# Tagged Questions

**8**

votes

**2**answers

786 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 ...

**1**

vote

**0**answers

280 views

### a question about contact manifolds

Let $\mathfrak{g}=< X_{f_1},X_{f_2},...,X_{f_n}> $ be a lie algebra of contact symmetries , $\mathfrak{g}\subset Sym(E_\omega )$ , such that the orbit spact $ B_\mathfrak{g}=E_\mathfrak{g}/ ...

**1**

vote

**1**answer

404 views

### Good books on stochastic partial differential equations?

I have a system of 2 PDEs, one with a probabilistic right side, and kind of stuck on what to read about those things.. Any good books around? Both analytical (if any) and numerical methods are ...

**1**

vote

**1**answer

140 views

### finding effective 2-form corresponding to an equation

What is the effective 2-form corresponding to the equation
$det Hess v=(v-q_1v_{q_1}-q_2v_{q_2})^4$
you can find the definition of effective forms here

**1**

vote

**0**answers

157 views

### multivalued solution of a equation

Definition: A scalar k-th order differential equation on a smooth manifold $M$ ,
is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $
for $\left | \sigma \right |\leqslant ...

**2**

votes

**3**answers

317 views

### Laurent series expansion for ODE.

OK, then I read Frobenius method in mathworld (I learned when I took ODE 2):
http://mathworld.wolfram.com/FrobeniusMethod.html
My question is:
Are there any ODEs where the solution is given by full ...

**2**

votes

**1**answer

950 views

### First Order PDE Solution Method Issues

Greetings, I have a rather pedantic problem that's honestly been holding me back for over a month & is something neither stackoverflow nor some of my professors nor some graduate students have ...

**2**

votes

**1**answer

401 views

### closed form solution of an ODEs

I posted this on mathstack and it is not a homework problem, I am doing a modeling problem and this ODE comes up. I don't know whether it could be considered to be a research problem yet. Please help ...

**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:
$r'^2+r^2=(kt)^2$, ...

**-1**

votes

**3**answers

356 views

### problem related to Airy functions [closed]

I have solved the Schrödinger equation for a triangular well potential and the solution comes in terms of Airy functions...now i am facing the following problems:
What are the normalization ...

**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 ...

**3**

votes

**2**answers

355 views

### Elaborating Mercer's theorem (RKHS) on Cameron-Martin space $k(x,y)=\min(x,y)$

Hi,
I'm trying to employ Mercer's theorem on the kernel $k(x,y)=\min(x,y)$. It is known (and easy to verify) that this is a nonnegative-definite kernel over $[0,T]$ for any $T>0$.
Fix $T>0$. ...

**1**

vote

**1**answer

246 views

### A 'conjecture' on critical elliptic pde

I conjecture the following.
Let $\Omega=\mathbb{R}^3-\overline{B_1(0)}$. Define
$$E_{\Omega}(u)=\frac{1}{2}\int_{\Omega}|\nabla u|^2dx-\frac{1}{6}\int_{\Omega}|u|^6dx.$$
$E_{\mathbb{R}^3}$ is defined ...

**1**

vote

**0**answers

91 views

### A critical elliptic PDE

I am considering the problem $-\Delta u=|u|^4u$, $x\in \Omega\subset \mathbb{R}^3$, $u|_{\partial \Omega}=0$. Where $\Omega$ is a unbounded domain. Some special case like $\Omega=\mathbb{R}^3-B_1(0)$, ...

**1**

vote

**1**answer

281 views

### Solution of a PDE and its uniqueness

Hallo,
consider $f: U \times I \rightarrow \mathbb{R}$, where $U \subset \mathbb{R}^{n}$ and $0 \in I \subset \mathbb{R}$ be two open sets. I am looking for the solution $f$ of the following PDE
...

**0**

votes

**2**answers

341 views

### Solution to differential equation

a) How to solve, or at least to prove the existence of a solution to differential equation for given initial condition $y(s)=y_0>0$ and $y'(s)=y_1$, $s<0$,
$$y''+(2-n)\coth(t) ...

**3**

votes

**1**answer

208 views

### A partial differential equation on $\mathbb{CP}^1$

Let $f$ be any complex function on $\mathbb{CP}^1$. Denote the local coordinates of $\mathbb{CP}^1$ as $z,\bar{z}$. Does the following equation
$\frac{\partial}{\partial z} f = ...

**0**

votes

**0**answers

415 views

### Does the following differential equation have a non-trivial solution:(f'')^4+(f)^4=2(f')^4 ? [closed]

Hello!
Currently,I meet the following differential equation:
${f''}^4+f^4=2f'^4$
we can easily find the trivial solutions:$a e^{bx}$,where $a$ is an arbitrary constant,$b=\pm i$ or $b=\pm 1$,does ...

**5**

votes

**1**answer

538 views

### Numerical calculation of Arnold tongue

Hello.
I am working on investigation of family of dynamical systems on the torus
$$\dot{x}=\cos(x)+b\cos(t)+a$$
$$\dot{t}=1$$
and it's Poincare map $$P:(x,0) \rightarrow (P(x),2\pi=0)$$
I need to find ...

**2**

votes

**0**answers

139 views

### Critical case linear autonomous functional differential equation

I am looking for asymptotic ($t\to\infty$) behavior of the general
solution $g(t)$ to a following linear functional differential equation
$$
\text{(1)} \quad\quad\quad g'(t)-g'(t-T)=-g(t)
$$
with ...

**3**

votes

**1**answer

508 views

### Are (Frobenius) integrability conditions covariant?

Starting with a system of partial differential equations for functions on $\mathbb{R}^n$, Frobenius' theorem gives a bunch of integrability conditions (on e.g. functions which are 'data' for the ...

**14**

votes

**4**answers

1k views

### ODE's without a Lipschitz condition

When teaching ODE's earlier this semester, one of my students asked the following question for which I didn't know the answer (and none of the textbooks I consulted seem to discuss it). It is ...

**0**

votes

**1**answer

66 views

### Self-inhibitions are diagonal matrix

In all the discontinuous neural networks models, the self-inhibitions are diagonal matrix, what is the reason for this assumption?

**1**

vote

**2**answers

84 views

### Finding Kuramoto Model coupling strength with limits?

The following is an equation describing the coupled phases of N oscillators according to the Kuramoto model:
$$
1 = K \int_{-\pi/2}^{\pi/2}\cos^{2}\left(\theta\right)\,
{\rm ...

**5**

votes

**1**answer

335 views

### total variation distance between two solutions of SDE

Suppose we have two stochastic differential equations with the same initial conditions:
$$d X_t^1= b_1(t,X_t^1)dt + dW_t$$
$$d X_t^2= b_2(t,X_t^2)dt + dW_t,$$
$X_0^1=X_0^2=x_0$; $W_\cdot$ is a ...

**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

127 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 ...

**7**

votes

**2**answers

447 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 ...

**1**

vote

**1**answer

274 views

### Books on Numerical Methods for Partial Differential Equations

Any good references for undergraduates?

**1**

vote

**1**answer

147 views

### Under which conditions Jacobi PDE system can be represented to symplectic monge Ampere equation?

We know we can reduce Symplectic monge ampere equations to Jacobi PDE system with some compatibility condition. I want to see when the vise versa is correct? and is there any theorem for it.
Here ...

**0**

votes

**0**answers

156 views

### Harmonic Function?

Hi,
Let $\varphi : U \rightarrow \mathbb{C}$ be a holomorphic function, where $U \subset \mathbb{C}^{n}$ containing $0$. Is the function $u(x_{1}, ..., x_{n}, y_{1}, ..., y_{n}) := Im( ...

**1**

vote

**0**answers

149 views

### two polynomial equations

Let $f:\mathbb R^2\rightarrow\mathbb R$ be a smooth function such that for every point $(x,y)\in\mathbb R^2$ the system
$$f_{11}+2tf_{12}+t^2f_{22}=0$$
$$f_{111}+3tf_{112}+3t^2f_{122}+t^3f_{222}=0$$
...

**1**

vote

**2**answers

220 views

### Invariant set of Lotka-Volterra equation

I have the Lotka-Volterra equation
$\dot{x}=x(1-y),$
$\dot{y}=y(x-1),$
where $x$ and $y$ are non-negative.
It is easy to see that the $x$- and $y$-axis are invariant sets. I can see from plots that ...

**6**

votes

**1**answer

510 views

### ANOTHER Exterior differential system on $SO(3;\mathbb R) \times \mathbb R$

I have another exterior differential system for one forms $U^i$, where the $\theta^i$ are a cotangent basis on $SO(3)$, i.e. they satisfy $d \theta^i = \epsilon_{ijk} \theta^j \wedge \theta^k$ for the ...

**1**

vote

**1**answer

175 views

### Exterior differential system on $SO(3;\mathbb R) \times \mathbb R$

I have the following exterior differential system for one forms $\alpha, \beta, \gamma$, where the $\theta^i$ are a cotangent basis on $SO(3)$, i.e. they satisfy $d \theta^i = \epsilon_{ijk} \theta^j ...

**10**

votes

**5**answers

859 views

### Is there any techniques for solving a differential equation including iterated function? f ' (x) = f( f( x ) )

I am trying to solve this differential equation but I have no idea how.
$f ' (x) = f( f( x ) ) $
Although I don't think this differential equation is solvable, I'd like to know if there is any ...

**0**

votes

**1**answer

336 views

### phase portrait of system of differential equations

Is there a full classification of the phase portraits of the following systems of differential equations
\begin{equation}
\cases{
\dot x=a_{11}x+a_{12}y+a_{13}z \\
\dot y=a_{2 1}x+a_{22}y+a_{23}z\\
...

**4**

votes

**1**answer

1k views

### Examples of separable ordinary differential equations in economics

I'm currently teaching an integral calculus course for business students, and we're just about to discuss differential equations. They've worked hard, and I'd like to reward them with some economic ...

**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

**1**answer

79 views

### A linear equation related to Camassa-Holm equation

I would like to study the well-posedness of the following equation
$u_t - u_{txx} + a u + b u_x + c u_{xx} + \gamma u_{xxx} = f$
with $u(0)=0$ and $f \in H^{s-1}(\mathbb{R})$, where $a, b, c, ...

**1**

vote

**1**answer

150 views

### Bessel and Neumann functions:ordering the zeroes

I realize the abundance of the literature on this theme which is a disservice in this case, because there are too many formulas in all the books I've found and no explicit answers. Maybe someone could ...

**3**

votes

**1**answer

173 views

### nonlinear delay differential equation

Consider the delay differential equation:
$ y_x(x) = \sqrt{y(x-\bar{x})} $
where $y$ is the unknown function of $x$, and where $\bar{x}$ is a fixed parameter.
This equation does not seem to have a ...

**1**

vote

**1**answer

116 views

### Boundary condition for a non linear schrodinger equation

I'm studying an anrticle on a non linear Schrodinger equation posed on $\Theta=(x\in R^2:|x|<1)$.
I read this: "we will only consider initial data of Sobolev regularity $s<1/2$ and thus we will ...

**6**

votes

**3**answers

333 views

### ordered exponential of unbounded operators

Let $H$ be a Hilbert space,
and let $A_t$ be a family of unbounded positive (self-adjoint) operators on $H$ parametrized by $\mathbb t\in R_{\ge 0}$. Consider the ordinary differential equation
$$
...

**2**

votes

**3**answers

751 views

### Jacobi method on first order partial differential equations

Hi,
I am interested in the Jacobi method to solve partial differential equation of first order. I would like to have a hint about a good book to study this subject.
Thanks in advance

**1**

vote

**0**answers

78 views

### Decay rate of Discrete Prolate Spheroidal Sequences in frequency

What is the decay rate of DPSS sequences in frequency?
Consider an interval $T\subset\mathbb{Z}$ of length N in time. Consider another interval $[-W,W]$ in frequency with $W<1/2$. Let $\phi_0$ ...

**1**

vote

**0**answers

149 views

### “Higher” Tangent spaces in char-p geometry - definition?

Hi, everyone!
I have some construction that requires exact definition.
I considered polynomial homomorphisms from $\Bbbk$ to $U_n(\Bbbk)$ (unitriangular matrices), where $char \Bbbk = p>0$ (more ...

**5**

votes

**1**answer

173 views

### Periodic Holomorphic ODE

Suppose I have an annulus $U\subset \mathbb{C}$ and a single-valued holomorphic function $V:U\to \mathbb{C}$.
I would like to know if there are (tractable) conditions on $V$ that ensure that the ...

**1**

vote

**1**answer

229 views

### Does this variable have an upper bound?

Let $x$ be a positive scalar variable whose time derivative satisfies
$$|\dot{x}(t)|\leq \exp \left\{\left(-\int_{0}^{t}\frac{1}{x(\tau)} \mathrm{d} \tau \right)\right\},$$
where $|\cdot|$ denotes the ...

**0**

votes

**0**answers

158 views

### search for a function satisfying some conditions

Hi everyone, I would like to find a function
$$\Psi\in\mathcal{C}^2: z\in\mathbb{R}\rightarrow\Psi(z)\in\mathbb{R_+}$$
satisfying the following conditions:
...