# Tagged Questions

**10**

votes

**2**answers

320 views

### ODE properties true in finite dimension but not in Banach spaces of infinite dimension

Some properties of Ordinary Differential Equations - ODE are true in finite dimension spaces but not in Banach spaces of infinite dimension.
The first one I know is the Peano existence theorem. I ...

**1**

vote

**0**answers

25 views

### Frobenius series of Fuchsian PDEs

I'm interested in the analyticity of Frobenius-like series solutions to a PDE in $z=(z_1,\ldots ,z_N)\in\mathbb{C}^N$ with regular singular behavior at $z_\alpha=0$ for all $\alpha=1,\ldots, N$.
For ...

**14**

votes

**0**answers

545 views

+200

### Unifying (& “justifying”) the various definitions for differential operators

Reading about differential operators in different sources I've picked up several definitions which are not obviously equivalent (to me). Here they are:
Definition 1 ("naive"): Let $X$ be a (real) ...

**6**

votes

**1**answer

264 views

### Solutions of equations characterizing a complex structure

Let $(S^n(1),g)$ be the round sphere and $J_{\delta , \beta}$ be an almost complex structure on $TS^n(1)$ with the definition
\begin{equation}
J_{\delta , \beta}(X^h)=\beta X^h + \alpha X^v,\\
...

**2**

votes

**0**answers

75 views

### 2nd order partial differential equation with non-constant coefficients

During my research I came across the following differential equation:
$$f(x,y) = \left(y^2 \partial_{x}^{2} + x^2 \partial_{y}^{2} \right)f(x,y)$$
Any ideas how to solve it without using series ...

**0**

votes

**0**answers

9 views

### Understanding the steps taken in a calculation of the maximum profile likelihood of a simple ODE, given some data [on hold]

I'm trying to understand a calculation made in a paper (section 2 from the supplementary contents of ...

**-2**

votes

**0**answers

8 views

### A specific question regarding a proof in Hassan Khalil's book, Nonlinear Systems [migrated]

I am trying to understand the proof of a Lemma in the book 'Nonlinear Systems' by Hasaan Khalil (3rd edition). In the Proof of Lemma 3.1, about Lipschitz continuity of vector valued functions, I am ...

**-1**

votes

**0**answers

71 views

### A simple question about ordinary diffential equations of first order [closed]

An ODE (Ordinary Differential Equation) of order $n$ becomes a relation:
$$ F(x,y,y',...,y^{(n)})=0$$
Then $F(x,y,y^{(1)})=0$ defines a an ODE of order one. In "basic standard texts", for purposes ...

**0**

votes

**1**answer

55 views

### Trying to solve this non-linear differential equation

I have a second order differential equation given by:
$x''(t) = \frac{exp(-\frac{x(t)^2}{4t})}{A \sqrt{t}}$
I would very much like to be able to obtain an analytic solution to this equation, which ...

**0**

votes

**0**answers

55 views

### Is the Rossler attractor globally stable?

The standard form of the Rossler system is $\frac{dx}{dt}=-y-z$, $\frac{dy}{dt}=-x+ay$, $\frac{dz}{dt}=b+z(x-c)$. For simplicity, consider the well known chaotic attractor that exists at a=b=0.2, ...

**0**

votes

**0**answers

22 views

### Is there a “natural” way to interpolate between a set of bound state wave functions?

Consider for example the Coulomb potential, $-Z/r$, for which there exist a set of bound states with energy $\epsilon_n := {-Z^2 \over 2 n^2}$ (in Hartree). If I want the "wavefunctions" for some ...

**2**

votes

**1**answer

160 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 ...

**29**

votes

**5**answers

3k views

### Differentiable functions with discontinuous derivatives

For years I've taught my honors calculus students about functions like (the continuous extension of) $x^2 \sin 1/x$, and for just as many years I've told them that they won't encounter functions like ...

**1**

vote

**1**answer

227 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

**1**answer

91 views

### Is there any solution for this PDE system?

Let $(\mathbb{R}^2,\langle .,.\rangle)$ be the Euclidean space and define the almost complex structure $J_{\delta,\beta}:TT\mathbb{R}^2\longrightarrow TT\mathbb{R}^2$ with
\begin{align}
...

**0**

votes

**0**answers

33 views

### Sufficient condition for a differential inclusion to have a global attractor

Let $\dot{x}(t) \in f(x(t))$ be a differential inclusion (d.i.). Is there any sufficient condition under which it will have a singleton global attractor ?
We know the condition if $f$ is single ...

**0**

votes

**0**answers

22 views

### numerical differentiation of sum of one-dimensional sinusoids with angular frequency close to Nyquist one

Suppose that $f(t) = \sum_i C_i e^{i\omega_i t}$, and $f$ is sampled at certain sampling angular frequency $\omega_s$. All $\omega_i$s are very close to $\omega_s/2$, and thus standard finite ...

**3**

votes

**0**answers

82 views

### Weak form of the Laplace-Beltrami operator on closed manifolds

Suppose we are dealing with diffusion over a boundaryless manifold $M$ (for simplicity let's say it's a surface). In that case, we have
\begin{align}
\int_M W \Delta U \mathrm{d} x & = -\int_M ...

**69**

votes

**13**answers

17k views

### why study Lie algebras?

I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics ...

**0**

votes

**1**answer

152 views

### An example for affine function [closed]

I'm looking for an example of a non-Euclidean non-compact Riemannian manifold $(M,g)$ such that we could define a non-constant affine function $f:M\rightarrow \mathbb{R}$, namely its gradient vector ...

**2**

votes

**2**answers

132 views

### Backgrounds of the p-Laplacian Operator

Motivation
I encountered the following partial differential equation (PDE) in a mathematical paper
$$\begin{array}{}
u_{tt}+\Delta^2u-\nabla\cdot\left(|\nabla u|^{p-2}\nabla u\right)-\Delta ...

**5**

votes

**1**answer

2k views

### Examples of separable ordinary differential equations in economics

I'm currently teaching an integral calculus course for business students, and we're just about to discuss differential equations. They've worked hard, and I'd like to reward them with some economic ...

**2**

votes

**0**answers

44 views

### Are singular critical points isolated for control systems on comapct semi-simple Lie groups

Given a control system on $SU(n)$ (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 algebra (to ...

**7**

votes

**2**answers

128 views

### Existance of Integrating Factors, a Constructive Proof

Being a novice with differential equations, I recently learned that if $\mu$ is an integrating factor for $\frac{dy}{dx}f(x,y)+ g(x,y)=0$, then the corresponding 1-form, $\mu fdy+\mu g dx$, is exact.
...

**1**

vote

**1**answer

77 views

### Exact solution to nonlinear differential equation sought

I am looking for an exact solution to equation:
$w''=aw^{-1/3}+b[f(y)]w^{-5/3}$, where $w=w(y), f(y)$ - arbitrary function (in this case $y^n$ with arbitrary $n$); $a,b$ - constants.
Of course I can ...

**6**

votes

**0**answers

80 views

### How to eliminate secular terms for perturbed non-oscillatory equations?

Even in a linear second order equation like $x''+x'+\epsilon x=0$ the standard asymptotic expansion has a secular term already in the first order of $\epsilon$, namely
...

**0**

votes

**0**answers

47 views

### solution of a mixed difference and differential equation

Is it possible to solve the following difference and differential equation:
$a\frac{d f(x)}{d x} = \frac{d f(2x)}{d x}$,
where $a<1$ and $\int_0^\infty f(x)=1$.

**1**

vote

**0**answers

64 views

### Finite blowup time for a simple ODE

Consider the ODE
$$ y'(x) = C y(x)^{2-x},$$
where $C$ is a positive constant. I suspect there is no closed-form solution. I want to understand the constant $C$ such that for a given $y(0) > ...

**2**

votes

**0**answers

61 views

### Is the Wave Function a “Smooth” Function of the Potential?

Consider the Schroedinger equation in a spherically symmetric system. In the unit system under which energy is measured in Hartree and length in Bohr radius $a_0$, the schroedinger equation can be ...

**0**

votes

**1**answer

54 views

### Stabilize the vector field of $y' = f (y) - \gamma H^T(HH^T)^{-1}h( y ) $ of ODE $y' = f(y)$

This question has been asked here but there is no answer:
http://math.stackexchange.com/questions/1585400/stabilize-the-vector-field-of-y-f-y-hthht-1h-y-of-ode-y
Consider autonomous ODE $y' = ...

**4**

votes

**1**answer

421 views

### conservation law and generalized Symplectic Monge-Ampere equation arising from 3-variables

If we have a Jacobi PDE system with conservation law $\theta \in \Omega^1(M)$ such that $d \theta$ is non-degenerate 2-form , then we know this fact that it can be written as symplectic 2D ...

**0**

votes

**1**answer

56 views

### Ordinary homogeneous differential equation [closed]

How to solve this one
$y''=(2xy - \frac{5}{x})y' + 4y^2 - \frac{4y}{x^2}$
I know it's homogeneous. I've made replacement $x = e^t$ and $y = ze^{-2t}$ but I had no result.

**1**

vote

**2**answers

70 views

### Index Reduction of Differential Algebraic Equations by Hand

I dont really understand how to reduce the index of DAEs ?
Does Reducing the index of DAE result in an ODE ?
How would I reduce the index of the DAE by Hand ?
Say I have :
$$
\begin{matrix}
...

**0**

votes

**1**answer

94 views

### gradient descent in space of functions

Differential equations of the form
$$\frac{d}{dt}\vec{x} = - \nabla E(\vec{x})$$
can be analyzed using phase portrait method. In particular, if the function $E$ (we call it energy) has local minimums, ...

**3**

votes

**0**answers

99 views

### Does there exist duality/symmetry between Fuchsian differential equations, like in the case of confluent hypergeometric?

My question is about whether certain dualities or symmetries hold between pairs of Fuchsian differential equations, but I need to give a bit of background to explain what I want to ask. Thank you for ...

**4**

votes

**1**answer

152 views

### Are all bidimensional second-order PDE at most quadratic in the top derivatives of Monge-Ampère type?

The general Monge-Ampère equation in $n$ independent variables is a quasi-linear combination of all the possible minors of the $n\times n$ Hessian matrix
$$
\left\|\frac{\partial^2u}{\partial ...

**3**

votes

**2**answers

127 views

### A Global Estimates for Linear Elliptic PDE

Let $\Omega$ be a bounded smooth region in $R^n$ and $u$ satisfy
$-\Delta u+a(x)u=f, \ \ u|_{\partial \Omega}=0$,
where $a(x)\geq 0$ and $f(x)$ are smooth functions. I wonder if the following ...

**84**

votes

**12**answers

17k 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" ...

**9**

votes

**3**answers

1k views

### A simple example where elliptic boundary regularity fails due to a kink in the domain

I'm seeking a simple example of where elliptic (preferably linear) boundary regularity fails due to a simple kink in the domain.
So far my gueses were to look at $-\Delta u = f$ on $[0,2\pi] \times ...

**1**

vote

**1**answer

139 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 ...

**4**

votes

**2**answers

455 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

38 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

343 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

306 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 ...

**10**

votes

**1**answer

450 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

105 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

567 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

60 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

431 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

74 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 ...