**0**

votes

**1**answer

23 views

### schauder estimate for the heat equation on compact manifolds

I asked this question on math.stackexchange.com, however I didn't get any answers so I'll try it here.
Let $M$ be a compact manifold without boundary. Consider $Lu:=\partial_tu-\Delta u$. Let $f\in ...

**1**

vote

**1**answer

28 views

### General solution to system of stochastic linear differential equations

Assume we are given the system of linear stochastic differential equations
$$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + \sum_{j=1}^n \sigma_{ij}(t) \cdot x_j \cdot dB_{ij,t} + b_j(t)\cdot ...

**0**

votes

**0**answers

43 views

### Solution of the system of differential equations

Consider that we are working in the polynomial ring $\mathbb{C}[x]$.
Suppose we have the following system of linear differential equations:
$$\left\{\begin{matrix}
L_1(y)=f_1\\
L_2(y)=f_2
...

**78**

votes

**12**answers

16k 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" ...

**0**

votes

**0**answers

59 views

### A PDE problem on the Heat-Like differential equations

I came across the following questions in part of my work:
Consider the Heat-Like equation of the form $\frac{\partial u}{\partial t}=\hat{H}u + f(x,t)u + g(x,t)$ where $\hat{H}$ is a Sturm-Liouville ...

**4**

votes

**2**answers

373 views

### What does it mean for a differential equation “to be integrable”? [duplicate]

What does it mean for a differential equation "to be integrable"?
Are "integrable" and "solvable" synonyms?
The first thing that comes to my mind is to say: it's integrable if we can find the ...

**1**

vote

**0**answers

30 views

### Algebraic invariants of linear ODE's with constant coefficients

Let consider linear ODE with constant coefficients:
$$y^{(n)}(x) + A_{n-1} y^{(n-1)}(x) + ... + A_1 y'(x) + A_0 y(x)= 0.$$
It admits some equivalence point transformations, that preserves its ...

**2**

votes

**0**answers

315 views

### A problem about Joint sine and cosine fourier transform

There is a problem on Page 60 of Book of B.M.Budak, A. A. Samarskii and A. N. Tikhonov, whose title is A Collection of Problems in Mathematical Physics (New York, Dover, 1964). The problem (the ...

**1**

vote

**1**answer

270 views

### On Harmonic Unit Vector Fields

When we restrict the Dirichlet energy functional to the set of all unit vector fields on a compact Riemannian manifold $(M,g)$, then the critical points of this functional are satisfied in $\Delta_g ...

**1**

vote

**1**answer

177 views

### The Biharmonic Eigenvalue Problem with Dirichlet Boundary Conditions

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

**9**

votes

**1**answer

402 views

### A tricky tractrix question about vertical tangents

This is raised by a recent question occurring in combinatorial geometry.
It is about a sort of tractrix, but instead of a line, the pulling end moves along a circle of radius $r>\frac12$ ...

**5**

votes

**1**answer

83 views

### A special function solution of a fourth-order ODE

I want to consider the solutions of the following fourth-order ODE:
$$
f^{(4)}(t)+a tf^{(1)}(t)+b f(t)=0,
\tag{$\ast$}$$
where $a,b$ are complex parameters. It turns out that with a Fourier ...

**1**

vote

**2**answers

529 views

### Inhomogeneous Bernoulli Equation

Does anybody suggest how to face the inhomogeneous Bernoulli differential equation
$y'+P(x)y=Q(x)y^n+f(x)$
for the simple case $f=const.$ and for the generic case.
I would like to know about ...

**1**

vote

**1**answer

49 views

### Holomorphic vector field with infinite separatrix

Let $V=\sum_{i=1}^{n}a_i(z_1,\ldots z_n)\frac{\partial}{\partial z_i}$ be a holomorphic vector field defined on a neighborhood $U\subset \mathbb{C}^n$ of the origin, such that the common zero point of ...

**5**

votes

**3**answers

345 views

### Determining geodesics between two points in curved space [closed]

In order to determine the geodesics between two points, one must solve the geodesic differential equations, which are as following
\begin{align}
p &= u'(s)\\
q &= v'(s)\\
p' + \Gamma^0_{00}p^2 ...

**0**

votes

**1**answer

57 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 ...

**5**

votes

**2**answers

212 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

**1**answer

159 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

96 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 ...

**0**

votes

**0**answers

51 views

### Tau functions for the KP and Toda lattice hierarchies

A statement, which is known to be true can be vaguely stated as "the tau function for the KP and Toda hierarchies are the same". I would just like to know exactly what it means as it is not obvious ...

**3**

votes

**0**answers

87 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 ...

**1**

vote

**0**answers

39 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

258 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, ...

**1**

vote

**0**answers

64 views

### Finding a stochastic differential equation as limit of a discrete stochastic equation

I'm dealing with the following problem:
Choose $Z_0 \in [0,1]$ and define a process governed by the following discrete stochastic equation:
$Z_{k+1}-Z_k=P_k(1-2Z_k)$
where $P_k=0$ with probability ...

**0**

votes

**1**answer

101 views

### Quadratic stability of linear time varying system

(This question was originally asked at Math.SE, where it didn't receive any answers.)
Consider the linear time-varying system
$$ \dot{x} = A(t) x, $$
where $x \in \mathbb{R}^n$ and $A: [0,+\infty) ...

**2**

votes

**1**answer

103 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 ...

**4**

votes

**0**answers

89 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 ...

**1**

vote

**0**answers

21 views

### Solution to Helmholtz equation with non-circular boundary

Let $D$ be an homogeneous 2D domain with non-circular boundary $\partial D$.
I am trying to solve the Helmholtz equation
$$
\nabla^2 u(r, \varphi) + k^2 u(r, \varphi) = - f(r, \varphi)
$$
in which ...

**0**

votes

**0**answers

30 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 ...

**80**

votes

**0**answers

6k views

### Dropping three bodies

Consider the usual three-body problem with Newtonian
$1/r^2$ force between masses. Let the three masses start off at rest,
and not collinear. Then they will become collinear a finite time ...

**0**

votes

**0**answers

37 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} ...

**4**

votes

**1**answer

64 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,
...

**5**

votes

**0**answers

425 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

71 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 ...

**10**

votes

**2**answers

1k views

### Reference for a nice proof of “undetermined coefficients”

I'm teaching an honors differential equations class and have been using linear algebra heavily. I thought it would be interesting to include a proof of the method of undetermined coefficients along ...

**3**

votes

**1**answer

79 views

### Bounds for smallest eigenvalue of Schrödinger equation with a superpotential and periodic boundary conditions

A very specific question, but posted on the off-chance that someone may be able to help. If we have a Schrödinger equation with arbitrary "superpotential"
\begin{equation}
-\frac{d^2 \psi}{dx^2} ...

**2**

votes

**0**answers

212 views

### Multivalued solution of PDE ${v_{xx}v_{yy}-v_{xy}^{2}}={(1+v_{x}^{2}+v_{y}^{2})^2}$

Let's start with a definition:
Definition: A scalar k-th order differential equation on a smooth manifold $M$,
is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $
for ...

**2**

votes

**0**answers

61 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 ...

**7**

votes

**2**answers

130 views

### What are the upper bound and stability conditions for the following simple linear system?

Consider the following linear system
$$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1)
$$
where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...

**7**

votes

**1**answer

321 views

### Uniqueness for a non-local differential equation

Question:Fix $\epsilon>0$. Consider the differential equation, defined for functions $f(t,x)\in C^\infty([0,\epsilon]\times[0,\epsilon])$ defined by
$$\frac{\partial}{\partial t} ...

**1**

vote

**0**answers

36 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. ...

**-1**

votes

**1**answer

85 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 ...

**4**

votes

**1**answer

57 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

43 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 ...

**1**

vote

**0**answers

119 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

30 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

107 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

**1**answer

201 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 ...

**0**

votes

**0**answers

59 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:
...

**12**

votes

**1**answer

119 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 + ...