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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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.
Lay's user avatar
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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} ...
Grunwalski's user avatar
4 votes
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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 x^i\...
Giovanni Moreno's user avatar
1 vote
1 answer
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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, ...
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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 ...
Matchmaticians's user avatar
1 vote
1 answer
585 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 C^...
alerte's user avatar
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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 dt+\...
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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 ...
Simone Gaiarin's user avatar
2 votes
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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 X=...
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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 ...
Bazin's user avatar
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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 $r>\frac12$ (...
Wolfgang's user avatar
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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 ...
Higgs-Boson's user avatar
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3 answers
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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 ...
imranal's user avatar
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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)...
Paramore's user avatar
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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} ...
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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 ...
Narasimham's user avatar
5 votes
2 answers
257 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 ...
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3 votes
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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 $...
Coffee's user avatar
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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 ...
Benjamin's user avatar
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8 votes
3 answers
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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/...
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The Biharmonic Eigenvalue Problem on a Rectangle with Dirichlet Boundary Conditions

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 \\ &...
Hosein Rahnama's user avatar
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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,...
hkju's user avatar
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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 ...
Ali Taghavi's user avatar
5 votes
1 answer
177 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)...
tobias's user avatar
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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 of ...
DC47's user avatar
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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 ...
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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: ...
Mary Star's user avatar
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2 votes
0 answers
141 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 ...
Lars Lau Raket's user avatar
1 vote
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205 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. ...
Uche Opara's user avatar
4 votes
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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 ...
Sosha's user avatar
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0 answers
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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 $Im(z)\neq 0$,...
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4 votes
3 answers
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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 $\...
guest's user avatar
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1 vote
0 answers
33 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 ?
obelix's user avatar
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1 vote
1 answer
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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 ...
Giorgio Ripani's user avatar
3 votes
0 answers
267 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: ...
H1ghfiv3's user avatar
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2 votes
1 answer
236 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 ...
Benjamin's user avatar
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2 votes
1 answer
401 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 ...
asv's user avatar
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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 ...
nikah amber's user avatar
9 votes
1 answer
160 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 + ...
user avatar
1 vote
0 answers
338 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 ...
H1ghfiv3's user avatar
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0 votes
1 answer
551 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,...
TomH's user avatar
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6 votes
0 answers
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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 ...
Chitrabhanu's user avatar
3 votes
2 answers
381 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 ...
7iat's user avatar
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0 votes
1 answer
114 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 ...
tobias's user avatar
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3 votes
1 answer
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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 ...
Miguel's user avatar
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2 votes
0 answers
79 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) \...
k3thomps's user avatar
  • 516
3 votes
4 answers
816 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 ...
Ravinder Singh's user avatar
1 vote
0 answers
100 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 ...
george's user avatar
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2 votes
1 answer
149 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 $g(t)$, ...
dff's user avatar
  • 230
1 vote
1 answer
135 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{...
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