Questions tagged [differential-equations]

Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.

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2
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0answers
366 views

Lifting a quadratic system to a non-vanishing vector field on $S^{3}$ or $T^{1} S^{2}$

Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non-vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...
24
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2answers
1k views

Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map $D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the differential operator ...
3
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1answer
277 views

A cubic system with two nested limit cycles with opposite orientations

What is an example of polynomial vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ such that two closed orbits $C_1,C_2$ of the system surrounds an annular region $R$ such that $...
138
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14answers
30k 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
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1answer
4k views

Non-uniqueness of solutions of the heat equation

For the heat equation $(\partial_t-\partial_x^2)f(t,x)=0$ defined on $[0,T)\times(-\infty,\infty)$, to obtain uniqueness of the initial value problem, usually it is required to limit the growth of the ...
10
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1answer
868 views

Solution of linear ODE

Let $A=A(t)$ be a smooth one parameter family of $n\times n$-matrices, $n\ge 2$. It seems that the solution of linear ODE $$\dot x= Ax$$ can not be written in a closed form using $\int$, $A$, $x(0)$ ...
5
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1answer
364 views

Fredholm index vs. Limit cycle theory

Let $A$ be the algebra of all smooth functions $f: \mathbb{R}^2 \to \mathbb{R}$ such that $f$ is flat at the origin and is real analytic on $\mathbb{R}^2 \setminus \{0\}$. Let $B $ be ...
14
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5answers
2k views

Are there any techniques for solving a differential equation of the form $f ' (x) = f( f( x ) )$?

I am trying to solve the following differential equation $$f ' (x) = f( f( x ) ),$$ but I have no idea how. I don't think the chain rule is useful for this. Although I don't think this differential ...
6
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0answers
1k views

The Perturbation of Non Hamiltonian algebraic Vector fields

In this question we are interested in the number of limit cycles which appear in the following perturbational system: \begin{equation}\cases{ x'=y -x^{2}+\epsilon P(x,y) \\ y'=-x+\epsilon Q(x,y) } \...
3
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0answers
147 views

Flat Riemannian metrics adapted to quadratic vector fields with center

Assume that $P(x,y),Q(x,y)\in \mathbb{R}[x,y]$ are two polynomials of degree $2$ with $P(0,0)=Q(0,0)=0.$ Suppose that the vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ has a center ...
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0answers
117 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 ...
63
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2answers
13k views

Why is differential Galois theory not widely used?

E.R.Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...
46
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9answers
4k 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 ...
37
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9answers
33k views

Is square of Delta function defined somewhere?

everyone. I am wondering whether anyone knows if the square of Dirac Delta function is defined somewhere. In the beginning, this question might look strange. But by restricting the space of the test ...
25
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1answer
821 views

Solving a delay-differential equation related to epidemiology

For some inexplicable reason, I have recently been interested in epidemiology. One of the classical and simplistic models in epidemiology is the SIR model given by the following system of first-order ...
14
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5answers
2k views

Books on the analysis of hyperbolic partial differential equations

Most of the present books on pde analysis deal with the elliptic partial differential equations. Is there some book related to rigorous analysis with hyperbolic pdes, and especially hyperbolic systems ...
10
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1answer
3k views

Existence/Uniqueness of solutions to quasi-Lipschitz ODEs

Would the Picard–Lindelöf theorem still be true if the requirement that f be Lipschitz continuous in y was replaced with the requirement that f be almost Lipschitz in y? If not, are there any moduli ...
12
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4answers
10k views

When is Sobolev space a subset of the continuous functions?

If we let $\Omega\subset\mathbb{R}^d$ with $d=1,2,3$ and define $\mathcal{H}^1(\Omega)=(w\in L_2(\Omega): \frac{\partial w}{\partial x_i}\in L_2(\Omega), i=1,...,d)$. My tutor has repeated several ...
30
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1answer
2k views

$f'=e^{f^{-1}}$, again

This question is a spin-off of this one, in which the OP asks whether there is a solution $f:\mathbb R\to\mathbb R$ of the functional equation (not exactly an ODE) $f'=e^{f^{-1}}$, where $f^{-1}$ is ...
16
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3answers
1k views

I would like to have a counter example that Peano's theorem does not apply to spaces with infinite dimension

Does Peano's theorem apply to spaces with infinite dimension? Or is there a counterexample? Here, Peano's theorem is: Let $E$ be a space with finite dimension. Consider a point $(t_0,x_0) \in \Re \...
17
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3answers
2k views

How do i solve this : $\displaystyle \ f'=e^{{f}^{-1}}$?

Let $f$ be a function such that :$f:\mathbb{R}\to \mathbb{R}$ and $f^{-1}$ is a compositional inverse of $f$. I would'd like to know how do I solve this class of differential equation : $$\...
11
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3answers
882 views

Vector field with holomorphic flow

Let $(M,J)$ be a complex manifold. Suppose that $X$ is a real vector field such that the flow of $X$ is by biholomorphisms.Question Show the flow of $JX$ is by biholomorphisms. I know one reference ...
9
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2answers
543 views

An algebraic Hamiltonian vector field with a finite number of periodic orbits(1)

Edit: The previous version of this question contained 2 part. In this new version, I deleted the first part and move it to a new question. Is There a polynomial Hamiltonian $H(x,y,z,w)=zP(x,y)+wQ(...
1
vote
1answer
166 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=-...
10
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1answer
737 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 $r>\frac12$ (...
3
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0answers
129 views

Solutions to $w_x=CA_x$, $w_y=CA_y$ other than $w=f(A)$ and $C=f'(A)$?

Let $R \in \{\mathbb{C}[t^2,t^3], \mathbb{Z}[\sqrt{5}]\}$, and let $A=A(x,y) \in R[x,y]$ with $\deg(A) \geq 1$ (total degree). I wish to prove or find a counterexample to the following claim: If ...
2
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1answer
130 views

Second order linear ODE question with boundary conditions

I'm working on a stochastic differential equations research problem and I have come across this second order ODE, my gut tells me it has an analytic or close to analytic solution, but I just can't ...
1
vote
2answers
201 views

Behavior of a non-linear differential equation

Let us consider the following differential equation $$ \dot{x}(t)=a - b\sin(x(t)), \quad a,b\in\mathbb{R}. $$ My question. Suppose $a>|b|$ and $x(0)=x_0\in\mathbb{R}$. Can the solution to the ...
116
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14answers
35k 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 ...
41
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4answers
12k views

Function satisfying $f^{-1} =f'$

How many functions are there which are differentiable on $(0,\infty)$ and that satisfy the relation $f^{-1}=f'$?
18
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9answers
9k views

Graduate ODE textbook

Suppose that a hypothetical math grad student was pretty comfortable with first-year real variables and algebra, and had even studied some other things (algebraic geometry, Riemannian geometry, ...
23
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4answers
4k views

Is there any way to rewrite a partial differential equation using language of differential forms, tensors, etc?

My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there any way (or any algorithm) ...
23
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5answers
4k views

Book Recommendation - PDE's for geometricians / topologists

I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as ...
14
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2answers
2k views

Solvability in differential Galois theory

It is well known that the function $f(x) = e^{-x^2}$ has no elementary anti-derivative. The proof I know goes as follows: Let $F = \mathbb{C}(X)$. Let $F \subseteq E$ be the Picard-Vessiot extension ...
25
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7answers
2k views

Does every ODE comes from something in physics?

Not sure if this is appropriate to Math Overflow, but I think there's some way to make this precise, even if I'm not sure how to do it myself. Say I have a nasty ODE, nonlinear, maybe extremely ...
13
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2answers
4k views

Elliptic regularity for the Neumann problem

I'm trying to understand how to establish regularity for elliptic equations on bounded domains with Neumann data. For simplicity, let's presume we are focusing on $-\Delta u = f$ in $\Omega$ and $\...
10
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3answers
664 views

What happens to Newtonian systems as the mass vanishes?

This question is closely related to another one I asked recently, and may be thought of as a warm-up to that one. Consider $\mathbb R^n$ with its usual metric, and pick a one-form $b$ and a function $...
7
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0answers
1k 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 ...
3
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2answers
9k 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 ...
19
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2answers
2k views

How to prove Lambert's W function is not elementary?

Liouville's theorem gives such a proof for antiderivatives of functions like $e^x/x$ or $e^{x^2}$, and differential Galois theory extends that to Bessel functions, say. But what tools exist for ...
16
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1answer
1k views

What braking strategy is most fuel-efficient?

You notice a stop-light ahead of you and it is currently red. You can't run the red light, so you will have to brake, but braking wastes energy and you want to be as fuel efficient as possible. What ...
10
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1answer
1k views

Solutions of equations characterizing a complex structure

Let $(S^n,g)$ denote the unit $n$-sphere endowed with its induced metric $g$ from its embedding into $\mathbb{R}^{n+1}$. The Levi-Civita connection of $g$ induces a splitting of the tangent bundle of $...
9
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1answer
1k views

Differential Equations Satisfied by Modular Forms

In Verrill's paper preprint here, she has the following theorem which is from a paper of Stiller. It states that Let $\Gamma$ be a discrete subgroup of $SL_{2}(\mathbb{R})$ commensurable with $SL_{...
8
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1answer
934 views

What is exactly the (singularity) confinement property ?

This property seems to be used both in the context of differential equations and several kinds of discrete equation systems or automata. It seems to be related in certain case to the Painlevé ...
8
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2answers
4k 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 ...
7
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1answer
3k views

Skellam distribution: Deep connection between Poisson distributions and Bessel function?

The probability mass function for the Skellam distribution for a count difference $k=n_1-n_2$ from two Poisson-distributed variables with means $\mu_1$ and $\mu_2$ is given by: $$ f(k;\mu_1,\mu_2)= ...
11
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2answers
1k views

Frobenius Theorem for subbundle of low regularity?

Frobenius Theorem says that a subbundle $E$ of the tangent bundle $TM$ of a manifold $M$ is tangent to a foliation if and only if for any two vector fields $X, Y \subset E$ the bracket $[X,Y]\subset E$...
10
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3answers
929 views

Diagonalization via the Toda flow

According to some almost indecipherable notes of a graduate Linear Algebra class, a symmetric matrix $A\in\mathbb R^{n\times n}$ can be diagonalised via the Toda flow. More specifically, if $X=X(t)\in\...
10
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1answer
852 views

Proof of the “Neo-classical Inequality”, a fractional extension of the binomial theorem

I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in $p\geq 1$ and $n\in\mathbb N$: $$\frac{1}{p^2}\sum_{j=0}^n\frac{a^{\frac{j}p}b^{\frac{n-j}p}}{\...
9
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2answers
953 views

What are first eigenfunctions of Laplacian for $CP^n$ with Fubini-Study metric?

I know the round $n$-sphere has $f_i=\cos(dist(e_i, x))$ as the set of first eigenfunctions for $e_i=(0, \cdots, 1, \cdots, 0)\in \mathbb R^{n+1}$. i.e. $\Delta f_i=\lambda_1 f$, where $\lambda_1$ is ...