 views

### explicit answer of and initial condition ODE with delta input

Assume this initial value problem ODE with constant coefficient:
$\mathcal{D}[u] = \sum_{n=0}^N {a_n u^{(n)}}=0$
$u(0)=u_0\hspace{0.2cm} ;\hspace{0.2cm} u'(0)=u_1\hspace{0.2cm} ;\hspace{0.2cm} ... \...

**4**

votes

**1**answer

125 views

### Which utility functions are linearly transformed by normal perturbations?

I'll ask this question as pure economics, as pure math, and showing the translation.
Economics (micro):
Which utilities have the property that whenever $EU(X)>EU(Y)$, the same is true after ...

**8**

votes

**0**answers

112 views

### An upper bound for the solution of an integro-differential equation

For those who are interested in "motivations" this has something to do with modeling flames in turbulent jets. However the question itself is irritatingly elementary and requires no mathematical or ...

**0**

votes

**0**answers

69 views

### Caratheodory Differential Equations ( Existence of Solution )

I'm working with filippov's book , "Differential Equations with Discontinuous Right Hand Sides ".
Theorem 1) For $ t_{0} \leq t \leq t_{0} + a ~~~,|x-x_{0}| \leq b $ let the function $ f(t,x) $ ...

**0**

votes

**0**answers

46 views

### Change of polynomial eigenvalues between polynomials

Given the polynomial eigenvalue problem
$$
p_t(z) = det ( P(z) + Q(t) ) = 0,
$$
where $P(z) = \sum_{i=0}^k P_i z^i$ with $P_i \in \mathbb{C}^{n \times n}$ and $Q(t) \in \mathbb{C}^{n \times n}$. The ...

**0**

votes

**0**answers

63 views

### Reference for Existence and uniqueness of an Integro-Differential Equation

I have an Integro-Differential Equation (IDE) of the following form:
$$
x'(t) = f(t,x(t)) + \int_0^t K(t-s, x(s), x(t)) ds,
$$
I have found this classical reference, but the IDEs considered therein ...

**0**

votes

**0**answers

61 views

### Perturbed trajectory of non-autonomous ode

Does the existence of strict Liapunov function guarantee that limit point of perturbed trajectory will be close to an isolated equilibrium for non-autonomous ode ? Can someone help me with some ...

**0**

votes

**1**answer

76 views

### On the (qualitative) behavior of a coupled differential equation

Let $\mathbf{x}(t):=[x_1(t),\dots,x_n(t)]^\top$, $n>1$, $A\in\mathbb{R}^{n\times n}$ be a nonnegative matrix and $\mathbf{b}\in\mathbb{R}^n$ be a positive vector. Consider the following ...

**0**

votes

**1**answer

113 views

### Continuity of the differential flow under a perturbation of the vector field

Suppose $v$ is a (possibly time-dependent) vector field on a compact manifold $M$.
Its flow is a mapping $g: M \times \mathbb{R} \rightarrow M$, where $g$ satisfies the following conditions (written ...

**1**

vote

**1**answer

166 views

### Asymptotic behavior of a solution of an ODE

I am not quite sure if this question is appropriate for this site as it might be not of a research level.
I am interested in the following ordinary differential equation on the real line
$$f’’(x)+(x-...

**2**

votes

**1**answer

116 views

### 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-...

**2**

votes

**0**answers

165 views

### Lemma 4.5.1 on page 77 in the book Averaging Methods in Nonlinear Dynamical Systems

I have a query regarding two equalities in the lemma in the book.
But first I'll provide two definitions that one needs for this lemma.
Definition 4.2.4: Consider the vector field $f(x,t)$ with $f:\...

**3**

votes

**2**answers

131 views

### Perturbed behavior of a differential equation

Let $a$, $b$ be two real positive parameters with $a>b$, and consider the following nonlinear differential equation:
\begin{align}
\dot{x}_{\varepsilon}(t) = a - b\sin(x_{\varepsilon}(t))+\...

**2**

votes

**0**answers

52 views

### Stability of ODEs with exponentials in the vector field

What is known about fine stability properties of ODEs of the following kind?
$$ \dot{x} = Ax + b + \phi(x),\quad x\in \mathbb{R}^d ,$$
where $d\geq 1$; $A$ is a constant matrix with all e.v. having ...

**4**

votes

**1**answer

161 views

### Is there a Feynman-Kac formula for vector-valued Schrödinger operators?

Given a vector function
$$f=(f_1,\ldots,f_n)\in L^2(\mathbb R,\mathbb R^n)$$
(for some $n\in\mathbb N$), let us define
$$\Delta f:=(\Delta f_1,\ldots,\Delta f_n),$$
where $\Delta$ is the Laplacian ...

**3**

votes

**0**answers

54 views

### Generalized viscosity sub(super)solution and it's convolution

Suppose that $\Gamma \subsetneq \mathbb{R}^n$ is an open symmetric convex cone containing positive orthant.
Note that $\Gamma \subset \left\{x=(x_1,...,x_n) \in \mathbb{R}^n | \sum_{i=1}^{n} x_i > ...

**4**

votes

**1**answer

158 views

### 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 $\varphi \in C^2$ ...

**2**

votes

**0**answers

69 views

### Differential equation's flow [closed]

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 ...

**1**

vote

**0**answers

53 views

### 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(...