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

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52
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585 views

Solving Special Function Equations Using Lie Symmetries

The Lie group and representation theory approach to special functions, and how they solve the ODEs arising in physics is absolutely amazing. I've given an example of its power below on Bessel's ...
34
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0answers
402 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 ...
15
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313 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 ...
12
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332 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 ...
12
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260 views

Kähler Geodesics

Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric $$\begin{pmatrix} ...
11
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285 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 ...
10
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137 views

Regularity of PDEs, general question (Example: Kolmogorov equation)

Let us consider a differential operator $L$ and the PDE $$Lu=0$$ with some boundary conditions (assume Neumann conditions). For example, Kolmogorov's equation $$(1) \quad Lu(t,x) = \partial_t u(t,x) + ...
10
votes
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331 views

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 ...
8
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175 views
+50

If any differential equation is given by $f''(x)+f'(x)+f^2(x) = x^2\;,$ Then $f(x)=$

If any differential equation is given by $f''(x)+f'(x)+f^2(x) = x^2\;,$ Then $f(x)=$ $\bf{My\; Try::}$ We can write above differential equation as $$e^xf''(x)+e^xf'(x)+e^x\cdot (f(x))^2 = ...
8
votes
0answers
134 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 ...
8
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0answers
403 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
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202 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
votes
0answers
225 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)= ...
7
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0answers
187 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
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1k 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 ...
6
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70 views

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 ...
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52 views

Stiff Nonlinear Differential Equations

As far as I know, the concept of stiffness is hard to define rigorously, but there are plenty of handwavy descriptions and motivating examples in the literature when it comes to linear differential ...
6
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0answers
142 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 = ...
6
votes
0answers
70 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 ...
6
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248 views

How to classify/ solve this PDE?

I am searching how to solve the PDE below but I can not seem to find a decent example online. My major did not focus much in solving PDEs so I feel very deficient. I know how to solve for the steady ...
6
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0answers
67 views

Differential equation $f'(x)=\alpha\cdot f(x-1)^\beta$

Is there a way to solve $$f'(x)=\alpha\cdot f(x-1)^\beta,$$ where $\alpha>0,\beta\neq0.$ I know that if the arguments matched, I could use separation of variables to get ...
6
votes
0answers
149 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 ...
6
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90 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 ...
6
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171 views

Rolling parabola & catenary

By rolling a rigid catenary on a straight line one obtains the locus of its center of curvature as a parabola. This is well known as the natural equation connecting arc length and radius of curvature ...
6
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0answers
86 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 ...
6
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0answers
144 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 ...
6
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0answers
427 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
235 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 ...
6
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0answers
410 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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0answers
212 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
57 views

Solution of Non-linear ODEs

I have a 3rd order non-linear differential equation. $$ y''' + k (y \cdot y''- y\,'^2+1) =0 $$ with boundary conditions $$ \lim_{x \rightarrow \infty} \frac{y}{x} =1 \\ y(0) = 0 \\ y'(0) = 0 $$ I ...
5
votes
0answers
44 views

Why does the taylor expansion of a nonlinear system of differential equations exist if it has continuous second order partial derivatives?

My textbook states that for a nonlinear autonomous system $$x^\prime = F(x,y)\qquad y^\prime = G(x,y)$$ The system is locally linear in the neighborhood of a critical point $(x_0,y_0)$ whenever ...
5
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0answers
100 views

Solution of an ODE

I am trying to solve the following ODE: $$y''(x)+\left( k_0^2-\frac{\lambda}{1+\cosh^2(ax)}\right)y(x)=0 \qquad k_0,\lambda,a>0$$ when as $x \rightarrow \infty$ the solution is of the form ...
5
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0answers
77 views

Referral of a Textbook or Book that teaches Intuition, focusing on Calculus.

I was wondering if there is a book out there that doesn't teach you how to do calculus, but teaches you how to apply it in the physical or social sciences. I know calculus, integration and ...
5
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0answers
91 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 ...
5
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0answers
67 views

Solving a Matrix DE involving the KL divergence

If we let $U_\mu$ be a vector field that associates a direction vector $U_\mu(\pi)$ with each $\pi \in $ unit simplex. Each such vector field is associated with a system of ODEs: $$ \pi'(u) = ...
5
votes
0answers
53 views

Differential equation with shifited term

I have a differential equation (Or integral equation) of the form: $$ f(x) = a e^{-x} + b \int_0^x f(cz+dx) e^{-z} dz$$ $a,b,c,d$ are constants. I am considering whether the above equation has a ...
5
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0answers
113 views

Examples on conceptual problems for eigenvalues in differential equations

I am currently holding a discussion class on diff eqs for engineers and I am looking for an interesting conceptual problem on eigenvalues in diff eqs. Most of the problems in 5 different books that I ...
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344 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
63 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
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0answers
69 views

How to solve this complicated ordinary differential equation?

Consider the following non-linear ODE: $$x^2 \frac{dy}{dx} + \exp{\left(x \, \frac{d^2 y}{dx^2} \right)} = \sin \left(\frac{d^3y}{dx^3} \, \cos \left( \frac{d^4y}{dx^4} \right) \right) $$ I have no ...
5
votes
0answers
134 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
90 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.
5
votes
0answers
238 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
79 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 ...
5
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0answers
113 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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128 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
86 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
139 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
73 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] $y(0) = 0$; $t_{0}=0$; $\alpha$, ...