**0**

votes

**1**answer

100 views

### Frobenius method for multiple singular points

As we know, if the equation
$$a(x)y''+b(x)y'+c(x)=0 \ \ \ \ \ \ \ \ \ (1)$$
has a regular singular point at $x=x_0$ then we seek solution of the equation as
$$y(x)=\sum_{n=0}^{\infty}\beta_n (x-x_0)...

**3**

votes

**1**answer

210 views

### stability of the Monge-Ampère equation

Is there any hope to prove this conjecture (or a similar one)?
Conjecture Let $\Omega_k$ be a family of convex (smooth) domains, and let $u_k$ be convex Alexandrov solution of $$ \begin{cases}
...

**2**

votes

**0**answers

103 views

### Modifying monkey saddles

We get negative Gauss curvature K surfaces of quasi polar symmetry surface form generated from:
$$ Re (x+ i y)^n = a^n $$ (n integer) with n humps above plane $ z =0$.
($ n =2,3,4 $ hyperbolic ...

**5**

votes

**2**answers

225 views

### Noninvariance for a specific nonlinear oscillator

Consider the nonlinear system
\begin{align*}
\frac{d}{dt} \begin{pmatrix} x_1(t) \\ x_2(t) \end{pmatrix} = \begin{pmatrix} x_2(t) \\ -4x_1(t) + x_1^2(t) \end{pmatrix},
\end{align*}
which admits ...

**3**

votes

**0**answers

101 views

### Equations of the form $((\Delta)^{2}+\lambda\Delta+\gamma)(f)=0$

Here is the problem: I am trying to figure out if there is a non-zero solution for the system of equations:
$(\Delta^{k-1}+a)(d^{\ast}\eta)=2d^{\ast}d\xi$
$(\Delta^{k}+b)(d\xi)=2dd^{\ast}\eta$
for $...

**2**

votes

**0**answers

49 views

### Attainability of Global Optima In Optimal Control

Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation:
$\frac{d x(t)}{dt} = F(x,u)$
one can consider the ...

**6**

votes

**3**answers

318 views

### Characteristic Variety of the Principal Symbol solves PDE system?

In the study of partial differential equations, it is often considered enough to analyze the principal symbols and their characteristic variety (see for example, http://www.sciencedirect.com/science/...

**1**

vote

**1**answer

267 views

### The Biharmonic Eigenvalue Problem with Dirichlet Boundary Conditions on a Rectangle

I am interested in solving the following biharmonic eigenvalue problem.
$$\begin{array}{cccc}
& \Delta ^2 \Psi (x,y) = \lambda \Psi (x,y), & - a \le x \le a & - b \le y \le b \\
&...

**0**

votes

**0**answers

43 views

### The Solution to the system of linear PDEs

I am looking for the solution to the following system:
$$ f_t(t,x) = -tx g(t,x), g_t(t,x) = (1-t)x f(t,x). $$
The equation comes from the integral equation
$$ f(t,x)=1+ x \int_{0}^{1-t} (1-s)f(s,x)ds,...

**5**

votes

**0**answers

95 views

### A finiteness question for integrable polynomial distributions on $\mathbb{R}^3$

This question is motivated by the finitness of limit cycles for polynomial vector fields on $\mathbb{R}^2$
Assume that $X,Y$ are two independent polynomial vector fields on $\mathbb{R}^{3}$ such ...

**4**

votes

**1**answer

71 views

### Getting out a system of linear ODEs by knowing the Magnus expansion

Assume we are given for a transition between two time points $t_0 = 0$ and $t_1$ a matrix relationship, eventually describing the solution of a system of linear with non-constant coefficients,
$$Y(t_1)...

**2**

votes

**0**answers

87 views

### What is known about topological equivalence of polynomial dynamical systems on two different domains in R^n?

The question is mainly about $\it flows$, not maps (i.e., continuous time, not discrete time).
Is it known if the study of polynomial dynamical systems on $\mathbb R^n$ can be reduced to the study ...

**2**

votes

**1**answer

129 views

### A question about viscosity solutions

Let $A:=[a,b]$ be a closed interval in $\mathbb{R}$. Let $F(x,p,q,r)$ be a function from $[a,b]\times \mathbb{R} \times \mathbb{R} \times\mathbb{R} \rightarrow \mathbb{R}$ describing a second order ...

**5**

votes

**0**answers

493 views

### Grothendieck problem

Could you suggest me a book or a link where I can find some information about the Grothendieck problem about differential equations?
The Grothendieck problem that I am reffering to is the following: ...

**2**

votes

**0**answers

85 views

### Green's functions on linear subspaces and relations to boundary conditions

Consider the Laplacian $-\Delta$ on (in a suitable sense) twice differentiable functions subject to homogeneous Dirichlet boundary conditions $\mathscr{H}=\{f : f(0)=f(1)=0\}$. We can identify the ...

**1**

vote

**0**answers

75 views

### Criterion for existence of solution to nonlinear second order ODE

Good day. I am asking this question as an issue of concern for determining whether an ordinary differential equation can be solved. Existence, and not uniqueness of solution, is what is required. ...

**4**

votes

**1**answer

63 views

### Problem on differential inclusion

For a differential inclusion $x'(t)\in h(x(t))$, is there any condition (of course, I don't want the map to be single-valued) under which we can say that for any trajectory $x(.)$ satisfying the ...

**2**

votes

**0**answers

62 views

### The motivation of Weyl-Titchmarsh function

Given a second linear differential operator,
$(Hf)(x)=-\frac{d^2}{dx^2}(x)+V(x)f(x)$,
where $V$ is a bounded and
real valued function, $f$ lies in $L^2(\mathbb{R})$.
For an $z$ with $Im(z)\neq 0$,...

**1**

vote

**0**answers

134 views

### Green's function on sphere

Consider radial (normal) coordinates on a sphere $S^n, n \geq 2$. Let the "origin" be the north pole $(0, 0,..., 1)$ and the coordinates be denoted by $(r, \theta)$. We know that the Laplacian $\...

**1**

vote

**0**answers

31 views

### Question about a specific differential equation [closed]

I have the following differential equation
(1+e^x+xe^xy)dx + (xe^x+2)dy=0
I need to check whether it is full and to find its general solution.
Can you help me ?

**1**

vote

**1**answer

115 views

### Solving Schroedinger Equation for the electronic energies of the Molecular Ion Hydrogen H2+ in the Elliptic coordinate system [closed]

Electronic Energies of Molecular Ion Hydrogen $H_2^{+}$
$r_1$ is the distance between the proton $1$ and the electron.
$r_2$ is the distance between the proton $2$ and the electron.
$R$ is the ...

**2**

votes

**0**answers

91 views

### Homogeneous regular manifolds

In order to solve the well-known Plateau-Problem on a general (non-compact) Riemannian 3-manifold, Morrey first introduced the condition of homogeneous regularity and defined it in the following way:
...

**2**

votes

**1**answer

207 views

### Simultaneous integral equation on $SU(n)$

Consider a smooth curve $U_s:[0,T] \rightarrow SU(4)$ which solves:
$\frac{d U_s}{ds} = (a + w(s)b)U_s$
for some given $a,b \in \mathfrak{su}(4)$ (which generate $\mathfrak{su}(n)$) and a smooth ...

**2**

votes

**1**answer

184 views

### Second order estimates of Monge-Ampere equations

In order to prove existence of solutions of real and complex Monge-Ampere equations in various modifications (e.g. as in the Calabi problem) one often uses the method of a priori estimates. One of the ...

**1**

vote

**0**answers

92 views

### Closed form answer to a naive integral [closed]

Let a and b be positive real numbers. How to find a closed form answer to the integral
$$\int_0^t \left(-a t + \big(1+ \dfrac{2bt}{3}\big)^{-3/2}\right)^{5/3} dt$$
If it is not possible to find a ...

**15**

votes

**1**answer

137 views

### Commuting ODE's implies existence of nonzero vanishing two variable polynomial?

Write $\partial := d/dt$, fix $m, n > 0$, and let$$F = \partial^n + f_1(t)\partial^{n-1} + \dots + f_{n-1}\partial + f_0,\text{ }G:= \partial^m + g_1(t)\partial^{m-1} + \dots + g_{m-1}\partial + ...

**1**

vote

**0**answers

94 views

### Comparing Dirichlet energy and area of a Surface-immersion

Let $(F,g)$ be a closed Surface, $(M,h)$ a Riemannian 3-Manifold and $f: F \to M$ a smooth immersion. Denote by $f^*(h)$ the pullback metric on $TF$ induced by $f$ and let $dV_g$ and $dV_{f^*(h)}$ be ...

**0**

votes

**0**answers

83 views

### Singular integral equation

Investigating a control problem for heat equation I stacked on solution of this integral equation which seems to be singular:
$$
\int_0^1\mathcal{K}(\tau)u(\tau)d\tau=\mathcal{M},
$$
in which $\...

**0**

votes

**1**answer

163 views

### How to prove that a non-linear differential equation has a solution

I want to prove that there exists $f:[0,1] \to [0,1]$ such that $f(0)=0$,
$$
\frac{d w(y-f(y))}{d y} = g(y) \frac{d v(f(y))}{d y}, \forall y \in [0,1],
$$
where $w:[0,1] \to [0,1]$ and $v:[0,1] \to [0,...

**0**

votes

**0**answers

102 views

### What is the Beltrami differential?

Let $R,S$ be Riemann surfaces and $f: R \to S$ an orientation preserving diffeomorphism. Then $f$ determines what is called a Beltrami differential denoted by $\mu \frac{d\bar{z}}{dz}$.
Local ...

**3**

votes

**2**answers

122 views

### Nonlinear ODE system: stability

I've got this 4x4 system that should model the wine fermentation process. All the $\mu, K_N, k_d$ etc are positive constants. Of course I have no idea of how to solve it. But at least I would like to ...

**0**

votes

**1**answer

87 views

### Fit a system of linear ODEs from several experiments

Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n=6$ (in any event a number below 10) , and the number of initial vectors $r$ is in the ...

**0**

votes

**0**answers

34 views

### Explicit solution for a variational inequality?

Assume $\eta_t \in \mathbb{R}$ is a given continuous function depending on time.
It is known that if we look for a continuous solution $\zeta_t \in \mathbb{R}$ depending on time of the following ...

**3**

votes

**1**answer

71 views

### Examples of systems with stable equilibria at the boundary of the phase space

Hopfield networks are gradient dynamical systems, used (among other things) to solve combinatorial optimization problems, because stable equilibria are at vertices of the hypercube $[-1,1]^n$. They ...

**2**

votes

**0**answers

70 views

### Point Spectrum of a Second Order System of Differential Equations

Consider the following operator acting on $H^1(\mathbb{R})$
$$
\mathcal{L} \left(\begin{array}{c} \phi \\ \psi \end{array}\right) = -\left(\begin{array}{c} \phi \\ \psi \end{array}\right)'' + V(x) \...

**3**

votes

**4**answers

477 views

### Solution of second order differential equation with singularities at 0,1, and ∞

I am trying to solve the following equation;
$$
U''+\left( \frac{1}{t}+\frac{3}{t-1}\right)U'+\left(\frac{1}{t}+C\right)\frac{U}{t(t-1)}=0
$$
where U is a function of t and C is constant.
The above ...

**1**

vote

**0**answers

60 views

### asymptotic behavior of the solution of an ordinary differential equation

I am a civil engineer with basic mathematics skills and need help for the following - perhaps simple - problem.
Consider the following autonomous system of two non-linear ordinary differential ...

**2**

votes

**1**answer

79 views

### Coupled differential equations

I'm looking at the following coupled set of differential equations. Because of the symmetry, I'm hoping to be able to write down the solution for $x_n(t)$ and $y_n(t)$ in terms of $f(t)$ and $g(t)$, ...

**1**

vote

**1**answer

80 views

### Runge-Kutta convergence [closed]

I am facing a problem solving a ODE with a Runge-Kutta 4th order method:
The expression in order to solve is :
\begin{equation}
Ay^{''}+By^{'}+Cy= Cu
\end{equation}
\begin{equation}
y =OUTPUT
\end{...

**1**

vote

**1**answer

59 views

### Distribution of Poles of solutions to the first Painleve equation

In the introduction of this paper the distribution of the poles of solutions to the first Painlevé equation is discussed.
In particular it is said that the poles form a deformed lattice that ...

**6**

votes

**1**answer

149 views

### Writing the quotient of solutions of linear ODEs as the solution of a nonlinear ODE

Let $A(x)$ and $B(x)$ be solutions to homogeneous linear ODEs with polynomial coefficients, i.e., $A(x)$ satisfies
$$p_KA^{(K)} + p_{K-1}A^{(K-1)} + \cdots + p_1A' + p_0A=0$$
and $B(x)$ satisfies
$$...

**2**

votes

**1**answer

186 views

### Partial differential equation from Kirchhoff system

I was seeking for the solution of following partial differential equation for two unknowns $\vec{u}(s,t), \vec{w}(s,t)$
$$\partial_t \vec{u} = \partial_s \vec{w} - [\vec{w} \times \vec{u}].$$
Using ...

**2**

votes

**1**answer

322 views

### Hilbert 16th problem, distribution of Limit cycles

Edit: Can one help for translation of the link in Russian(comment by Dimitry Todorov)(Or at least a summary of it)?
It seems that the second part of the Hilbert 16th problem is solved or is going to ...

**5**

votes

**3**answers

663 views

### Make mathematical sense of the Dirac well Potential Equation

A classical problem in quantum mechanics involving the Dirac Delta function is given by
$$
y''+(\delta(x)-\lambda^2)y=0.
$$
Then, to find ''bound states'', you solve on the right and find the ...

**0**

votes

**2**answers

108 views

### Class of analytically-integrable divergence-free vector fields?

Is there an "interesting" class of analytically-integrable, divergence-free vector fields over $\mathbb{R}^2$ and/or $\mathbb{R}^3$?
That is, I'm looking for a large class of vector fields given by $...

**0**

votes

**1**answer

95 views

### Method of characteristics [closed]

I have difficulties understanding how to solve a PDE in $\mathbb{R}^{4}$ using the method of characteristics. I have a limited background in solving PDEs. I have seen only 2-dim examples and none for ...

**0**

votes

**0**answers

114 views

### The complex leaves containing real limit cycles of Lienard equation

According to the answer and comments to this question we realize that a useful approach to such type of questions is to consider algebraic vector field which possess algebraic solutions. On the ...

**1**

vote

**1**answer

234 views

### Two limit cycles which lie on the same leaf

Edit 1: For a related discussion see this MSE post
I apologize in advance, if this question is obvious:
1)What is an example of a polynomial vector field on $\mathbb{R}^{2}$ with at least two ...

**1**

vote

**0**answers

87 views

### An algebraics Hamiltonian vector field with a finite number of periodic orbits(2)

Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different ...

**3**

votes

**2**answers

123 views

### Given $x$ in a path-connected open set $S$ on the plane, are there non-crossing paths from $x$ to every point in $\partial S$?

A have a (non-simply) bounded path-connected open set $S\subset\mathbb{R}^2$.
Given $x\in S$, there are paths in $S$ from $x$ to any point in the boundary $\partial S$.
However, can all these paths ...