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

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25
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218 views

Solving Special Function Equations Using Lie Symmetries

The lie group + representation theory approach to special functions & how they solve the ode's arising in physics is absolutely amazing. I've given an example of it's power below on Bessel's ...
19
votes
0answers
172 views

Geometric & Intuitive Meaning of $SL(2,R)$, $SU(2)$, etc… & Representation Theory of Special Functions

Many special functions of mathematical physics can be understood from the point of view of the representation theory of lie groups. An example of the power of this viewpoint is given in my question ...
16
votes
0answers
241 views

Solution of 2nd order linear ODE with regular singular points, and complex exponents at singularity

The steady state temperature distribution of a rod given by: \begin{equation} \frac{\textrm{d}p(x)y'}{\textrm{d}x} - y = 0,\; 0 \leq x \leq 1,\; \text{and} \;y(0) = 0, \end{equation} ...
12
votes
0answers
202 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 ...
11
votes
0answers
286 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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0answers
226 views

Kähler Geodesics

Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric $$\begin{pmatrix} ...
10
votes
0answers
199 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 ...
9
votes
0answers
198 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 ...
8
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0answers
273 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 $V(x)$ ...
8
votes
0answers
165 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. ...
6
votes
0answers
131 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)= ...
6
votes
0answers
101 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
votes
0answers
210 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 ...
6
votes
0answers
320 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 : ...
6
votes
0answers
141 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
votes
0answers
207 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
votes
0answers
215 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 ...
6
votes
0answers
399 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
votes
0answers
204 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 ...
6
votes
0answers
81 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 ...
6
votes
0answers
284 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 ...
5
votes
0answers
229 views

For what values of t, the solution for this equation exist

I need help in finding maximal solution for the problem: $$ \cases {{\dot{x} = x^2+t}\\{x(0)=0}}$$ I know that because $x(1) \geq \frac{1}{2}$ and that every solution $x(t)$ of the problem is greater ...
5
votes
0answers
282 views

Finding the inverse of a function.

Let $f:\mathbb{R}\to \mathbb{R}_+$ with $f\geq\epsilon>0$ be smooth and define $G:\mathbb{R}\to\mathbb{R}$ thus $$G(x):=\int_0^x\frac{1}{f(u)}\mathrm{d}u$$ Then it is clear that $G$ is ...
5
votes
0answers
67 views

Introduction to differential equations for pure mathematicians

Is there a good reference for learning about differential equations for people who are mainly interested in the theoretical tools (especially in differential geometry/topology) that use them? I ...
5
votes
0answers
42 views

Vector Laplace equation with constraint

I want to solve Laplace equation for a vector $\boldsymbol v=(v_x,v_y)$: $$\nabla^2 \boldsymbol{v}=0$$ but under the constraint that $$(1+v_x)^2+v_y^2=1$$ which becomes $v_y = -(2v_x+v_x^2)^{1/2}$. ...
5
votes
0answers
88 views

Boundary Layer, leading order, Pertubation Theory, Differential Equations

I have got the following problem, taken from Multiple Scale and singular perturbation methods, Kevorkian & Cole book, page 94, exercise 1.b.: Find the leading order of the problem: $\varepsilon ...
5
votes
0answers
63 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 ...
5
votes
0answers
157 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 ...
5
votes
0answers
104 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
votes
0answers
120 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
votes
0answers
96 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
0answers
77 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 ...
5
votes
0answers
105 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
197 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
0answers
66 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
votes
0answers
838 views

Questions about the Picard–Lindelöf theorem for an ODE

In the sketch proof of Picard–Lindelöf existence theorem for an ODE: It can then be shown, by using the Banach fixed point theorem, that the sequence of "Picard iterates" $φ_k$ is convergent and ...
5
votes
0answers
132 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
votes
0answers
195 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
5
votes
0answers
110 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}} + ...
4
votes
0answers
56 views

Why does $\frac{1}{r}\frac{dr}{d\theta} = \cot \psi$?

In the discussion of linear fractional equations in Birkhoff and Rota's Ordinary Differential Equations, the authors assert that if we convert a DE of the form $y' = F\left(\frac{y}{x}\right)$ to ...
4
votes
0answers
65 views

Polar representation of conic sections $r(\theta)=\frac1{1 + e \cos\theta}$

Consider a curve given in polar coordinates by $r(\theta) = \dfrac1{1 + e \cos\theta}$, where $e\ge0$. a) Show that the distance of each point on this curve to the line $x=\frac1e$ is a constant ...
4
votes
0answers
60 views

Intuition behind variation of parameters method for solving differential equations

I have used the variation of parameters method (and have been taught it, although not hugely in depth) and I was wondering if I've understood the intuition behind it. In particular I've been thinking ...
4
votes
0answers
58 views

To show a given function is not the viscosity solution.

For the equation $ F(x,u,u',u'') = -au''-1 =0$ for $ x\in (0,2)$ with $ u(0) = 0 = u(2) $ and $a(x)$ is $1$ for $x\in (0,1)$ and $2$ for $x\in [1,2)$. Need to show that the function $$ u(x) = ...
4
votes
0answers
81 views

Green's function in a moving frame for a constant heat source

I am looking for the Green's function of the problem in two dimensions $r =(x,z)$, \begin{equation} \nabla^2g + \frac{v}{D}\frac{\partial g}{\partial z} = -\delta (r-r_0) \end{equation} Which ...
4
votes
0answers
45 views

Solution techniques for f'(x)=f(g(x))

I stumbled over this seemingly natural question and was surprised, that I couldn't find a satisfying answer. Differential equations of the type $f'(x)=g(f(x))$ are studied for all kind of classes of ...
4
votes
0answers
39 views

How to show that a leaf is topologically a cone.

I am trying to understand the topological behaviour of foliations around irreducible singularities, specially in the case of singularities in the Poincaré domain. I am using the third chapter of this ...
4
votes
0answers
61 views

Quasilinear second order ODE

Consider a smooth $u\colon\mathbb{R}\rightarrow\mathbb{R}$ satisfying $$ u^{\prime\prime}+a\left(u^{\prime}\right)^{2}+bu=0\text{ on }\mathbb{R} $$ with $$ ...
4
votes
0answers
33 views

Solving $y^{(n)}(t)=f(t); t>0$ with initial conditions

I will use the notation $\frac{d^n y}{dt^n} \equiv y^{(n)}$. How do I solve this ODE? $$y^{(n)}(t)=f(t); t>0;\\ y(0)=y_0, y'(0)=y_1, ..., y^{(n-1)}(0)=y_{n-1}$$ What I did: The ODE is in the ...
4
votes
0answers
71 views

Question about solutions of $x''+(1+r(t))x=0$ when $\int_1^\infty |r(t)| dx <\infty$ .

Let $x''+(1+r(t))x=0$ where $r(t)$ is continous and $\int_1^\infty |r(t)| dx <\infty$ show that the equation has solutions $\phi_1$ and $\phi_2$ such that $$\lim_{t\to\infty} ...
4
votes
0answers
37 views

Help with an ODE

I need some help, I have this ODE but can't solve it for $y(x)$, I try every method I know, but with no succes,please, somebody can help me? $(\varepsilon-x)y=y'(-x+y^2-2x^2)$ Thanks.