Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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11
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235 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 ...
11
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193 views

Kähler Geodesics

Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric $$\begin{pmatrix} ...
8
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103 views

Notions of stability for differential equations

Consider a system of differential equations $$\dot{x} = f(x,u)$$ $$y = h(x,u)$$ where $x(t), u(t)$ are vectors in some $\mathbb{R}^n$. We define the infinity norm of a function in more-or-less in the ...
8
votes
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435 views

Complicated exercise on ODE

I have this exercise extracted from a examination of qualitative theory of ODE (in which we study the existence and uniqueness of solutions, and stability using the function of Lyapunov) I don't know ...
7
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98 views

Reference request: a differential equation arising in geometry

$$ \frac{d\beta}{d\alpha} = \frac {\sin\beta}{\sin\alpha} $$ In what contexts (if any) is this equation known to occur?
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130 views

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. ...
7
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142 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 ...
6
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59 views

Proving that two functions involving integrals with Legendre polynomials are equal

I have two functions that I expect to be equal (where $P_{2l}$ are the even Legendre Polynomials): $$F_{2l}(x)=x\, \tanh(\pi x/2)\left|\int_0^1 u^{i x-1}P_{2l}(u)\,du\right|^2$$ ...
6
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122 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 $V(x)$ ...
6
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116 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 ...
6
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126 views

Algorithm for obtaining the surface of a mirror

My colleague and I have been trying to implement an algorithm described in the paper "Recovering local shape of a mirror surface from reflection of a regular grid", primary author of which being ...
6
votes
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172 views

Non-linear second order DE, with no x term in it

Okay, I have a second order non linear de, which has no term containing the variable x. assuming $$ y = f(x) $$ , the equation is $$ y'' - Ay' = \cos{y} - B\sin{y} $$ I tried solving it by ...
6
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374 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, ...
6
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177 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 ...
5
votes
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88 views

Solve PDE by getting two ODEs

My goal is to solve this PDE for $f:[-1,1] \times \mathbb{R}_{\ge 0}\rightarrow \mathbb{C}$ $$ \partial_t f(x,t) = -\partial_x^2 f(x,t) + g(t)V(x)f(x,t).$$ I would consider this PDE to be solved if ...
5
votes
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77 views

general solution of the equation $\frac{dy}{dx} =\exp(y/x)$

How can i get the general solution of the equation a) $\frac{dy}{dx} = \exp(y/x)$ b) $\frac{dy}{dx} = \exp(x-y)$ and $y=2$ when $x = 0$ I tried b) first: This is a first-order nonlinear ordinary ...
5
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161 views

Propper algorithm to integrate ODE's numerically.

I have studied in a course several algorithms to integrate ODE's numerical: Runge-Kutta, Predictor-Corrector methods, Taylor... However the teacher failed to show which is the best for every ...
5
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172 views

The Stable Manifold Theorem Applications

Definition: Let $\phi_t(x)$ be the flow of the nonlinear system $x'=f(x)$. The global stable manifold of $x'=f(x)$ at $0$ is defined by: $$W^s(0)=\bigcup_{t\leq 0}\phi_t(S)$$ Where $S$ is a ...
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81 views

Examples of potentials for which Schrödinger equation lacks discrete points in continuous spectrum

In Landau, Lifshitz, "Quantum Mechanics, non-relativistic theory" in $\S18$ "The fundamental properties of Schrödinger's equation" the following is said in a footnote: it must be mentioned that, ...
5
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102 views

Uniqueness solutions of $dx/dt = f^2(x) + e^{-t}$.

Someone can help me in the following problem? Is a question of Zhang. Let $f(x)$ be continuous for $x \in \mathbb{R}$, show that $dx/dt = f^2(x) + e^{-t}$ has the property of uniqueness of ...
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64 views

Earnshaw's theorem

Proposition Suppose $U\colon\Omega\to\mathbb R$ is a non-constant harmonic function, i.e. $U\in\mathcal C^\omega$, i.e. analytic, and $\Delta U=0$, where $\Omega\subseteq\mathbb R^n$ is a region. ...
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96 views

Solving numerically the equation of motion of D7 brane perturbation

I want to solve this equation $$ \partial_{\rho}^{2}\phi+\frac{3}{\rho}\partial_{\rho}\phi+\left(\frac{M^{2}}{(1+\rho^{2})^{2}}-\frac{l(l+2)}{\rho^{2}}\right)\phi=0 $$ numerically. I know that ...
5
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153 views

Stability for Nonlinear System

I am trying to assess the (Liapunov) stability of the equilibrium at $(0,0)$ of the system \begin{align*} x_1' &= -4x_2 + x_1^2 \\ x_2' &= 4x_1 + x_2^2. \end{align*} I plotted the phase ...
5
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55 views

Separable non-linear ODE (with radicals)

I am trying to solve the equation $$ \frac{dy}{dt}=\sqrt{(\gamma-1+\frac{2\alpha\beta}{2\alpha-1})e^{-2\alpha y}-\frac{2\alpha\beta}{2\alpha-1}e^{-y}+1} $$ [1] $y(0) = 0$; $t_{0}=0$; $\alpha$, ...
5
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167 views

Bifurcation of integral curves

Consider the following first order ODE: $$\frac{\operatorname{d}\!y}{\operatorname{d}\!x} = x^2 - y^2$$ Despite the fact that this ODE has a very simple expression, it is not solvable in terms of ...
5
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123 views

Clarification in a paper

This is regarding a clarification in page 384 of a paper published in Annals of Statistics by Amari. In page no. 384, he defines $$R_i(t)=\frac{\partial}{\partial \theta_i} ...
5
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74 views

Solution to differential equation $f^{(n)}-(n+1)f^{(n-1)}-(n+1)nf^{(n-2)}-\dotsc-(n+1)!f=g$

Let $n$ be a given positive integer and $g$ be a continuous function. We are looking for a function $f \in C^n(\mathbb{R})$ such that $$f^{(n)}-(n+1)f^{(n-1)}-(n+1)nf^{(n-2)}-\dotsc-(n+1)!f=g.$$ It ...
5
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104 views

Uniqueness result in linear differential equation of degree $n$.

Suppose that $f$ is such that $$f^{(n)}=\sum_{j=0}^{n-1}a_jf^{(j)}$$ Some little work is needed to get to ($a_j=0$ if $j<0$) $${f^{(n + 1)}} = \sum\limits_{j = 0}^{n - 1} {\left( {{a_{j - 1}} + ...
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1k views

Restricted Three-Body Problem

The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity ...
4
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0answers
68 views

How to solve $\frac{dy}{dx} = \frac{x^2-y^2}{x^2(y^2+1)}$

I tried to solve this using the solution of a first order differential equation but I don't think this can be reduced to that form. How to approach this problem and find $y$? Please help.
4
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49 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 ...
4
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0answers
97 views

Operators such that $\langle Ax,x \rangle=-\langle x,Ax \rangle$

Let $X$ be a Banach space. We consider the differential equation: $$x'(t)=Ax(t), \ \ \ t\in\mathbb{R}$$ where $A$ is a bounded operator on $X$. If $X$ is a Hilbert space, and $x(t)$ is a solution of ...
4
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0answers
91 views

What differential equation might model this almost-harmonic oscillator?

I need to precisely control the motion of a damped, driven (nearly) harmonic oscillator: $$ \ddot x(t) + \alpha\dot x(t) + \omega_0^2 x(t) \approx V(t) $$ I use the $\approx$ symbol because this is ...
4
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66 views

Solving second order nonlinear ODE

Having the following second order ordinary differential equation: $$ \ddot{x} = a \cos(x) $$ where, $a$ is a constant. What's an approach to solve this kind of equation?
4
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62 views

$y=0$ is the differential equation $\frac{dy}{dx}=E(y)$ singular solution ,equivalent improper integral $\int_{0}^{1}\frac{dy}{E(y)}$ is convergence

Assmue that continuous function $E(y)$ such $$E(0)=0,E(y)\neq 0,0<y\le 1$$ $y=0$ is differential equation $$\dfrac{dy}{dx}=E(y)$$signular solution, iff and only if the improper integral ...
4
votes
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30 views

Wronskian different from zero and solutions of ODE.

Let $a_0, \ldots , a_{n-1}$ continuous functions in an interval $I$.Consider the equation $$x^{(n)} = a_{n-1}(t)x^{(n-1)}+\cdots+a_0(t)x. \tag 1$$ Let $\phi_1, \phi_2, \ldots,\phi_n$ $n$ are ...
4
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0answers
71 views

Scalar Autonomous Differential Equation?

What precisely is a scalar autonomous differential equation? I'm confused about what this precisely means, more so because we did not discuss this in any lectures nor is it, as far as I can tell, ...
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228 views

Modelling a Water Rocket. Requires Some Validation and Help. ( WARNING : Extremely Long but Interesting Post )

Good day people of math.stackexchange.com This is a pet project that I plan to use to convince my Prof that I would rather try something similar to this than to do the prescribed project. Edit : ...
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66 views

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 ...
4
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0answers
36 views

One-parameter group for spheres

Let $S^1$ be the unit sphere $x_1^2+x_2^2=1$ in $\mathbb{R}^2$ and let $X=S^1\times S^1\in\mathbb{R}^4$ with defining equations $f_1=x_1^2+x_2^2-1=0, f_2=x_3^2+x_4^2-1=0$. The vector field ...
4
votes
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114 views

Help me understand this differential equation solution

I found a differential equation in an old paper, where the solution is a bit hard to understand. Given this equation: $$\frac{1}{2} r^2 \left(\frac{d \phi}{dr}\right)^2 + c^2 \left(r ...
4
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138 views

Heat Kernel Asymptotics on Manifold with Boundary

On a closed Riemannian manifold $M$, the heat kernel $k_t(x, y)$ of the Laplace-Beltrami operator (or more general of any generalized symmetric Laplace-type operator acting on sections of a vector ...
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189 views

How to solve this differential equation for $y$ in terms of $x$ and $k$

$$yy'+\frac yx+k=0$$ How to solve this differential equation for $y$ in terms of $x$ and $k$ where $k$ is a parameter of $x$? $y(x)=y$ is a function and $x(k)=x$ is a gamma function
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238 views

Solving inhomogenous bessel equation

I have the following differential equation to be solved $\dfrac{d^2\psi}{dr^2}+\dfrac{d\psi}{rdr}+4\left(\omega^2-k_0^2-\dfrac{n^2}{r^2}\right)\psi=AJ_n^2(kr)+\dfrac{k}{r}J_n(kr)J_{n+1}(kr)-\omega ...
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0answers
354 views

Confused by a proof in Rudin *Functional Analysis*

I am reading Rudin's Functional Analysis and got quite confused by his proof of Thm 8.5, that is, the existence of fundamental solutions for differential operator $P(D)$, where $P$ is a polynomial. ...
4
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225 views

A solution of $-y'' + q(x)y= \lambda y$

Could you help me with the following problem (from Poschel and Trubowitz)? I am looking for a solution of the differential equation $-y'' + q(x)y= \lambda y$, for $0 \leq x \leq 1$ with ...
4
votes
0answers
220 views

How to solve a differential equation associated with square wheels?

I'm looking for a general solution for $f(t)$ given an unrelated function $g(t)$ in $$f(t)^2 - 2g(t)f(t)\sin(t) - 2f'(t) + g(t)^2 - 2g(t)\cos(t) + 1 = 0$$ Is it possible to solve without knowing ...
4
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0answers
161 views

How to analysis the stability of these ODE?

Study whether the null solution of the system: $$\begin{cases} \frac{dx_1}{dt}=x_2(t)\\ \frac{dx_2}{dt}=-w(t)^2 x_1(t)\\ \end{cases} $$ is Lyapunov stable, where $$ w(t)= \begin{cases} 0.4 ...
4
votes
0answers
88 views

Efficiently solving a large, sparse linear system $M(s)ab(s)=c(s)$ (determined by smooth functions) over some range of $s$

I'm looking at a differential equation on the edges of a graph (the application is neuroscience), and the Laplace transform of the solution on most of the edges has a general solution more-or-less of ...
3
votes
0answers
60 views

Ordinary differential equation­

$$\dfrac{dy}{dx}-\dfrac{\tan y}{1+x}=(1+x)e^x\sin y$$ I tried $\sin y=t$ but failed. It seems to immune to methods I know of or I am just unable to make the right substitution... Wolfram alpha ...