**2**

votes

**2**answers

4k views

### Looking for the solution of first order non-linear differential equation ($y ′+y^{2}=f(x)$) without knowing a particular solution.

I have been working on Riccati Equation. I have tried many different methods to find a closed form for the solution of first order non-linear differential equation ($y'+y^{2}=f(x)$) without knowing a ...

**2**

votes

**0**answers

44 views

### How do spectrums interact with bi-Lipschitz maps?

If it makes things simple, we can just stick to bi-Lipschitz maps from $S^k \rightarrow \mathbb{R}^d$ (w.r.t geodesic distance on the sphere with the standard round metric and the $2-$norm on the ...

**3**

votes

**1**answer

66 views

### When is a limit cycle generated by a Hamiltonian oval stable?

Consider a real polynomial $H$ of degree $n+1$ in the plane. A closed, connected component of a level curve $H=t$ is denoted by $\gamma(t)$ and called an oval of $H$. Let $\omega$ be a real 1-form ...

**0**

votes

**0**answers

47 views

### is it possible to solve this nonlinear differential equation analytically? [closed]

I have a hard time trying to solve a first order nonlinear equation, I posted it in math.stackexchange but I did not get any answer.
I have the following first order differential equation:
$$y^2\...

**90**

votes

**12**answers

18k views

### What is an integrable system

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" ...

**1**

vote

**1**answer

185 views

### An answer to this system of PDE's

Planning of the question:
Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle
The isotropic almost complex structures $J_{\delta , \sigma}$ were introduced by Aguilar on the ...

**0**

votes

**0**answers

19 views

### Measure of sub level of a torsion energy

Given a domain $\Omega$ (not necessarily open, but bounded. We can take quasi open domain). And let $u_{\Omega}$ be the minimizer of the torsion energy,
$$
\int_{\Omega}|\nabla u |^2\, -\, \int_{\...

**0**

votes

**1**answer

33 views

### Approximate largest eigenvalue of Monodromy matrix

Does anyone know the procedure (or have pseudo code) to approximating the largest eigenvalue of a monodromy matrix? Or even to approximate the monodromy matrix itself?
There is no explicit solution ...

**6**

votes

**1**answer

122 views

### The “Peano phenomenon” for differential equations

Consider the following statement:
If $f:\mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, for the autonomous equation $$x' = f (x)$$
the "Peano phenomenon" can arise only at those values ...

**-1**

votes

**0**answers

27 views

### How to calculate the monodromy matrix of the following ODE system [migrated]

I have the following equation:
$$
\frac{dw}{dt} = (-V(t)+\frac{1}{\lambda}F(t)) w,
$$
with $t>0$ and parameter $\lambda>0$
The matrices $F(t)$ and $V(t)$ take the form
$$
F(t)= \left( \begin{...

**0**

votes

**1**answer

71 views

### Uniqueness of Newton (modulo a constant) series on a compact set

Good morning everybody. My question is as follows:
Let $K$ be a compact subset of $\mathbb R$ and let $v\in C^\infty(K)$.
Consider the finite difference operator $\Delta v(x)\doteq v(x+1)-v(x)$. It ...

**5**

votes

**3**answers

275 views

### Nonlinear ODE: $y'=(1+axy)/(1+bxy)$

Consider the first order nonlinear ODE problem:
$$
y'(x)=\frac{1+ay(x)x}{1+by(x)x}, \quad x>0
$$
where $a, b>0$ are some constants. I would like to know if these kind of equations were ...

**0**

votes

**1**answer

76 views

### Behavior of a Solution of a Nonlinear ODE

In my work, I encountered the following equation:
$$
(a'(x)+1)^2+k^2(x) a^2(x)=1,\;\;k(x)=2 {\mbox{sech}}(x).
$$
I would like to know as much as possible about the solution. More particularly, I would ...

**5**

votes

**1**answer

296 views

### Boundary regularity of Dirichlet Eigenfunction on bounded domains

Consider a bounded, connected and open subset $\Omega\subset \mathbb{R}^d$ and the Dirichlet Laplacian $-\Delta$ acting in $L^2(\Omega)$.
Then we know that the eigenvalues of $-\Delta$ form an ...

**0**

votes

**0**answers

63 views

### existence of an initial-boundary value problem with nonhomogeneous boundary conditions

Let $n\geq 2$ be an integer and $\Omega\in R^n$ be a bounded domain with boundary $\partial\Omega$ . Consider the following IBVP:
$u_t=\Delta u$, for $x\in \Omega$, $t>0$;
$u(x, 0)=f(x), x\in\...

**1**

vote

**1**answer

256 views

### Reference to the Existence and Uniqueness of the PDE system

I've the following Problem on systems of Partial Differential Equations. I have "$ N $" Physical variables. and Finally I form the equation on a bounded domain having regular boundary in $R^d$ ($d=2$...

**6**

votes

**1**answer

347 views

### Techniques to solve a non linear differential equation related to curvature

Many years ago, I considered the following non linear differential equation:
$y=y''.(1+y'^{2})^{-3/2}$
This equation expresses the equality between the value of a given function $y\in C^{2}(R)$ and ...

**2**

votes

**1**answer

82 views

### Smooth dependence on the initial condition of the integral of an ODE

I am considering an ODE $\dot{x}=f(x)$, with $x\in\mathbb{R}^d$ and $d<\infty$. $f$ is a $C^k$ function and I denote by $\Phi_t x$ be the value of the solution at time $t$.
I assume that my ODE ...

**1**

vote

**0**answers

32 views

### Differential inequalities for a strictly diagonal dominant system of linear ODEs

Let $A$ be a real $d\times d$ matrix. The diagonal elements are strictly negative ($a_{ii}<0$) and the off-diagonal elements are non-negative ($a_{ij}\geq 0$ for $i\neq j$). $A$ is strictly column ...

**7**

votes

**0**answers

242 views

### Partial differential equations outside of academia [closed]

I've seen a number of career/jobs questions on mathoverflow before, so I thought I would ask. Please excuse me if this isn't the best place for this specific question. Lately I've been really ...

**5**

votes

**1**answer

258 views

### Algebro-geometric version of {vector fields} $\longleftrightarrow$ {flows} correspondence?

Main Question: What Is the correpondence between flows and vector
fields in algebraic geometry?
Here is a more precise statement could be an answer If it was true (I have no idea it is):
"...

**7**

votes

**0**answers

833 views

### The integral of torsion

I found the following * exercise (exercise *9) in page 407 of the book of do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book ...

**4**

votes

**0**answers

219 views

### Models used for the Zika virus?

I am currently teaching an ordinary differential equations course, and am thinking about doing a module on infectious disease models, e.g. SIR/SIRS. I thought, if possible, it would be nice to ...

**3**

votes

**0**answers

74 views

### Green's Function for a Kernel with Symmetric Fourier Transform $\nabla^2-x^2$

I am trying to find the inverse of the following kernel in 3 dimensions
$$
\nabla^2-x^2,
$$
where,
$$
x^2=\vec{x}.\vec{x}
$$
It seems quit simple and one would think there should already be solutions ...

**0**

votes

**0**answers

201 views

### Isothermal coordinates

Is there an application or interest in studying the isothermal surfaces where the metric is $ds^2=E∗(du^2+dv^2)$ and where $E>0$ is an harmonic function?
I know that this metric is a special kind ...

**4**

votes

**0**answers

71 views

### Periodicity of KdV equation in relation to zero-curvature equation

In most of the resources that I have read, integrable systems described by a PDE posses a zero-curvature equation
$$
\partial_t U - \partial_x V + [U,V] = 0
$$
which gives rise to the monodromy matrix
...

**0**

votes

**0**answers

33 views

### Reparametrisation of a PDE with arclength

Suppose I have the following PDE:
$\frac{\partial y}{\partial t} = \frac{\partial}{\partial x}\left[(1-y)^3y^3\left(\sin \theta + \frac{\partial y}{\partial x}\cos \theta\right) \right]$
I notice ...

**1**

vote

**0**answers

57 views

### Are affine maps (wrt to a connection), which preserve a tensor field, given by a PDE?

Let $(M, \nabla)$ be a manifold together with a connection on $TM$ and let $T$ be a tensor field on $M$. Suppose the pseudogroup $\Gamma$ of locally defined smooth maps, that simultanously preserve ...

**1**

vote

**1**answer

46 views

### Is this non-linear system of differential equations tractable by other means than numeric approximation and dynamic analysis?

Is there any way to solve the following system of non-linear differential equations exactly?
$x'(t) = x\times(y - \frac{1}{3(t + C)})$
$y'(t) = -\frac{1}{3}x^2 - \frac{y}{t + C}$
Here $x$ and $y$ ...

**0**

votes

**0**answers

58 views

### Linearization of specific plane vector field

I have a vector field $v = (f(x,y), \alpha y)$, such that $f(0, 0) = 0$ and $df (0, 0) = (1, 0)$. When is smooth linearization possible and when is it not?
I only see obstacles in the form like this:...

**2**

votes

**0**answers

100 views

### Are singular critical points isolated for control systems on compact semisimple Lie groups

Given a control system on $\mathrm{SU}(n)$ (or any other compact, semi-simple Lie group I suspect) of the form:
$\frac{d U_t}{dt} = (A + w(t)B)U_t$
where $A,B \in \mathfrak{su}(n)$ generate the ...

**-4**

votes

**1**answer

115 views

### Existence and uniqueness of solutions for a system of first order PDEs [closed]

Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs:
A$\dfrac{\partial}{\partial t}\pmb{v}(t,x)=B(t,x,\pmb{...

**0**

votes

**1**answer

68 views

### Equivalent ways to study a semilinear parabolic equation as a perturbed abstract Cauchy problem

I am considering the following abstract Cauchy problem on Banach space $X$:
\begin{cases}
u'(t)=Au(t)+\big(f(t)+Bu(t)\big),&t\in[0,T],\\
u(0)=x_0,
\end{cases}
Suppose $A$ generates an analyitc ...

**1**

vote

**1**answer

120 views

### Is there an entire solution for the Van der pol equation?

Is there a non constant entire function $\gamma(t)=(x(t),y(t)): \mathbb{C} \to \mathbb{C}^{2}$ which satisfy the following Vander pol dififferential equation?
$$\begin{cases}\dot{x}=y-x^{3}\\\dot y=-...

**1**

vote

**0**answers

58 views

### Dynamics of pairwise distances in the $n$-body problem

Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the $n$-body problem, so I decided to cross post here as well.
...

**5**

votes

**1**answer

181 views

### Injectivity/Surjectivity of $F_A :=\frac{d}{dt} +A(t) $ for a hyperbolic path of matrices $A(t)$ on $H^1 $

I am looking for a reference to the following problem:
Given a hyperbolic (no purely imaginary eigenvalues), continuous path of matrices $A(t)$ in $\mathbb{R}$ with hyperolic limits at $\pm \infty $.
...

**4**

votes

**0**answers

92 views

### Connection between cardiac equations and untangling knots?

I was surprised to learn that there is (conjecturally) a connection between a cardiac muscle model known as the FitzHugh-Nagumo equations, and untangling knots:
Maucher, Fabian, and Paul Sutcliffe....

**3**

votes

**1**answer

211 views

### How to learn concepts of Functional Analysis which are common in PDE

I am a master student and working in PDE area. I am trying to gain deep understanding of some of the concepts in functional analysis which are common tools in PDE research, such as weak*-topology, ...

**1**

vote

**2**answers

921 views

### Solution of Heat equation with Neumann BC in an arbitrary domain

Consider the heat equation $u_t=\Delta u$ with Neumann boundary condition and initial condition $u(x,0)=u^0(x)$ in a bounded domain $\Omega$ with smooth boundary.
Is this true:
Any solution $u(x,t)\...

**1**

vote

**1**answer

56 views

### Analytic conjugacy of vanishing holonomy groups implies analytic conjugacy of foliations

i am reading and trying to do some exercises and problems of the book Lectures on Analytic Differential Equations- Y. Ilyashenko, S. Yakovenko.
I can not solve the problem 11.6 that says
Consider ...

**-1**

votes

**1**answer

84 views

### PDE with harmonic function [closed]

I'm looking for the solution of the following equation
$$2cE^3=E_u^2+ E_v^2$$ where $c$ is a constant and $E$ is an harmonic function w.r. to the variables $u$ and $v$.

**0**

votes

**1**answer

54 views

### What is the relationship between solutions for the parameterised second order differential equations

Let us consider the following parameterised complex-valued second order differential equations, and $u(x,\lambda)$ be the solution for
$$
u''+u'-i\lambda V(x)u=0, \, x\in [0,1],
$$
What is the ...

**0**

votes

**0**answers

28 views

### Uniqueness and Properties of Nonlinear 2nd Order ODE with Asymptotically Constant Coefficients

I have the following differential equation:
$$ V(z) = e^z [1-\varphi(\gamma)]+(\gamma-g)V'(z)+\frac{1}{2}\kappa^2 V''(z)$$ with
$$ e^z \varphi'(\gamma) = V'(z)$$ where $\varphi(\cdot)$ is a ...

**4**

votes

**0**answers

256 views

### Scattering for rapidly decaying solutions of NLS

Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions
of the Nonlinear Schrödinger Equation" the following property.
Given the problem
\begin{equation}
\left\{
\begin{array}{rl}
...

**2**

votes

**1**answer

139 views

### Differential equations → predicate logic mapping

I was watching Tom LaGatta's talk on category theory recently, and one of the audience brought up a statement that caught my attention (around the 56:15 mark):
I was gonna say, there was a book I ...

**2**

votes

**0**answers

42 views

### 1D inhomogeneous linear Schrodinger equation

I have the following problem:
$iu_t - u_{xx} = f$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=0$. I can show that $\|u\|_{L^2(x,t)}$ (space-time) is controlled by the norm $\|f\|_{L^2(...

**2**

votes

**0**answers

28 views

### A question about Carleman linearization

Carleman linearization is a technique used to embed a finite dimensional system of analytic ordinary differential equations into an infinite system of linear differential equations:¹⁻²
Let $f$ be ...

**1**

vote

**0**answers

37 views

### Monotonocity of non-trivial solution of ODE [closed]

Consider the differential equation $$y^{''}+q(x)y=0,q(x)<0$$ where $q(x)$ is a continuous function. Let $y$ be a non-trivial solution of ODE. How to prove that $y,y^{'}$ are strictly monotone ...

**8**

votes

**1**answer

6k views

### A particularly hard nonlinear first order ordinary differential equation

I hope this is an appropriate place to post this question, however, I am stuck on solving an apparently simple ODE.
I have checked numerous texts, references, software packages and colleagues before ...

**1**

vote

**0**answers

80 views

### A specific type of first-order nonlinear ordinary differential equation

I am trying to divide $\mathbb{R}^+\times \mathbb{R}^+$ into some curves so that the integration of the function $ h(x)h(y) $(where $h(x)$ is a $C^1$ function from $\mathbb{R}^+\to \mathbb{R}^+$ that ...