**0**

votes

**0**answers

59 views

### An innocuous second order linear ODE [on hold]

Is there much work done on equations of the form
$$ y'' + \alpha(t)y = 0,$$
where $\alpha(t) \in C^\infty([0,\infty))$ and $\alpha(t) > 0$? In particular, I am looking for some blow-up results. I ...

**-2**

votes

**0**answers

36 views

### I can't derive the integrating factor of this first order linear Equation [on hold]

I can't derive the integrating factor of this first order linear Equation
(x2 - y2 - y) dx - (x2 - y2 - x) dy = O.
the answer is: integrating factor = 1/(x2 - y2)

**4**

votes

**0**answers

76 views

### Symbol of differential operator and change of variables [on hold]

Recently I posted the following question on stack exchange, but it remained with no answer http://math.stackexchange.com/questions/1863658/symbol-of-differential-operator-and-change-of-coordinates
I ...

**0**

votes

**0**answers

41 views

### Questions about the regularity of the solution of the heat equation in a bounded domain [closed]

I have questions about the proof of the following theorem:
Theorem 8 (Smoothness). Suppose $u\in C^2_1(U_T)$ solves the heat equation in $U_T$. Then $u\in C^\infty(U_T)$
Here is the statement and ...

**3**

votes

**1**answer

60 views

### Gap-opening perturbations of the periodic Schrödinger operator

I am trying to understand this short paper and I am getting stuck right at the end.
Let $V(x)$ be $C^\infty$ and 1-periodic (that is, $V(x)=V(x+1)$).
We are considering the operator
$$A=-\dfrac{d^2}...

**2**

votes

**1**answer

84 views

### Differential equation with inequality constraints

Does there exist a $\gamma > 0$ and functions $g$ and $f$ s.t. we have the following for all $x \in [0, 1]$:
$$
g(x) \in [0, 1] \\
f(x) \in [0, x] \\
x \frac{d}{dx}g(x) - \frac{d}{dx}(g(x)f(x)) = 0 ...

**0**

votes

**0**answers

21 views

### Integral kernel connecting dual differential equation solutions?

Consider a partial differential operator $D_{\{p\}}(x)$ in several variables $x=(x_1,x_2,...,x_n)$ in $\mathbb{R}^n$ and a set of a few parameters $\{p\}$. The solution $F_{\{p\}}(x)$ such that
$$D_{\...

**0**

votes

**1**answer

92 views

### Endless Transformation in finding Particular Solution of Riccati Equation

I have a question regarding "Endless Transformation. I'm actually working on Riccati Equation and I found a Post here on MO (Looking for the solution of first order non-linear differential equation ($...

**0**

votes

**0**answers

35 views

### Adiabatic elimination of a variable in a system of nonlinear stochastic ODEs?

If this is too basic for MathOverflow... say the word and I shall move it to Math.SE
First consider this system of ODEs. Say I have two variables $u$ and $a$, following
$$
\dot u = -u + f(a)
$$
$$
\...

**2**

votes

**0**answers

50 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

81 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 ...

**1**

vote

**1**answer

203 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

20 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_{\...

**1**

vote

**1**answer

42 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

127 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 ...

**5**

votes

**3**answers

284 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

77 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 ...

**0**

votes

**0**answers

64 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\...

**6**

votes

**1**answer

355 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

94 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

**1**answer

41 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 ...

**5**

votes

**1**answer

275 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

249 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 ...

**0**

votes

**0**answers

204 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 ...

**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 ...

**4**

votes

**0**answers

74 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

58 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

48 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

59 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:...

**-4**

votes

**1**answer

117 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

124 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

59 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.
...

**4**

votes

**0**answers

96 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....

**5**

votes

**1**answer

182 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 $.
...

**3**

votes

**1**answer

229 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

**1**answer

57 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

**0**answers

30 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 ...

**2**

votes

**0**answers

44 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

31 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 ...

**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 ...

**4**

votes

**0**answers

239 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 ...

**1**

vote

**0**answers

38 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 ...

**1**

vote

**0**answers

82 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 ...

**3**

votes

**0**answers

69 views

### Deriving Milne's predictor of order four from extrapolation polynomial [closed]

I am trying to derive the following Milne's predictor formula of order four for the differential equation $\frac{dy}{dx}=f(x,y)$ from an extrapolation polynomial of degree four. $$y_{n+1} = y_{n-3}+\...

**36**

votes

**5**answers

1k views

### Undergraduate ODE textbook following Rota

I imagine many people are familiar with the extremely entertaining article "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations" by Gian-Carlo Rota. (If you're not, do ...

**2**

votes

**0**answers

78 views

### How can I prove that this D-module is free?

I have the following setup, I expect that it is studied in the theory of $D$-modules, and I apologize in advance if I am wrong.
First, I have an algebra $A$ of differential operators on $n$ ...

**6**

votes

**0**answers

205 views

### Nonzero solutions to the functional ODE $f'(x)=f(f(x))$

Does $\frac{df}{dx}=f(f(x))$ have nonzero solutions? And if so, what analytic/numerical methods can be used to characterize them?