# Questions tagged [differential-equations]

Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.

1,191
questions

**-5**

votes

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25 views

### A mistake in the solution of First-order system of linear differential equations [closed]

My solution:
enter image description here
enter image description here
Wolfram Alpha solution:
enter image description here
Please help me to find a mistake.

**3**

votes

**0**answers

48 views

### Complex Monge-Ampere equation with degenerate right hand side

Given a Kahler manifold $(X, \omega_0)$, and a smooth function $f$, suppose that I have a smooth solution to the following complex Monge-Ampere equation:
$(\omega_0 +i \partial \bar \partial \varphi)^...

**0**

votes

**0**answers

19 views

### Polynomial approximations of the vector field and distance between generated flows

Let $\textbf{h} = (h_1,...,h_n)$ be a $C^1$ system of ODEs defined everywhere on on some compact subspace $\mathbb{X} \subset \mathbb{R}^n$.
Suppose we have a polynomial approximation $\textbf{p} = (...

**2**

votes

**1**answer

86 views

### Dynamical system described by coupled nonlinear differential equations

Suppose a dynamical system is described by two variables, $x$ and $y$, and they change over time according to the following two coupled nonlinear differential equations:
\begin{equation}
\begin{split}
...

**1**

vote

**0**answers

30 views

### Set of eigenvalues of the boundary problem

I'm looking for the results about the set of eigenvalues of boundary problem for differential equation
\begin{equation}
\bigl(p(x) u'(x; \lambda) \bigr)' + q(x) u(x; \lambda) = -\lambda w(x) u(x; \...

**30**

votes

**3**answers

2k views

### What do we learn from the Wronskian in the theory of linear ODEs?

For a real interval $I$ and a continuous function $A: I \to \mathbb{R}^{d\times d}$, let $(x_1, \dots, x_d)$ denote a basis of the solution space of the non-autonomous ODE
$$
\dot x(t) = A(t) x(t) \...

**1**

vote

**1**answer

74 views

### Alternate proof of uniqueness of integral curves to vector fields

Let $V$ be a continuous vector field on an open set $U \subset \mathbb{R}^n$ and let $p_0 \in U$. There are many ways to construct local integral curves of $V$ through $p_0$, i.e. differentiable maps ...

**1**

vote

**0**answers

73 views

### Solutions of the differential equation $f'=(f^{-1})^{[n]}$

For clarity, we use the notations $f^{[n]}=f\circ \dots\circ f$ for nesting and $f^{(n)}=\frac d{dx}\left(\frac{d}{dx}\dots\right)f$ for differentiation.
After reading these two posts (here and here)...

**0**

votes

**0**answers

22 views

### What is the analog of the symmetrized Jacobi matrix for delay equations?

For a linear system of ODEs in $\mathbb{R}^{n}$ (with the usual inner product), say $\dot{V}(t) = A(t) V(t)$, we know that if $\xi_{1},\ldots,\xi_{k} \in \mathbb{R}^{n}$ and $V_{j}(t)=V_{j}(t,\xi_{j})$...

**7**

votes

**1**answer

140 views

### Non-perturbative Renormalization in the sense of Polchinski's equation. Do we have a mathematical formulation?

My question is about mathematical treatment of exact renormalization group in the sense of Polchinski's flow equation. In a heuristic form, Polchinski's equation looks like: $\partial_t S[\phi] = \...

**2**

votes

**1**answer

32 views

### Does a scalar LTV system with odd-periodic coefficients and even-periodic inputs have no periodic solutions?

Problem Setup
Suppose we have the following scalar, linear time-varying (LTV) system with parameter $\mu \in [0,\pi[$:
\begin{cases}
\dot{x_1}(t,\mu) = a(t,\mu)x_1(t,\mu) + b(t,\mu) \\
x_1(0,\mu) = ...

**0**

votes

**1**answer

70 views

### A time dependent variational problem coming from a second order linear PDE

Fix $u_0\in H^1(\Omega)$ and $f=f(x,y,t)\in L^2(\Omega\times [0,T])$ where $\Omega$ is a sufficiently smooth bounded domain in $\mathbb{R}^2$.
Consider the problem of finding $u:\Omega\times[0,T]\to\...

**0**

votes

**0**answers

27 views

### Floquet theory and Poincaré theorem on the continuation of periodic orbit

I read about the Floquet theory and a theorem that it named Poincaré's theorem of the continuation of periodic orbit.
Poincaré's Theorem: Consider a dynamical system depending on the parameter $\...

**-1**

votes

**0**answers

61 views

### how to show that the operator $\mathcal {L}$ has only one negative eigenvalue?

Consider the operator $\mathcal {L}: H^2_{per}([0,L])\subset L^2_{per}([0,L]) \longrightarrow L^2_{per}([0,L])$ given by
$$\mathcal{L}(y)=w\cdot y''+(3\varphi-1)y, \; \forall \; H^2_{per}([0,L]),$$
...

**3**

votes

**0**answers

45 views

### Asymptotic behaviour of solutions to system of ODEs

Let $Y:(0,+\infty)\to\mathbb{R}^n$ be a solution to the system of ODEs
$$
L[Y]=0,
$$
where $L$ is a linear operator which behaves, in a neighbourhood of 0, as
$$
L[Y](r)\simeq-Y''(r)-\frac{1}{r}Y'(r)...

**0**

votes

**1**answer

124 views

### How to solve the integral equation $f(x)=\frac{\lambda\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_1}{\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_2}$?

Recently, I have asked a question about variational analysis (First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space). Such the question can be addressed in some cases by ...

**4**

votes

**0**answers

33 views

### Results on: (path/initial condition)-dependent variant of exponential map generates compactly supported diffeomorphisms

Let $M$ be a connected and simply connected Riemannian manifold, non-compact, and suppose that $\{V_p\}_{p \in M}$ is a family of vector fields on $M$ indexed by $M$. Suppose moreover that the map
$$
...

**3**

votes

**0**answers

240 views

### Can a polynomial family of curves be identified by a polynomial satisfied by its derivatives?

Suppose we have a 1-parameter family of curves that foliate a subset of the plane, defined by a function that is a polynomial in the coordinates $(x,y)$ and a parameter that identifies the individual ...

**1**

vote

**0**answers

66 views

### Snoidal wave solutions of the $\phi^4$ model

I want to prove the existence of snoidal wave solutions of the $\phi^4$ model, given by
$$u_{tt}-u_{xx}=u-u^3,\; (x,t) \in \mathbb{R}\times \mathbb{R}.$$
So, we are looking for solutions in the form $...

**2**

votes

**1**answer

199 views

### A second order nonlinear ODE

In my research (in differential geometry) I recently came across the following nonlinear second order ode:
$$\frac{f''(x)}{f'(x)}-\frac{2}{x}+\frac{f'(x)+1}{2f(x)-x-1}+\frac{f'(x)-1}{2f(x)+x}=0$$
It ...

**5**

votes

**1**answer

1k views

### Analysis of solutions to a nonlinear ODE

Consider the following ODEs:
$\phi^2=\phi''\sqrt{1-\phi'^2}$, or $\phi^2=-\phi''\sqrt{1-\phi'^2}$.
Is there any theory (e.g. comparison theorems) which analyzes solutions of the above ODEs? I am only ...

**4**

votes

**0**answers

102 views

### Intrinsic numerical methods on Riemannian manifolds

I am interested in numerical methods for ordinary differential equations on a Riemannian manifold $M$. The general form of such an equation is $\dot x(t)=V(x(t)), x(0)=x_0 \in M$, where $V$ is a ...

**1**

vote

**0**answers

75 views

### Solving numerically a linear ODE

I start by saying that I do not have a strong background in numerical analysis, so I may miss some basic things or make trivial mistakes.
Motivated by some problems in digital signal processing, I ...

**3**

votes

**2**answers

176 views

### Usefulness of the $\sigma(L^\infty,L^1)$ topology in the context of differential equations

In Brezis's Functional Analysis book through chapters 3-4, I've seen the $\sigma(L^\infty,L^1)$ topology on $L^\infty$ but did not see (so far) any application of it in differential equations. Is ...

**20**

votes

**2**answers

1k views

### Intercept the missile

A stealth missile $M$ is launched from space station. You, at another space station far away, are trusted with the mission of intercepting $M$ using a single cruise missile $C$ at your disposal .
...

**1**

vote

**0**answers

61 views

### Solution existence for two-dimensional parabolic PDEs

I am looking for a solution $(f,g) \in C^{1,2}([0;T]\times\mathbb R;\mathbb R^2)$ to the following PDE system
$$
f_t(t,x) + a_1(t,x) f_x(t,x) + a_2(t,x) f_{xx}(t,x) - b_1 f(t,x)^2 + c(t) g(t,x) - d(t,...

**4**

votes

**1**answer

99 views

### Continuous dependence on initial parameters of an ODE for non-Lipschitz functions?

For ODEs, the standard theorem of continuous dependence of initial parameters deals only with functions that are Lipschitz. Do there exist more general results holding for non-Lipschitz functions? If ...

**2**

votes

**1**answer

171 views

### Seeking a Lyapunov function for a SIR model with immunity loss

We add the immunity loss to the SIR model and obtain the following autonomous system.
$$
\begin{align}
s' &= -is+\alpha r \\
i' &= i s - \gamma i\\
r' &= \gamma i-\alpha r
\end{align}
\...

**0**

votes

**0**answers

39 views

### How to solve a system of second order ODE from time t = T to t = 0

I have a system of second-order ODEs
$$
\mathbf{M\ddot{x} + C\dot{x} + Kx = f}
$$
I want to know some good numerical methods to solve this system of the equation given the initial conditions at time $...

**1**

vote

**0**answers

39 views

### Obtaining generating function for multivariate recurrence with non-constant coefficients

Consider a second order recurrence of the form below for some fixed $n$
\begin{align}
(n+s-1) A_{n}(k,s) &= (n-k+1) A_{n}(k-1,s-1) + (k+s-3) A_{n}(k,s-1) \\
A_n(k,s) &= 0 \tag{when $s < k$ ...

**5**

votes

**2**answers

273 views

### Name for a standard trick to construct a diffeomorphism

The following construction is standard, and it deserves a name.
Suppose we need to construct a diffeomorphism from a manifold $M$ to itself with some additional properties.
Observe that the flow $\...

**4**

votes

**1**answer

89 views

### Singular Sturm-Liouville problems: criterion for discrete spectrum for zero potential ($q=0$) and Hermite Polynomials

There are some known criteria for the Sturm-Liouville Problem
\begin{equation} \tag{1}
\frac {\mathrm {d} }{\mathrm {d} x}\left[p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=-\lambda w(x)y
\...

**4**

votes

**0**answers

73 views

### Closed subgroup (Cartan) theorem without transversality nor Lipschitz condition within Banach algebras

Yesterday, I came across the following preliminary theorem.
Theorem Let $\mathcal{B}$ be a Banach algebra (with unit $e$) and $G$ be a closed subgroup
of $\mathcal{B}^{-1}$ (the group of ...

**24**

votes

**1**answer

768 views

### Solving a delay-differential equation related to epidemiology

For some inexplicable reason, I have recently been interested in epidemiology. One of the classical and simplistic models in epidemiology is the SIR model given by the following system of first-order ...

**0**

votes

**0**answers

66 views

### ODE operator splitting with second order time discretization not possible?

I am trying to solve an ordinary differential equation (ODE) using an operator splitting approach:
$\frac{\partial f}{\partial t} = A(f) + B(f)$
Let's assume that $A$ and $B$ are very simple:
$\...

**0**

votes

**1**answer

94 views

### restricted three body problem equations of motion using particle distances and one angle variable

If we solve numerically a three (or $N$) body planar problem, it's easy to calculate the distances of the bodies as function of time.
Conversely if we know the interparticle distances as functions of ...

**0**

votes

**1**answer

79 views

### A solution of a system of equations that involve directional derivatives

[Edited on 29-March-2020 to make the question clearer]
Let $f, g : [0,1]^2 \to \mathbb{R}$ be two smooth functions, which are strictly increasing and concave in each coordinate. That is, for every $0 ...

**3**

votes

**0**answers

96 views

### Second derivative estimates

I am in big trouble since I don't see how to proceed (I don't need the exact calculation) with the following estimates.
In one of his papers, Lin proves the following result:
Let's consider a ...

**2**

votes

**0**answers

57 views

### Convergence rate of Toda/Morse flow

Let $A(t), A_0$ be a $n\times n$ hermitian complex matrices and consider the following matrix flow
\begin{align}
\frac{dA}{dt} &= \left [ C\circ A , A \right ] \\
A(0) &= A_0 \ .
\end{align}
...

**2**

votes

**0**answers

35 views

### When does the PDE $F\cdot \nabla u+Gu+H=0$ admit a global solution $u:\mathbb R^3\to\mathbb R$?

Let $F:\mathbb R^3\to \mathbb R^3$ and $G,H:\mathbb R^3\to\mathbb R$ be some given smooth maps. In order for the PDE $F\cdot \nabla u+Gu+H=0$ to admit a global solution $u:\mathbb R^3\to\mathbb R$, ...

**4**

votes

**1**answer

106 views

### Obstruction to the existence of a globally defined integrating factor

Let $U$ be an open subset of $\Bbb{R}^n$ and take $\omega$ to be a nowhere-vanishing smooth $1$-form on $U$. The Frobenius Theorem implies that, near each point of $U$, $\omega$ may be written as $g\,{...

**1**

vote

**0**answers

63 views

### Intuition from Hopf lemma (boundary point lemma )

Consider the classical boundary point lemma:
Let $L$ be an elliptic operator.
Boundary Point Lemma Suppose $\Omega$ has the interior sphere property and that $u\in C^2(\Omega)\cap C^1(\bar\Omega)$ ...

**3**

votes

**1**answer

194 views

### Aleksandrov maximum principle for semi-convex function

Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$.
Note. Saying that $u$ is semiconvex is ...

**1**

vote

**0**answers

43 views

### wave equation with non-smooth coefficients

Let us consider the equation
$$ \sum_{j,k=0}^n \partial_j ( a^{jk}(t,x)\partial_k u) =F, \quad \text{on $(-\infty,T)\times \mathbb{R}^n$}$$
subject to $u|_{t<0}=0$. Here, $F \in L_{\text{comp}}^2((...

**1**

vote

**2**answers

255 views

### Simple bound on $\log(x)/x$

I would like to pick $x$ as small as possible while guaranteeing that $\log(x)/x \leq \epsilon$ where $\epsilon \ll 1$. Clearly $x$ should (roughly) be of the order $1/\epsilon$; I would like simple ...

**7**

votes

**0**answers

152 views

### No intermediate denominators growth for holonomic functions?

My question concerns holonomic sequences of rational numbers, meaning assignments $a(n) : \mathbb{N} \to \mathbb{Q}$ fulfilling a linear recurrent relation of the form
$$
a(n+k) = \sum_{i=0}^{k-1} p_i(...

**13**

votes

**2**answers

639 views

### Solution to differential equation $f^2(x) f''(x) = -x$ on [0,1]

I'd like to solve a differential equation $$ f^2(x) f''(x)=-x $$ where $f(x)$ is defined on $[0,1]$ and has a boundary condition $f(0)=f(1)=0$.
I somehow found out that the solution is fairly close ...

**-2**

votes

**1**answer

57 views

### Lorenz ODEs with negative parameters

Consider the Lorenz system
$$\dot{x}(t) = \sigma(y-x) \, ,$$
$$\dot{y}(t) = x(\rho-z) - y \, ,$$
$$\dot{z}(t) = xy-\beta z \, .$$
Usually one considers the parameters $\sigma, \rho,$ and $\beta$ to be ...

**1**

vote

**0**answers

61 views

### Fixing constants of a series solution of a fourth-order PDE

The following is the PDE I want to solve,
$$\left(1+x^{2}\right)^{2}y_{xxxx}+8x\left(1+x^{2}\right)y_{xxx} + 4\left(1+3x^{2}\right)y_{xx} + K\left[2x yy_{xx}+\left(1+x^{2}\right)\left(yy_{xxx} + y_{x}...

**0**

votes

**1**answer

128 views

### Conditions to determine sign of real roots

From a delay system, I obtain the following as part of a characteristic equation:
$$f(\lambda) = \lambda - a + be^{-c\lambda},$$
where $a, b,$ and $c$ are positive number and $a<b, ac<1$. My ...