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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219 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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175 views

Kähler Geodesics

Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric $$\begin{pmatrix} ...
10
votes
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164 views

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 ...
9
votes
0answers
358 views

Osgood condition

Let $h$ and $g$ be continuous, non-decreasing and concave functions in the interval $[0,\infty)$ with $h(0)=g(0)=0$ and $h(x)>0$ and $g(x)>0$ for $x>0$ such that both satisfy the Osgood ...
7
votes
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101 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 ...
7
votes
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388 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 Liyapunov) I don't ...
7
votes
0answers
116 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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123 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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104 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
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166 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
0answers
85 views

Function whose inverse is also its derivative?

What are some good examples of a function $f : \Re \to \Re$ where its derivative is equal to its inverse? I attempted to find a monomial that satisfied it by starting with $f(x) = ax^b$ and showing ...
5
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74 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 ...
5
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58 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. ...
5
votes
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87 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
votes
0answers
136 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
votes
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48 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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154 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 ...
5
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154 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
votes
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326 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, ...
5
votes
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71 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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100 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}} + ...
5
votes
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231 views

ODE Laplace Transform an impulse bring oscillating system to rest

$2y''+y'+2y=\delta(t-5)$ $y(0)=0, y'(0)=0$ Consider the system given by ODE above in which an oscillation is excited by a unit impulse at $t=5$. Suppose that it is desired to bring the system to ...
5
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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
votes
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92 views

Euler-Lagrange Equation example

I have been working on solving Euler-Lagrange Equation problems in differential equations, specifically in Calculus of Variations, but this one example has me stuck. I am probably making mistakes in ...
4
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0answers
69 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, ...
4
votes
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57 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
votes
0answers
35 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
0answers
112 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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105 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} ...
4
votes
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144 views

Nice corollaries to Poincaré-Bendixson theorem

I am interested in the applications of Poincaré-Bendixson theorem not (explicitely) related to differential equations. For convenience, recall the theorem: Let $X : \mathbb{R}^2 \to \mathbb{R}^2$ be ...
4
votes
0answers
114 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 ...
4
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186 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
4
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207 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 ...
4
votes
0answers
331 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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0answers
219 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
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0answers
203 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
votes
0answers
158 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
86 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
31 views

Cauchy-Euler Equation of order $n$

What I wish to prove is that for a Cauchy-Euler equation of order $n$, the substitution $x=e^{t}$ transforms it into a linear differential equation with constant coefficients. To put it as a theorem: ...
3
votes
0answers
21 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 ...
3
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42 views

Differential Equation has a unique solution periodic

Let $A(t)$ continuous and periodic of period $S$ in $\Re$. Suppose $x' = Ax$ has $\varphi \equiv 0$ as the only periodic solution of period $S$. Show that there exists $\delta> 0$ such that for ...
3
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23 views

Invertible e converges series.

If T is a linear transformation on $R^n$ with $||T - I||<1$, prove that $T$ is invertible and that the series $\sum_{k=0}^\infty(I-T)^k$ converges absolutely to $T^{-1}.$ (Use the geometric series) ...
3
votes
0answers
39 views

WKB and asymptotic behavior of second order differential equation

I want to study the large $x$ solution to a Riccati equation. After listening to the lectures on Mathematical Physics by Carl Bender, I have fallen in love with asymptotic analysis. But, by no means ...
3
votes
0answers
40 views

Are any solutions lost when solving non-exact differential equations?

I have just started studying differential equations, one of the problems I found while I was practicing is "Consider the equation $$ (y^2 + 2xy)dx - x^2 dy=0 $$ (a) Show that this equation is not ...
3
votes
0answers
111 views

Proving Nonhomogeneous ODE is Bounded

I am trying to prove the following: Given a solution $x(t)$ of the IVP $\dot x=A(t)x+h(t)$, where $A(t), h(t)$ are continuous on $0<t<\infty$, prove that x(t) is bounded for $t\ge1$ if both ...
3
votes
0answers
63 views

Laplace Transform of the Wave Equation

I am given a damped wave equation $u_{tt}(t,x)+2u_t(t,x)=u_{xx}(t,x); \forall t>0$ Now I know the laplace transform of this given the initial conditions, $u(0,x)=\sin x, u_t(0,x)=0;$ is ...
3
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0answers
197 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 : ...
3
votes
0answers
39 views

Solve ODE by Fourier transform, and versus by Laplace transform?

Regarding solving ODE by Fourier transform, I read a nice reply by O.L.. After applying Fourier transform to an ODE to obtain an algebraic equation, the reply showed that some terms involving the ...
3
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0answers
24 views

Show that this initial-value problem has a unique solution

I am trying to show that the following initial-value problem $$\frac{dx}{dt} = - x + tx^{1/2}; \quad x(2) = 2$$ has a unique solution on $I = [2,3]$. By letting $f(t,x) = - x + tx^{1/2}$ and $(t_0 ...
3
votes
0answers
42 views

How to integrate/differentiate parameters in differential equations

I have an ODE, which I would be fine about solving, were it not for the parameter: $$(\omega^2+x^2)\frac{dy}{dx}=y$$ I'm given that $\omega>0$ is a parameter. Separating the variables gives: ...