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Questions tagged [ordinary-differential-equations]

For questions about ordinary differential equations, which are differential equations containing only derivatives w.r.t. one variable. For questions specifically concerning partial differential equations, use the [tag:pde] instead.

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975 views

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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329 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 ...
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391 views

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)), \...
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292 views

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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207 views

$\tau$ structure of the sixth Painlevé equation

I am studying the isomonodromic deformations theory, which leads in the case of a $\mathcal{C}_{0,4}$ Riemann surface to the sixth Painlevé equation. I read that this equation had a $\tau$-structure,...
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839 views

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 $V(x)$ ...
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389 views

Fabius function and equivalent

The Fabius function $F$ can be defined on $[0,1]$ by $F(0)=0$ $F(1)=1$ on $[0,\frac{1}{2}]$ $F'(x)=2.F(2x)$ on $[\frac{1}{2},1]$ $F'(x)=2.F(2(1-x))$ It's a known example of a not analytic $C^\...
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Can (linear) differential equations of infinite order be recast into equations of first order?

In most analysis courses one sees that differential equations of order $n$ are basically a subset of higher dimensional differential equations of order $1$, for example the equation: $$f^{(n)}(t)=F\...
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251 views

The diferential equation $y' = \frac{\ln(x^2+y^2)}{x^2 + y^2}$

In my University, the integral calculus teacher gave me this diferential equation to solve. $$ y' = \frac{\ln(x^2+y^2)}{x^2 + y^2} $$ I dont have any clue of what form has the solution of this ...
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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\...
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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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114 views

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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250 views

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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402 views

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}...
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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 ...
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108 views

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 -...
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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{...
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137 views

Solving a matrix differential equation

I am trying to solve: $\frac{d U_t}{dt} = Tr(G^{\dagger}U_t)G - Tr(U_t^{\dagger}G)U_t G^{\dagger} U_t$ Where $U_t \in SU(4)$ and $G \in SU(4)$ is given and constant. Is it possible to solve this ...
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170 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 ...
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313 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 ...
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157 views

Separable non-linear ODE (with radicals)

I am trying to solve the equation $$ \frac{dy}{dt}=\sqrt{(\gamma-1+\frac{2\alpha\beta}{2\alpha-1})e^{-2\alpha y}-\frac{2\alpha\beta}{2\alpha-1}e^{-y}+1}\tag{1} $$ $y(0) = 0$; $t_{0}=0$; $\alpha$, $...
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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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176 views

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) ...
7
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68 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 ...
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147 views

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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0answers
164 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}...
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117 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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91 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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154 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&#...
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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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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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147 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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260 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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214 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 ...
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Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form $L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$ for ...
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95 views

Qualitative dependence of solution to second-order matrix differential equation on eigenvalues

Suppose we have a matrix differential equation in $\vec{x}(t)=\left(\begin{smallmatrix}x_{1}(t) \\ \vdots \\ x_{n}(t)\end{smallmatrix}\right)$, such that: $$\frac{\mathrm{d}^{2}\vec{x}}{\mathrm{d}t^{...
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499 views

Proof Strategy for a Dynamical System of Points on the Plane

I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, ...
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275 views

Hints/Help studying an Abel Differential Equation

I want to know more than qualitative information about the Abel differential equation $\frac{dy}{dx}+y^3+x=0$. $\qquad ... \;(1)$ Since I don´t know how to solve this and as far as could see, this ...
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148 views

$\Sigma$-equivalence between $(y,z,1)$ and $(y+\mathcal{O}(2),z+\mathcal{O}(2),1)$

Consider the sets $\mathfrak{X}(\mathbb{R}^3) = \{X: \mathbb{R}^3 \to \mathbb{R}^3; X \mbox{ is smooth}\}$ and $\Sigma = \{0\}\times\mathbb{R}^2$. Let $X, Y$ be vector fields in $\mathfrak{X}(\mathbb{...
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85 views

Analytic or perturbative solution in any limits?

Consider the system of 3 ordinary differential equations $$\dot{x}=v$$ $$\dot{v}=a$$ $$\dot{a}=-Aa+v^{2}-x$$ which can also be written as a single 3rd order ODE $$\dddot{x}=-A\ddot{x}+\dot{x}^{2}-...
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0answers
71 views

Non-unique solution of first order PDE

Question: $$\frac{\partial u}{\partial x} \frac{\partial u}{\partial y}=1 \qquad \qquad u=0 \; \text{ when } \; x+y=1$$ Find all possible solutions and state where each one exists. Attempt: Using ...
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44 views

Identifying ODE types for solving by hand and when to use computers instead

So this questions relates to my specific ODE but also ODEs in general. I am a big fan of solving ODEs by hand, but I also know when to give up and use, say, Mathematica to solve it for me. Having ...
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0answers
245 views

Prove that doesn't exist a Lyapunov function

Given the following ODE in polar coordinates \begin{array}{lcl} \frac{dr}{dt} = r\sin(\frac{1}{r}) \\ \frac{d\theta}{dt} = 1\end{array} 1) Show that the origin $(0,0)$ is Lyapunov stable ...
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335 views

Useful reformulation of Goldbach's conjecture?

Let us assume there exists some infinite order differential equation whose solution is: $$ y= \sum_{n=1}^\infty A_n \exp(p_n^sx) $$ Where $p_n$ is the $n$'th prime. Substituting $ y=\exp(\lambda x)$...
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78 views

Solving linear nonhomogenous system of differential equations

I have a question, concerning one special solution of the following system. First I will sum up my results, to give a context for my question. My question will be seperated by a line, if you want to ...
6
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0answers
92 views

Generalization of Liouville's formula to other coefficients of the characteristic polynomial

If $X(t)$ is an $n \times n$ matrix solving linear homogeneous ODE $$ \frac{d}{dt} X(t) = A(t)X(t), $$ then for $\det X(t)$ we have Liouville's formula: $$ \frac{d}{dt} \det X(t) = \text{tr} A(t) \det ...
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0answers
143 views

Linearization which is a Sturm-Liouville problem: Stability questions

Consider the scalar phase equation $$ \theta_t=\theta_{xx}+f(\theta),\qquad f(\theta+2\pi).\qquad (1) $$ Traveling waves profiles $\theta(x-ct)$ can be found using phase-plane analysis for $$ \theta_{\...
6
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0answers
93 views

Closed form solution to an ordinary differential equaiton

How to solve the following ordinary differential equation? $$y'(x)= \frac{C_1}{y(x)} +C_2 C_3 \cos\left(C_3 x\right) +C_4$$ where $C_1, C_2, C_3, C_4\in \mathbb{R}$ are all constants. It looks ...
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0answers
168 views

Transforming the solutions of $\dot x = f(\mathbf{x}, t)$ and $\dot x = B(\mathbf{x}, t)f(\mathbf{x}, t)$ into each other

How can I prove the following theorem? If the function $B(\mathbf{x}, t)$ is strictly positive, then the solutions of the two differential equations $\dot x = f(\mathbf{x}, t)$ and $\dot x = B(\...
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0answers
821 views

Contour Integral solution to differential equations, Euler transformation?

In Spain's book, Functions of mathematical physics he introduces the contour integral method of solving ODEs. The baseic idea is: given an ODE $\sum_0^m a_r(t) \frac {d^rf}{dt^r} = 0$, a solution may ...