# 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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### Has this Peculiar Property of Unit Circles Already been Noticed?

Yesterday I needed to do some calculations with circles and "ventured" to calculate the arc length via the $\int{\sqrt{1+\left(f'(x)\right)^2}}$ formula and was baffled to see that in the case of unit ...

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**1**answer

222 views

### Analytic solutions to algebraic differential equation

Dear Colleagues and Friends,
Here I need to find some good reference on a subject that seems very much studied: sorry, if the rest of this question is too naive.
I believe that it's known that if a ...

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51 views

### Existence of a solution to a boundary value problem

Consider an ODE of the form
$$y''=f(x,y,y') $$
with $x \geq 0$, and initial conditions of the form
$$y(0)=y_0>0, \\y'(0)=m_0. $$
I want to claim that under reasonable conditions (which I can't ...

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**1**answer

223 views

### A cubic system with two nested limit cycles with opposite orientations

What is an example of polynomial vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ such that two closed orbits $C_1,C_2$ of the system surrounds an annular region $R$ such that $...

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242 views

### Flow of a nowhere vanishing complete vector field

Let X be a nowhere vanishing complete vector field on a manifold M, $\gamma: \mathbb{R} \to M$ be its flow with $\gamma(0)=p \in M$ and suppose it is not periodic. If $\gamma(\mathbb{R})$ is closed, ...

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189 views

### what is about the corresponding power series?

According to the papers The absolutely continuous spectrum of Jacobi matrices and these lecture notes:
periodicity ~ potential well or lattice (order)
lack of absolutely continued spectrum ~ Anderson ...

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285 views

### Symbol of the Laplace-Beltrami on $\mathbb{S}^2$

This question is about how the principal part (or symbol) is defined on a manifold?-I assume that the answer is: As in $\mathbb{R}^n$ using local coordinates, i.e.
A differential operator $P=\sum_{|\...

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79 views

### Formal way to prove existence and continuity in an integral equation

In the paper "A Boundary Value Problem Associated with the Second Painlevé Transcendent and the Korteweg-de-Vries Equation" by Hastings and McLeod, the authors study the ODE
$$\frac{\mathrm{d}^2 y}{\...

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238 views

### Simple-looking nonlinear ODE with fractional power

I am trying to solve the following nonlinear ODE for a function $P(x)$:
$$\left(1-x^2\right)P''(x)+k(k+1)P(x)=cP(x)^\frac{k-1}{k+1}.$$
Here, $k$ and $c$ are arbitrary parameters. By rescaling $P(x)$,...

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**1**answer

37 views

### Purpose of using a saturable logistic like term

I would like to know what is the purpose of using the term $P\over (k+P)$ in the following. I found it when reading the article found here but it was commonly used in few other related articles .
Is ...

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493 views

### Sobolev trace theorem on Lipschitz domains

Supposing that D is a bounded Lipschitz domain (and not smooth) in $\mathbb{R}^d$. From what I know, it is known that the trace operator is well-defined and continuous from $H^s(D)$ to $H^l(\partial D)...

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77 views

### Does there exist a comparison principle for vector differential equations?

As is well known, there is a comparison principle for scalar differential equations $\dot x(t)=a(x(t))$ and $\dot y(t)=b(y(t))$ with $x(t_0)=y(t_0)$ and $a(\cdot)\leq b(\cdot)$, with the result being ...

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75 views

### Diffraction across an absorbing wall

I need help finding the procedure for the solution of the following differential equation.
This is equation is: Find $u:\mathbb{R}^2 \to \mathbb{C}$ such that for $C>0$
$ \begin{cases} u_{xx}+ ...

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**1**answer

158 views

### “Canonical” form for gauge equivalence classes of matrices in $\mathfrak{gl}_n(x)$

Let $\mathfrak{gl}_n(x)= \mathfrak{gl}_n \otimes_\mathbb{C}\mathbb{C}(x)$ be the algebra of matrices taking values in rational functions.
Definition: Two matrices $A, B \in \mathfrak{gl}_n(x)$ are ...

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406 views

### Random N-body problem

Suppose there are $N$ unit-mass particles whose initial positions
are uniformly distributed in a unit-radius disk.
Each particle is assigned a randomly oriented initial velocity vector $v_i$ of length ...

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**1**answer

81 views

### Higher order Lyapunov equation

Let $A$ be a (finite) Hurwitz matrix.
In this related question of mine, (see also https://en.wikipedia.org/wiki/Lyapunov_equation) it is shown that
$$
\int_0^\infty \sum_{j,k} (e^{At})_{ij} Q_{jk} (...

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**1**answer

182 views

### Regular holonomic D-modules as generalisation of regular singular points

I'm trying to understand why the definition of a regular holonomic D-module is a good generalisation of the usual definition of a regular singular point for a differential equation. More precisely, ...

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105 views

### Wolff's article: Note on counterexamples in strong unique continuation problems

I am reading Wolff's Note on counterexamples in strong unique continuation problems:
http://www.ams.org/journals/proc/1992-114-02/S0002-9939-1992-1014648-2/S0002-9939-1992-1014648-2.pdf
On Page 3, ...

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215 views

### Continuation (Extension) of Harmonic Functions

Suppose $(M,g)$ is a simply connected smooth Riemannian manifold with smooth boundary and suppose that $U \subset \partial M$ is a smooth open connected subset of the boundary. Now my question is the ...

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84 views

### How can I show the principal symbol is not elliptic?

Let us assume the principal symbol of a nonlinear differential operator $E$ at a point $p$ is
$$\sigma_{E}(p):\Gamma (T^*M)\times \Gamma (T^*M)\to \Gamma (T^*M)$$
which acts as follows:
...

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**1**answer

176 views

### Ground state for a double well potential (Schrödinger)

Consider $V(x)$ a one dimensional polynomial, confining, symmetric double well potential i.e.
$V(x)=V(-x)$ for all $x\in\mathbb{R}$
$\displaystyle \lim_{x\to\pm\infty} V(x)=+\infty$
$V(x)\in \mathbb{...

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199 views

### Solutions for an ODE

I would like to know if someone could provide me the solution(s) for the following equation:
\begin{align*}
\frac{d^{2}w}{dx^{2}}\cdot\frac{dw}{dx} = e^{-x}w
\end{align*}
Where $w > 0$, $w''>0$ ...

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62 views

### Well-posedness of a differential equation: Cauchy vs Dirichlet [closed]

I am struggling to understand what makes some differential equations well-posed Cauchy (initial value) problem rather than a Dirichlet (boundary value) problem. Consider, for example, a simple ...

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**1**answer

164 views

### Riemann surface from Riccati equation

I have quite a practical question motivated by physics.
Consider the Riccati equation whose solution gives a quantum-mechanical (QM) analogue of the classical momentum:
$$
(p(x))^2 + \dfrac{\hbar}{i}...

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275 views

### On Wilson's claim that Lyapunov function level sets are not exotic spheres

In Wilson's paper "The structure of the level surfaces of a Lyapunov function," he states in Corollary 1.3 that the level sets of a smooth Lyapunov function are diffeomorphic to a standard sphere. (...

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494 views

### What is a Green's function in the language of $\mathcal{D}$-modules?

Let $P$ be an analytic linear differential operator defined on some open interval $X=(a,b)$ and $\mathcal{M}=\mathcal{D}_X / \mathcal{D}_X \bullet P$ the corresponding $\mathcal{D}$-module. I'm trying ...

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640 views

### Vector field with holomorphic flow

Let $(M,J)$ be a complex manifold. Suppose that $X$ is a real vector field such that the flow of $X$ is by biholomorphisms.Question Show the flow of $JX$ is by biholomorphisms.
I know one reference ...

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342 views

### Fredholm index vs. Limit cycle theory

Let $A$ be the algebra of all smooth functions $f: \mathbb{R}^2 \to \mathbb{R}$ such that $f$ is flat at the origin and is real analytic on $\mathbb{R}^2 \setminus \{0\}$.
Let $B $ be ...

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445 views

### Is there a closed-form solution for $\frac{dy}{dx} = 1 + \frac{a}{y} + \frac{b}{x}$?

I am looking for an exact solution for the following special case of Chini Equation with $2\geq a > 1 > b > 0, x, y \in \mathbb{R}^+$,
$$\frac{dy}{dx} = 1 + \frac{a}{y} + \frac{b}{x}$$
I ...

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160 views

### Reference request: Original source of Yosida approximation

Numerous papers/books(citation needed) refer to the operator $$A_\lambda := \lambda AR_\lambda (A) = \lambda^2 R_\lambda(A) - \lambda I$$ where $R_\lambda(A)=(I+\lambda A)^{-1}$ is the resolvent, as ...

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509 views

### An ordinary differential equation

While I was working on a variational problem, I met this equation as its Euler-Lagrange equation, but I cannot solve it:
$ x= \frac{af'(x)}{\sqrt{1+af'(x)^{2}}} + \frac{bf'(x)}{\sqrt{1+bf'(x)^{2}}} \ (...

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261 views

### Gauge equivalence between operators

I have tried to figure out the following problem for some time now, but with little success:
Let $ \mathcal{L} $ be a third order linear differential operator with coefficients in $ \mathbb{C}(X) $. ...

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215 views

### Are all solutions to an ordinary differential equation continuous solutions to the associated implied differential equation and vice versa?

Now I have to heavily emphasize the fact that I have never studied differential algebra or the concept of other types of differentiation (which is what I believe is the concept behind a differential ...

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35 views

### A uniqueness result for a BVP over a semi-infinite interval

Let $q:[0,\infty) \to \mathbb{R}$, and consider the ODE
$$u''(x)=q(x)u(x) $$
with the boundary conditions $u(0)=\lim_{x \to \infty} u(x)=0$. Under what conditions on $q$ is $u \equiv 0$ the only ...

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285 views

### On modified Bessel solutions to complex ODE's using Kummer's series

I am trying to reduce the following ODE to Bessel's ODE form and hence solve it:
$$x^{2}y''(x)+x(4x^{3}-3)y'(x)+(4x^{8}-5x^{2}+3)y(x)=0\tag{1} \, .$$
I tried to solve it via the standard method, i.e.,...

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34 views

### Uniqueness of positive solutions to the n-vortex type equation

The $n$-vortex equation, in the context of optical vortex solitons, is of the form
\begin{equation}
-(ru_{r})_r+\dfrac{n^2}{r}u+\beta ru= f(u^2)ru,\quad r\in (0,\infty),\\
u(0)=0=u(\infty),
\end{...

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**1**answer

91 views

### Does this equation have an explicit solution? [closed]

For a positive constant $C$:
\begin{align}
y(x)+C\ln y(x)=f(x).
\end{align}
At least from specific $f(x)$, such as piece-wise linear function, is there an explicit solution for $y(x)$?

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206 views

### Minimum eigenvalue of One-dimensional Schrodinger Operator

Consider the One dimensional Schrodinger Operator
$$
-\frac{d^2}{dx^2} + V(x)
$$
Where the Potential Function $V$ is of the form $V(x) = x^4 - a^2x^2$ , $a \in \mathbb{R} $.
Now of course,the ...

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64 views

### Convergence of Bessel (Sturm-Liouville) Expansions at the End Points

I have asked this question before on MSE but received no answer at all. So I assume that it is proper to ask it here. I am not a mathematician so my language may not be too precise, please correct me ...

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**1**answer

136 views

### A non vanishing vector field on $S^3$ with a periodic attractor

Is there a non vanishing real analytic vector field $X$ on $S^3$ such that $X$ has an attractor periodic orbit(An asymptotically stable periodic orbit) ? What about the smooth case?

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189 views

### Does the divergent solution of this equation :$f'=e^{f^{-1}}$ of Gevrey type and could be Borel summation applied for it?

This question was asked here in MO by someone seeking for the solution of the functional -differential:$f'=e^{f^{-1}}$ not exactly an O.D.E, and again here seeking for the growth rate of it solution ...

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**1**answer

150 views

### Solutions to linear SDE with many noise sources

It is well known how to solve the linear stochastic ODEs with one source of noise
$$dX_t=(a(t)X_t+c(t))dt+(b(t)X_t+d(t))dW_t$$
See, for instance, https://math.stackexchange.com/questions/1788853/...

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216 views

### Hörmander's hypoellipticity theorem for complex coefficients

Hörmander's theorem says that if $L = \sum _{i=1} ^r X_i ^2+ X_0 + f$ on some open subset $U \subseteq \Bbb R$ has the property that the Lie algebra generated by $\{X_0, \dots, X_r\}$ at every point ...

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87 views

### A global geometric formulation of the fundamental theorem of Picard Vessiot theory?

Let $X$ be a smooth curve over an algebraically closed field of characteristic 0. The category of quasi-coherent $D$-modules $\mathcal{D}_X$-$Mod$ is a symmetric monoidal abelian category. We can ...

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117 views

### Geometric interpretation of energy-momentum tensor and Lagrangian associated to a soliton equation

I have a question for you.
BACKGROUND
Consider an immersion $F\,:\;\mathscr S\;\longrightarrow\;\mathbb E^3$ of a surface $\mathscr S$ in the $3$-D euclidean plane $\mathbb E^3$ with canonical ...

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198 views

### Spectral growth of One dimensional Schrodinger Operator

Conside the One dimentional Schrodinger Operator
$$
-\frac{d^2}{dx^2} + ( V(x) + E )
$$
Where the Potential Function $V$ is of the form $V(x) = ax^2 + b^2x^4$ , $a,b \in \mathbb{R} $.
What is known ...

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147 views

### Is the closed orbit of the Vander pol equation a stable periodic orbit?

We consider the Vander Pol vector field $$(1) \;\;\;\;\;\; \begin{cases} x'=y-(x^3-x)\\ y'=-x\end{cases}$$ on $\mathbb{R}^2.$
It is well known that this equation has a unique limit ...

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49 views

### Control of the Hessian of the distance function at infinity

Let $(M,g)$ be a complete noncompact manifold and $r$ is the distance function to a fixed point. We assume that the sectional curvature $|Rm|(x) \le K(r(x))$ where $K(r)=\frac{1}{1+r^k}$ for some $k&...

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120 views

### An indefinite integral containing functions that are solutions to a 2nd order linear ODE

I am trying to evaluate an indefinite integral of the form
$\int \frac{dz}{A u_1^2 + Bu_2^2 + Cu_1u_2}$
where $u_1$ and $u_2$ are two independent solutions to the ODE
$u'' + F(z)u = 0$
This ...

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127 views

### Symmetry-finding with SAGE?

On pp. 152-3 of Hydon's Symmetry Methods for Differential Equations (2000 ed.), he lists some computer packages for symmetry-finding. This related Mathematica StackExchange question mentions the SYM ...