Questions tagged [differential-equations]
Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.
1,645
questions
0
votes
1
answer
97
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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.
0
votes
2
answers
2k
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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 :
$$
\begin{matrix}
...
4
votes
1
answer
218
views
Are all bidimensional second-order PDE at most quadratic in the top derivatives of Monge-Ampère type?
The general Monge-Ampère equation in $n$ independent variables is a quasi-linear combination of all the possible minors of the $n\times n$ Hessian matrix
$$
\left\|\frac{\partial^2u}{\partial x^i\...
1
vote
1
answer
179
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
2
answers
264
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 ...
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^...
2
votes
1
answer
560
views
General solution to system of stochastic linear differential equations
Assume we are given the system of linear stochastic differential equations
$$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + \sum_{j=1}^n \sigma_{ij}(t) \cdot x_j \cdot dB_{ij,t} + b_j(t)\cdot dt+\...
10
votes
2
answers
7k
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What does it mean for a differential equation "to be integrable"? [duplicate]
What does it mean for a differential equation "to be integrable"?
Are "integrable" and "solvable" synonyms?
The first thing that comes to my mind is to say: it's integrable if we can find the ...
2
votes
1
answer
488
views
On Harmonic Unit Vector Fields
When we restrict the Dirichlet energy functional to the set of all unit vector fields on a compact Riemannian manifold $(M,g)$, then the critical points of this functional are satisfied in $\Delta_g X=...
5
votes
1
answer
333
views
A special function solution of a fourth-order ODE
I want to consider the solutions of the following fourth-order ODE:
$$
f^{(4)}(t)+a tf^{(1)}(t)+b f(t)=0,
\tag{$\ast$}$$
where $a,b$ are complex parameters. It turns out that with a Fourier ...
10
votes
1
answer
870
views
A tricky tractrix question about vertical tangents
This is raised by a recent question occurring in combinatorial geometry.
It is about a sort of tractrix, but instead of a line, the pulling end moves along a circle of radius $r>\frac12$ (...
1
vote
1
answer
171
views
Holomorphic vector field with infinite separatrix
Let $V=\sum_{i=1}^{n}a_i(z_1,\ldots z_n)\frac{\partial}{\partial z_i}$ be a holomorphic vector field defined on a neighborhood $U\subset \mathbb{C}^n$ of the origin, such that the common zero point of ...
5
votes
3
answers
2k
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Determining geodesics between two points in curved space [closed]
In order to determine the geodesics between two points, one must solve the geodesic differential equations, which are as following
\begin{align}
p &= u'(s)\\
q &= v'(s)\\
p' + \Gamma^0_{00}p^2 ...
0
votes
1
answer
748
views
Frobenius method for multiple singular points
As we know, if the equation
$$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
409
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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 $$ \begin{cases}
...
2
votes
0
answers
165
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Modifying monkey saddles
We get negative Gauss curvature K surfaces of quasi polar symmetry surface form generated from:
$$ Re (x+ i y)^n = a^n $$ (n integer) with n humps above plane $ z =0$.
($ n =2,3,4 $ hyperbolic ...
5
votes
2
answers
257
views
Noninvariance for a specific nonlinear oscillator
Consider the nonlinear system
\begin{align*}
\frac{d}{dt} \begin{pmatrix} x_1(t) \\ x_2(t) \end{pmatrix} = \begin{pmatrix} x_2(t) \\ -4x_1(t) + x_1^2(t) \end{pmatrix},
\end{align*}
which admits ...
3
votes
0
answers
134
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Equations of the form $((\Delta)^{2}+\lambda\Delta+\gamma)(f)=0$
Here is the problem: I am trying to figure out if there is a non-zero solution for the system of equations:
$(\Delta^{k-1}+a)(d^{\ast}\eta)=2d^{\ast}d\xi$
$(\Delta^{k}+b)(d\xi)=2dd^{\ast}\eta$
for $...
3
votes
0
answers
69
views
Attainability of Global Optima In Optimal Control
Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation:
$\frac{d x(t)}{dt} = F(x,u)$
one can consider the ...
8
votes
3
answers
1k
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Characteristic Variety of the Principal Symbol solves PDE system?
In the study of partial differential equations, it is often considered enough to analyze the principal symbols and their characteristic variety (see for example, http://www.sciencedirect.com/science/...
2
votes
1
answer
815
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The Biharmonic Eigenvalue Problem on a Rectangle with Dirichlet Boundary Conditions
I am interested in solving the following biharmonic eigenvalue problem.
$$\begin{array}{cccc}
& \Delta ^2 \Psi (x,y) = \lambda \Psi (x,y), & - a \le x \le a & - b \le y \le b \\
&...
0
votes
0
answers
56
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
$$ f(t,x)=1+ x \int_{0}^{1-t} (1-s)f(s,x)ds,...
5
votes
0
answers
114
views
A finiteness question for integrable polynomial distributions on $\mathbb{R}^3$
This question is motivated by the finitness of limit cycles for polynomial vector fields on $\mathbb{R}^2$
Assume that $X,Y$ are two independent polynomial vector fields on $\mathbb{R}^{3}$ such ...
5
votes
1
answer
177
views
Getting out a system of linear ODEs by knowing the Magnus expansion
Assume we are given for a transition between two time points $t_0 = 0$ and $t_1$ a matrix relationship, eventually describing the solution of a system of linear with non-constant coefficients,
$$Y(t_1)...
3
votes
0
answers
136
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 of ...
3
votes
1
answer
310
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 ...
6
votes
0
answers
774
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
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 ...
1
vote
0
answers
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. ...
4
votes
1
answer
98
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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 ...
2
votes
0
answers
184
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 $Im(z)\neq 0$,...
4
votes
3
answers
4k
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
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 ?
1
vote
1
answer
340
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 ...
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:
...
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 ...
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 ...
1
vote
0
answers
119
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 ...
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$$F = \partial^n + f_1(t)\partial^{n-1} + \dots + f_{n-1}\partial + f_0,\text{ }G:= \partial^m + g_1(t)\partial^{m-1} + \dots + g_{m-1}\partial + ...
1
vote
0
answers
338
views
Comparing Dirichlet energy and area of a Surface-immersion
Let $(F,g)$ be a closed Surface, $(M,h)$ a Riemannian 3-Manifold and $f: F \to M$ a smooth immersion. Denote by $f^*(h)$ the pullback metric on $TF$ induced by $f$ and let $dV_g$ and $dV_{f^*(h)}$ be ...
0
votes
1
answer
551
views
How to prove that a non-linear differential equation has a solution
I want to prove that there exists $f:[0,1] \to [0,1]$ such that $f(0)=0$,
$$
\frac{d w(y-f(y))}{d y} = g(y) \frac{d v(f(y))}{d y}, \forall y \in [0,1],
$$
where $w:[0,1] \to [0,1]$ and $v:[0,1] \to [0,...
6
votes
0
answers
3k
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What is the Beltrami differential?
Let $R,S$ be Riemann surfaces and $f: R \to S$ an orientation preserving diffeomorphism. Then $f$ determines what is called a Beltrami differential denoted by $\mu \frac{d\bar{z}}{dz}$.
Local ...
3
votes
2
answers
381
views
Nonlinear ODE system: stability
I've got this 4x4 system that should model the wine fermentation process. All the $\mu, K_N, k_d$ etc are positive constants. Of course I have no idea of how to solve it. But at least I would like to ...
0
votes
1
answer
114
views
Fit a system of linear ODEs from several experiments
Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n=6$ (in any event a number below 10) , and the number of initial vectors $r$ is in the ...
3
votes
1
answer
97
views
Examples of systems with stable equilibria at the boundary of the phase space
Hopfield networks are gradient dynamical systems, used (among other things) to solve combinatorial optimization problems, because stable equilibria are at vertices of the hypercube $[-1,1]^n$. They ...
2
votes
0
answers
79
views
Point Spectrum of a Second Order System of Differential Equations
Consider the following operator acting on $H^1(\mathbb{R})$
$$
\mathcal{L} \left(\begin{array}{c} \phi \\ \psi \end{array}\right) = -\left(\begin{array}{c} \phi \\ \psi \end{array}\right)'' + V(x) \...
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 ...
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 ...
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 $g(t)$, ...
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{...