# Tagged Questions

**11**

votes

**2**answers

1k views

### How much can one say about this differential equation?

Consider the ODE $y^{\prime \prime}(x) = \cos(x) y(x)$ with boundary value conditions $y(0)=1$, $y(1)=2$. Solving it results in a linear combination of Mathieu functions, but what I find more ...

**1**

vote

**0**answers

30 views

### Backlund transformation related to two NL differential equations

I'm looking for a Backlund transformation linking the following two nonlinear differential equations for real $t$:
$$\dfrac{d^2}{dt^2}f(t)=\cos\left[f(t)\right]$$
...

**1**

vote

**1**answer

76 views

### A Non-homogeneous, Linear (Matrix) System of ODEs: What's Known About it? [closed]

Consider the following system of ODEs
$$
Y^{'}(t) = - \left[ A Y(t) + Y(t) A \right] + B(t)
,
$$
where $Y(t)$,$A$,$B(t)$ are all matrices, with the properties $A=A^T$, $Y=Y^T$. $Y(t)$ is the matrix ...

**3**

votes

**2**answers

209 views

### Existence of an integral equation (Faedo-Galerkin, Banach fixed point, Picard-Lindelof)

Let $m \geq 2$ and let $m'$ be its conjugate. Let $w_j$ for $j=1, ..., k$ be a basis of $H_1 \cap L^{m'}$. The task is to show that there is a $u(t) \in \text{span}(w_1, ..., w_k)=:A$ such that
...

**1**

vote

**0**answers

66 views

### Linear dynamical system with discontinuous coefficients

I am solving a linear dynamical system $X'=A(t) X$, where $t$ is the independent variable and $A(t)$ is a square matrix. Some of the coefficients of $A(t)$ have a discontinuity at a certain value of ...

**2**

votes

**0**answers

82 views

### Diffusion equation on mixing of diffusing particles

I am trying to study mixing of diffusing particles like it was done by E. Ben-Naim On the Mixing of Diﬀusing Particles.
The picture below shows the idea how permutations and inversion numbers reflect ...

**2**

votes

**1**answer

121 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

**1**answer

127 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

**1**answer

212 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 ...

**4**

votes

**1**answer

95 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

**1**answer

558 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

**1**answer

157 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

**1**answer

172 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

**1**answer

142 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

**1**answer

46 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

**1**answer

187 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

**2**answers

161 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

**1**answer

233 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 ...

**1**

vote

**0**answers

91 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

**1**answer

324 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

**0**answers

164 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

**1**answer

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

**0**answers

286 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

**2**answers

183 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 ...

**1**

vote

**0**answers

226 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

**2**answers

785 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

**1**answer

681 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

**2**answers

220 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 ...

**5**

votes

**0**answers

93 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

**1**answer

126 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

**0**answers

266 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

**2**answers

267 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

**1**answer

265 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

**0**answers

81 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

**1**answer

400 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

**1**answer

123 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 ...

**1**

vote

**1**answer

421 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

**3**answers

627 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

**2**answers

508 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

**1**answer

336 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.
...

**9**

votes

**2**answers

994 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

**0**answers

823 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

**2**answers

1k 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

**3**answers

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

**1**answer

734 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

**2**answers

532 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

**1**answer

299 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 ...

**3**

votes

**5**answers

458 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

**1**answer

356 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

**3**answers

626 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. ...