-1
votes
0answers
36 views

Proving the solution of one non-linear first order ODE has value 'e-1' at point 1

Consider the following first order non-linear ODE defined on interval $[0,1]$: $$F(x)=f(x)\ln\left(\frac{f^2(x)}{f^2(x)-1}\right)$$ where $f(x)=\frac{\partial F(x)}{\partial x}$, and the initial ...
-2
votes
0answers
35 views

Is there an explicit ODE solution for this system?

y'(t)=A(t)*y(t)+g(t), A(t)=[-t^2, t; sin(t) cos(t)] g(t)=(1+t/2; -1) y(0)=(3; 1) I actually do not know how to solve an ODE with variable constants, but when I use the ...
2
votes
1answer
83 views

Global Solutions of Ordinary Differential Equations

Background Let $f: [0, \infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n$ be a jointly measurable function satisfying, $f(t, \cdot)$ is locally Lipschitz for every $t \geqslant 0$, for every ...
0
votes
1answer
112 views

Solution of an Ordinary Differential Equation

I'd like to know if there is an analytical solution of the following Ordinary Differential Equation : $\displaystyle{\dfrac{d}{dt}x_i(t) = D_i + \sum_{j=1}^{n}L_{ij}x_j(t) + ...
3
votes
1answer
195 views

Exponential mapping versus flow

In Hamilton's article on the Nash-Moser Theorem, he gives the map that maps a vector field $X$ to its flow $e^{tX}$ in $\mathrm{Diff}(M)$ as an example where the implicit function theorem in Frechet ...
3
votes
1answer
61 views

Delay Differential Equations Numerical methods

I have a general question about delay differential equations. I know that even simple ones hardly have analytic solutions and mine clearly doesn't have any as it is a system of non-linear delay ...
5
votes
1answer
517 views

Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$

Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$. When $f$ is a function, it looks like the only solution is $f(x)=0$. But what if we allow distributions, such as the Dirac delta?
4
votes
1answer
116 views

Local positivity of solutions to linear differential inequalities (Chaplygin's theorem)

According to the entry "Differential inequality" of the Encyclopedia of Mathematics http://www.encyclopediaofmath.org/index.php/Differential_inequality the following result is due to Chaplygin ...
1
vote
1answer
165 views

Number of solutions of a system of equation!

Let $\Theta =(\theta_1,\ldots, \theta_n)\in {\mathbb T}^n$. I want to show that the system of equations $$ \sum_j 2\sin(\theta_i -\theta_j)+\sin(2\theta_i -2\theta_j) =0,\ \ i=1,\ldots, n, $$ has ...
2
votes
1answer
134 views

Parameter dependent differential equation in a Lie group

It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference ...
3
votes
1answer
40 views

Hopf bifurcation for systems where the dynamics is homogeneous of degree 1

Consider dynamical system in dimension 3 $$x'(t)=f(x(t),d)$$ where the dynamics f is homogeneous of degree 1 and there is exactly one line of equilibrium points. This line is independent of the ...
3
votes
1answer
170 views

Non-linear first order ODE

This is a two part question. On one hand, I am trying to find positive solutions of the following equation: $$1-\frac{d}{dx}f=\frac{f\log(f)}{x}$$ for $x>1$. If that is not possible, I would at ...
2
votes
2answers
155 views

Regularity of a nonlinear ODE [Traveling wave solutions of parabolic systems]

In the book of Volpert on Traveling wave solutions of Parabolic Systems (AMS), one reads "the following assertion is readily proved and we shall not discuss it in detail". The same result is tacitely ...
4
votes
1answer
205 views

Exact Differential Equations of Order n via Pfaffian Differential Equations?

I'm wondering if somebody could both shed some light on, & offer references for more details about, this interesting quote: The derivation of the conditions of exact integrability of an ...
2
votes
0answers
87 views

growth bound for solution of an ordinary integro-differential equation

I am considering the following ordinary integro-differential equation $$ A g = \sigma^2(y) + \int_\mathbb{R} \nu(y,dz) z^2 $$ where $$ A = b(y) \partial + \frac{1}{2} \sigma^2(y) \partial^2 + ...
6
votes
1answer
248 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 $\int$, $A$, $x(0)$ ...
2
votes
0answers
139 views

Differential Equations vs Difference Equations

My question is: Is there a duality between a solution of an ODE,PDE,SDE or integral equations with their analog counterpart in the discrete domain? I mean if I know a solution to the difference ...
0
votes
1answer
109 views

The solvability of a Hölder ODE [closed]

The problem is follow, I want to know weather there is a function $u\in \text{C}^{0}\left((-1,1)\right)$ but not $\text{C}^1((-1,1))$, such that $\lim\limits_{h\rightarrow ...
0
votes
0answers
264 views

Green's function of coupled ODEs

For functions $a(x)$ and $b(x)$ and "sources" $S_1(f,g)$, $S_2(f,g)$ and $S_3(f,g)$ lets say one has the differential equations for functions $f(x)$ and $g(x)$, $f' + af + bg = S_1(f,g) + S_2(f,g)$ ...
5
votes
2answers
165 views

Constructing a linear ODE for a product of two holonomic functions without introducing additional singularities

A function $f$ is called holonomic if it satisfies some linear differential equation with polynomial coefficients $$p_n(x) f^{(n)}(x)+\dots+p_1(x)f'(x)+p_0(x)f(x)=0.$$ Now if $f,g$ are holonomic then ...
0
votes
0answers
223 views

Whether does the following equation have only one finite zero?

Dear MOs, Here is a calculus problem which bored me for sometime. Let $a>0$ and $b<0$ be fixed.Define the following function (EDIT: Following the comment by Barry Cipra, you may only consider ...
8
votes
2answers
741 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 ...
4
votes
1answer
628 views

What is symmetry group of non-linear equation?

I am not very sure if this is a proper question, but I'm trying to investigate what the area of math can offer in researching the differential equation in polar coordinates: $r'^2+r^2=(kt)^2$, ...
5
votes
2answers
212 views

When is a solution to an ODE determined by its average value?

Fix a smooth vector field $\vec v$ on $\mathbb R^n$. It is well-known that any trajectory $\vec x : [0,1]\to \mathbb R^n$ solving the ODE $\dot x(t) = \vec v(\vec x(t))$ is determined by its ...
6
votes
0answers
88 views

Lagrangean uniqueness versus Eulerian uniqueness

(1) Lagrangean description. Let us consider a $N\times N$ system of autonomous ODE: $$ \dot x=a(x),\quad \mathbb R\ni t\mapsto x(t)\in \mathbb R^N,\quad a:\mathbb R^N\rightarrow \mathbb R^N. $$ ...
4
votes
1answer
103 views

Dependence of the blow-up time of existence of an ODE with respect to initial condition.

Let $V$ be a smooth vector field on $\mathbb{R}^n$. Assume that the maximal solution to the Cauchy problem $x'=V(x), x(0)=x_0$ exist only for $t\in [0,T)$, where $T$ is finite, denote this time by ...
6
votes
0answers
251 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
2answers
247 views

how to solve a singular integral equation involving the kernel $1/x$

Dear all, Suppose we know that $f(x)$ is nonnegative and Hölder continuous at zero with exponents $1/2$. We also know that $$ f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0, $$ ...
2
votes
1answer
240 views

Leibniz rule for Pseudo-differential operators of negative order

Does anyone know of some good references for a fractional Leibniz rule for pseudo-differential operators of negative order? As a specific example, I would like to compute $\partial_{x}^{-1}(uv)$, ...
1
vote
0answers
73 views

Semi implicit DAE integration using an implicit Runge Kutta scheme

I'm looking for some references regarding integration of DAEs in the form $M(t) \frac{dy}{dt} + G(y(t),t) + f(t) = 0, \quad y \in R^n, M(t) \in R^{nxn}$ using a high order implicit Runge Kutta ...
0
votes
1answer
381 views

Continuous variation from solution of easy problem to solution of hard problem

I asked this question a week ago over on math.stackexchange and got no reply, so I am asking here with slightly different wording. I am trying to prove that there exists a solution to a problem. I ...
0
votes
1answer
122 views

a question about asymptotics of solutions to a ordinary differential equation

This might be an easy question in the theory of ordinary differential equation. But since I know very little about it, I posted it here and hope to get some answers or references. Consider the ...
0
votes
0answers
607 views

First order differential equation of vector fields

Given vector fields $Y$, $Z$ on a (possibly compact) manifold $M$, I would like to know about the existence of solutions $X$ to the differential equation $$ \nabla_Y X + a \cdot \mathrm{div}(Y)\cdot X ...
1
vote
1answer
387 views

On properties of Wronskians of ODE

Hello everyone, I'm currently working over a certain class of ODE of the form $D_N \; \phi(x) = \lambda^N \phi(x) \quad, \quad N>1$ where $D_N = \delta_N \delta_{N-1} \cdots \delta_1$ and ...
1
vote
3answers
579 views

Solving $x\partial_x f = 0$ over distributions

Solving $x\partial_x f = 0$ over 'normal' functions is the same as solving $\partial_x f = 0$, i.e. one gets $f(x)=c_1$ as the complete answer. But over distributions (if my calculations are ...
5
votes
2answers
507 views

Runge-Kutta method with c<1

In trying to solve an ODE $y'=f(y,t)$ with a function f that is discontinuous at a subset (codim=1) of $\mathbb R^n$, I am looking for a Runge-Kutta ODE method whose stages do not evaluate $f(x,t)$ at ...
2
votes
1answer
329 views

Regularization of Zygmund functions

Dear community. I would like to derive a "good" estimate on $\frac{d}{dt}f_\epsilon(t)$, where $f_\epsilon$ is a regularization of a Zygmund-continuous function $f$, i.e. ...
8
votes
2answers
927 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 ...
3
votes
0answers
779 views

(Approximate) analytic solutions to the Mathieu equation

I'm trying to solve the driven Mathieu equation $x''+\beta x'+(a-2q\cos{\Omega t})\frac{\Omega^2}{4}x=f(t)$ for both zero and non-zero $\beta$. I can write down an analytic solution using the ...
6
votes
2answers
901 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 ...
4
votes
3answers
1k views

Analytical solutions of a differential equation (from Archimedes' Spiral)

There is a differential equation in polar coordinates: $r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const. I've found that a) if $\phi \in (0,t)$, t is quite small, then $r(\phi) \approx k/2 *\phi^2$ b) if ...
1
vote
1answer
708 views

The upper semicontinuous envelope of a lower semicontinuous function

We call $u^{*}$ is the upper semicontinuous envelope of $u$ if it is the smallest upper semicontinuous function satisfying $u\le u^*$. My question is that is there any good properties of the upper ...
0
votes
2answers
525 views

Approach to solving a differential-functional equation

What could be an approach to solving such equations? $$f'(x)=C \prod_{k=0}^x f(k)$$ $$\frac{g'(x)}{g(x)}=C+ \sum_{k=0}^{x-1} g(k)$$ Here the product and the sum are understood as indefinite sum and ...
1
vote
1answer
297 views

Does any iterative equation of n-th order have exactly n independent solutions?

Does any iterative equation of n-th order which does not inclute derivatives of order higher than 1 have exactly n independent solutions? Let's designate n-th iterate of a function $y(x)$ as ...
2
votes
5answers
435 views

Analytic hypoellipticity of linear ordinary differential operators

Let $P = a_n(x) D_x^n + a_{n-1}(x) D_x^{n-1} + \ldots + a_0(x)$ be a linear ordinary differential operator with polynomial (or real analytic) coefficients $a_j(x)$. Suppose that $a_n(x)$ doesn't ...
3
votes
1answer
354 views

Spectral Galerkin method for a semi-linear parabolic PDE

I'm trying to understand how to apply the Galerkin method to $u_t - \Delta u = u^3$. I understand how to obtain all of the a-priori estimates using Sobolev embeddings and such but my question concerns ...
1
vote
3answers
612 views

Do boundary conditions for elliptic PDE need to be homogenous to use spectral theory?

Question 1: It appears that when studying an elliptic equation $Lu=f$ in $\Omega$ with $u = g$ on $\partial \Omega$ we need to have $g=0$ in order that the inverse operator, $K=L^{-1}$ is linear. ...
7
votes
3answers
764 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 ...
2
votes
1answer
557 views

Reference for existence and uniqueness of differential equations for low differentiability?

My specific situation is that I have a non-spacelike continuous future directed curve $\gamma:[0,a)\to M$ in a Lorentzian manifold. The curve must necessarily satisfy a local Lipschitz condition and ...
1
vote
1answer
229 views

variational formulation: boundedness of the bilinear form

The simplest case of the problem I'm thinking about involves an elliptic differential operator, $Lu = -u'' + qu$, on the interval $(0,1)$, with homogeneous Dirichlet boundary conditions. I want to ...