Questions tagged [ordinary-differential-equations]

For questions about ordinary differential equations, which are differential equations involving ordinary derivatives of one or more dependent variables with respect to a single independent variables. For questions specifically concerning partial differential equations, use the [tag:pde] instead.

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72
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On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
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How to solve $\dot{x} = \frac{f(x)}{\|f(x)\|}$?

How to solve the following ODE? $$\dot{x} = \frac{f(x)}{\|f(x)\|},$$ where $x : \mathbb{R} \to \mathbb{R}^n$, i.e., $x(t)$ is the trajectory. The right-hand side $f : \mathbb{R}^n \to \mathbb{R}^n$ ...
22
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495 views

Witt's proof of Gelfand-Mazur / Ostrowski's theorem

Now asked on MathOverflow. Background: It seems that, after his groundbreaking work on quadratic forms and inventing Witt vectors, Ernst Witt developed the hobby of giving extremely short proofs to ...
20
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1answer
523 views

Kähler Geodesics

Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric $$\begin{pmatrix} \frac{1}{1-|a|^2}&\frac{1}{1-a\bar{b}}\\\frac{1}{1-\bar{a}b}&\frac{1}{1-|b|^2}\end{...
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Find a function such that $f^{-1}=f'$

Let $f:\Bbb{R}^+\rightarrow\Bbb{R}^+$ be a differentiable bijection and let $f$ satisfy: $f'=f^{-1}$ (where $f^{-1}$ denotes the inverse of $f$). Find $f$. This comes from a facebook page "...
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Relation between non-vanishing Vector Fields on $\mathbb{T}^2$ and Fundamental Group Maps

Let X be a vector field on $\mathbb {T}^2$, we say that $\varphi: \mathbb {R} \to \mathbb {T}^2$ is a periodic orbit of $X $, if $\varphi $ is a periodic function and $\varphi'(t) = X (\varphi(t)), \...
14
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1answer
236 views

Eigenvalue problem for $−\psi''(x) − (ix)^ N \psi(x) = E\psi(x)$ in complex plane

To find the eigenvalue $E$ in the complex plane of $x$ for one dimensional Schrodinger equation $$ −\psi''(x) − (ix)^ N \psi(x) = E\psi(x). $$ where $N$ can be any real number, the boundary condition ...
13
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3answers
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Damped Harmonic Oscillator and Response Function

This is another one of those questions that I feel like I am almost there, but not quite, and it's the math that gets me. But here goes: For a driven damped harmonic oscillator, show that the full ...
12
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1answer
635 views

Measure-driven differential equations

Background: I need some help to understand the concept behind measure-driven differential equations. The solution of an ordinary differential equation is continuous. In order to describe discontinuous ...
11
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388 views

Discretization formula for system of differential equations. “Solution to one of these is the initial condition of the other”. In which sense?

Consider the following stochastic differential equation \begin{equation} dy=\left(A-\left(A+B\right)y\right)dt+C\sqrt{y\left(1-y\right)}dW\tag{1} \end{equation} where $A$, $B$ and $C$ are parameters ...
11
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138 views

When is the derivative of $f(g(x))$ equal to $g(f'(x))$?

By the chain rule, we know that the derivative of $f(g(x))$ is $f'(g(x))g'(x)$. Question: When is the derivative of $f(g(x))$ equal to $g(f'(x))$? Trivial solutions include the following: Let $f$ ...
11
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1answer
513 views

Counter example for uniqueness of second order differential equation

I have a second order differential equation, \begin{eqnarray} \dfrac{d^2 y}{d x^2} = H\left(x\right) \hspace{0.05ex}y \label{*}\tag{*} \end{eqnarray} where, $\,H\left(x\right) = \dfrac{\mathop{\rm ...
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How to solve a time-dependent Schrodinger equation in periodic Dirac delta potential

I'm trying to solve a 1D time-dependent Schrodinger equation: $$ i\,\frac{\partial \psi(x,t)}{\partial t}=\left[-\frac{1}{2} \frac{\partial^2}{\partial x^2}+V(x)+F(t)\,x\right]\!\psi(x,t) $$ where $...
11
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Uniqueness of an infinite system of linear ODEs

How to prove that $\dot{x}=ax,\space x(0)=1$ has a unique solution if $a,x$ are infinite dimensional matrices? More specifically, let $Q$ be a bounded infinitesimal generator, i.e. $Q=(q_{i,j})_{i,j\...
11
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1answer
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Invariant submanifolds

Let $M$ be a smooth manifold, and let $N$ be a submanifold. Let $V$ be a smooth vector field on $M$ which generates a flow $\Phi_t$ on $M$. My intuition tells me (perhaps modulo some technical ...
10
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Application of Thoms transversality theorem

I try to verify example 20.4.10 from Wiggins - Introduction to Applied Nonlinear Dynamical Systems and Chaos and I am quite new to the topic so please be patient. In the book is written that the ...
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In which commutative algebras does any derivation possess a flow?

Suppose $A$ is a commutative algebra over $\mathbb{R}$ with unity. $\mathbb{R}$-linear map $\xi\colon A\to A$ is a derivation of $A$ iff $\xi(ab)=a\xi(b)+\xi(a)b$ for any $a,b\in A$. If $\gamma\colon \...
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1answer
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Is it possible to create a trajectory going through these two points in the phase plane?

Consider the ODE \begin{align} u_{xx}+\cos(u)+\frac{1}{2}=0\tag{E}\label{ode} \end{align} which has stable equilibria $s_n, n\in\mathbb{N}_0$ and unstable equilibria $q_n, n\in\mathbb{N}_0$, where \...
9
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1answer
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Solve ODE $y'' + (y')^2 + y = \ln(x)$

I want to solve $y'' + (y')^2 + y = \ln(x)$ with boundary conditions $y(1) = 0$ and $y(2) = \ln(2)$. The solution is $y = \ln(x)$ but I don't know how to start the problem.
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Finite group of “linear substitutions”

From what I can tell, a linear substitution is an operation on a set of variables $x_1,\ldots,x_n$ which sends them to a new set of variables $y_1,\ldots, y_n$ via a linear transformation $$\vec{y} = ...
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Heat equation proving smoothness

I have a question regarding a PDE course: Let $T$ be the strongly continuous semigroup which belongs to the heat equation, thus with generator $A$ is the Laplacian. Suppose we have $g \in C^{\infty}...
9
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1answer
286 views

Uniqueness of solutions to $u_{tt} - c^{2}u_{xxxx} + au_{t} = 0$

The problem I am working on is to show that there is a unique compactly supported solution to the PDE $u_{tt} - c^{2}u_{xxxx} + au_{t} = 0$, $(x, t) \in \mathbb{R} \times [0, \infty)$ with $u(x, 0)= \...
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Modelling a Water Rocket. Requires Some Validation and Help. ( WARNING : Extremely Long but Interesting Post )

Good day people of math.stackexchange.com UPDATE: Version 2 can be found here: https://physics.stackexchange.com/questions/275284/modelling-a-water-bottle-rocket-version-2-long-post-warning. This is ...
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Separable non-linear ODE (with radicals)

I am trying to solve the equation $$ \frac{dy}{dt}=\sqrt{\left(\gamma-1+\frac{2\alpha\beta}{2\alpha-1}\right)e^{-2\alpha y}-\frac{2\alpha\beta}{2\alpha-1}e^{-y}+1}\tag{1} $$ $y(0) = 0$; $t_{0}=0$; ...
8
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When do differential equations induce maps between algebraic varieties and how to find these varieties

The intuition: Consider a single variable polynomial differential equation with integer-polynomial coefficients, for example $$ y '' = -y $$ Then consider a pair of algebraic varieties (that is ...
8
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Is this physical model exactly solvable?

There exists a popular model in the Physics of heavy quark bound systems, called the Cornell potential model, in which the inter-quark potential is modeled to vary with radial distance $r$ as $$V(r) ...
8
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An attractor for blow-up solutions to a cubic oscillator

(Related to this MathOverflow question). Consider the nonlinear ODE $$\tag{1} \frac{d^2u}{dt^2}+u=u^3, \qquad t\in\mathbb R,$$ which has the conserved quantity $$\tag{2} E=\frac12 u'^2+\frac12 u^2 -...
8
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Linear differential equation on $\mathbb{R}^2.$

Let $l_1, l_2, r_1, r_2 $ be negative real numbers such that $r_1r_2 \neq 0$ and consider the two matrices $$L = \left(\begin{matrix} l_1&0\\ 0& l_2 \end{matrix} \right),\,\, R= \left(\begin{...
8
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1answer
230 views

Singular linear systems of ODEs

A classical problem in quantum mechanics involving the Dirac Delta function is given by $$ y''+(\delta(x)-\lambda^2)y=0. $$ Then, to find ''bound states'', you solve on the right and find the ...
8
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Separable Differential Equation (Check Answer)

Question: Determine all differentiable functions in the form $y$ = $f(x)$ which have the properties: $f'(x)$=$(f(x))^3$ and $f(0)=2$ What I have done I set up the differential equation as such $$ \...
8
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344 views

Numerically solving a non-linear PDE by an ODE on the Fourier coefficients

I need to solve numerically a PDE of the form $$ u_t(x,t)=u_{xx}(x,t)+u_x(x,t)^2-a(x)u_x(x,t)-a_x(x) $$ with initial condition $u(x,0)=u_0(x)$. I can assume that both $u(\cdot,t)$ and $a(\cdot)$ are ...
8
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1answer
221 views

Density given by variable-coefficient PDE

I am looking for a time-dependent probability density $f(x,y,t)$ solving the equation $$-\frac{\partial f}{\partial t} = \alpha\cdot \big(y - F(x)\big)\frac{\partial f}{\partial x}+\beta\cdot \big(G(y)...
8
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179 views

Solution of a nonlinear first order ODE

Is it possible to find an analytic solution to the following ODE: $$y\ln(xy)y'+x=0 $$ It is neither separable nor can be made an exact one. I cannot seem to work any substitution either. I've also ...
8
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367 views

ODE system and Lie symmetries

The ODE system (see below), where $F$ is a given function together the algebraic condition (SYM) imply that $$\boxed{y(t)=k-x(t)} \tag{*}$$ for some $k$ constant. The result that $y$ is a translation ...
8
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1answer
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Uniqueness of Solution to a Boundary Value Problem

Question Let $f:\mathbb R_+ \to \mathbb R_+$ be a function twice continuously differentiable (with derivative $f'$ and second derivative $f''$), and $a$, and $b$ be parameters in $\mathbb R_+$. ...
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analysis/ode question

Prove that there does not exist any twice continuously differentiable function $f:[0,1]\to \mathbb{R}$ with $f(0)=0$ and $f(1)=1$ such that $$ - e^{-f'} f'' + 2 f = 0. $$ As an attempt, I drew a graph ...
7
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Solving Euler's equation for rigid-body rotational velocity in 4D

A rigid object, with no torques applied to it, rotates with constant angular momentum. But its angular velocity $\omega$ is not constant in general; it follows the differential equations $$\frac{d\...
7
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1answer
252 views

ODE for the differential of a flow on a manifold

This question is a consequence of my horrible knowledge in differential geometry. It can be stated as follows. Consider the solution $y_p(t)$ to the ODE $$\partial_t y_p(t) = X(y_p(t)), \qquad y_p(0) ...
7
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0answers
120 views

Understanding the notation when finding action-angle coordinates

I'm trying to learn the basics of KAM theory and I wanted to first get to grips with Liouville integrability for Hamiltonian systems but I'm getting rather confused by the notation which seems to be ...
7
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Methods to solve $\int_{0}^{\infty} \frac{\cos\left(kx^n\right)}{x^n + a}\:dx$

Spurred on by this question, I decided to investigate for different functions on the numerator. Here, I went from $\exp(..)$ to $\sin(..) / \cos(..)$. I initially thought I could modify the result ...
7
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173 views

Finding a particular solution to a linear PDE

I want to solve the PDE $$\frac{\partial u}{\partial t}+x_1(x_2-x_3) \frac{\partial u}{\partial x_1}+x_2(x_3-x_1) \frac{\partial u}{\partial x_2}+x_3(x_1-x_2) \frac{\partial u}{\partial x_3}=\sum_{i=1}...
7
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129 views

Solving a dual integral equation involving a zeroth-order Bessel function

Consider the following dual integral equations \begin{align} \int_0^\infty q^3 f_0(q) J_0 (qr) \, \mathrm{d} q &= g(r) \qquad\qquad\quad (0<r<1) , \\ \int_0^\infty f_0(q) J_0 (qr) \, \...
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129 views

Recommendation for intro to geometric integrators?

Explicit Request Looking for book or lecture note recommendations on numerical optimization that (ideally) have the following: Emphasis on geometric and physical intuition Emphasis on symplectic ...
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226 views

Region of attraction of simple ODE with perturbation

There are a few nice discussions about ROA covering a few subtopics: Region of attraction of : $x'=-y-x^3,y'=x-y^3$ via Lyapunov Function Region of attraction and stability via liapunov&#...
7
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77 views

How Turing instability explains the patterns present on the animal skin?

Alan Turing in his original paper presents the following system of differential equations: $\frac{\delta X_r}{\delta t} = f(X_r, Y_r) + \mu(X_{r+1} -2X_r + X_{r-1})$ $\frac{\delta Y_r}{\delta t} = g(...
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96 views

How far has a chasing wasp flown as her target walks around a square?

I take a walk each morning along the sides of a square; each side is one mile. I start at one corner and walk at a constant speed. As I start on the walk, an unfriendly wasp always starts at the ...
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349 views

The origin of the name homological equation

Let $$\dot{y} = Ay + \cdots \, ,$$ where the dots represent higher order terms in $y$. Make the change of variables $y \mapsto x - h(y)$, where $h$ is a vector valued polynomial of order $r$ in $y$....
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154 views

Asymptotics for solutions of a version of Lienard's differential equation

Consider the second order differential equation $ x'' + f(x)x' + g(x) = 0 $ with $$ f(x) = -\lambda + x^2, \quad g(x) = (-1 + x^2)x \, . $$ with $\lambda > 0$. Note: The original post had a ...
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559 views

What is meant by a “hyperbolic periodic orbit”?

At the end of the Wikipedia article on "hyperbolic sets" (https://en.wikipedia.org/wiki/Hyperbolic_set) there is a reference to a periodic orbit being "hyperbolic", i.e. a periodic orbit of a ...
7
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303 views

Two Matlab ODE solvers, two different results

I am solving a system of ODEs using Matlab. One particular set of parameters caused the solver to fail, so I worked my way through the different solvers Matlab provides. I was surprised to find that ...

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