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

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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} ...
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250 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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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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696 views

Differential Equations of Infinite Order

As a physicist I was playing with some QM problem and stumbled upon an ordinary differential equation of infinite order (coefficients are polynomials) that could be cast in the form: ...
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654 views

Finding the modified Green function for the Helmholtz equation

I've been wrestling with this question for quite some time now, and the result was like 20 leaves of paper packed with scribbling...anyway, here's the question: I need to find the solution to the ...
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Backgrounds of the p-Laplacian Operator

Motivation I encountered the following partial differential equation (PDE) in a mathematical paper $$\begin{array}{} u_{tt}+\Delta^2u-\nabla\cdot\left(|\nabla u|^{p-2}\nabla u\right)-\Delta ...
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Classification of non-hyperbolic equilibrium and its global manifolds

Given $$ \begin{cases} \dot{x} = x^2\\ \dot{y} = -y\\ \dot{z} = z \end{cases} $$ Classify the type of equilibrium for the point $(0,0,0)$ when $u=0$ as well as its stability. Also, describe the ...
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Why do these Integration-by-Parts Evaluation Terms Vanish?

The Associated Legendre operator is $$ L_mf = -\frac{d}{dx}\left((1-x^{2})\frac{df}{dx}\right)+\frac{m^{2}}{1-x^{2}}f, $$ where $m$ is a positive integer. For the purposes here, define ...
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43 views

No local optima in quantum control?

Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation: $\frac{d x(t)}{dt} = F(x,u)$ one can consider the ...
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Linear algebra machinery for differentiation of families of functions.

So I know that since differentiation is linear, for many types of functions we can represent it using linear algebra. Famous examples include polynomials, if we represent them with their coefficients ...
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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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Boundedness of the solution of class of ODEs

Let's have linear $$ \tag 1 y''(t) + b(t)y(t) = 0, \quad t \in (t_{0}, \infty ) $$ Here $|b(t)|$ is monotonically decreased function of time, and $\lim_{t\to \infty }b(t) = \frac{a}{t} \to 0$. How ...
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Assumption in PDE theory

I have an exercise in PDE theory. Let $w \in C^2(U)\cap C(\overline{U})$ where $U$ is open, bounded and connected and $c \in C(\overline{U},\mathbb{R})$ with $c(x) \le 0$ everywhere. Moreover, ...
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Constructing a metric for the tautological line bundle of $\mathbb C P^2$

I'm doing some independent reading in differential geometry, and the following is my attempt to work out the details of the construction of the tautological bundle on $\mathbb CP^2$ and the induced ...
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Effect of adding a constant to the torsion of a 3D curve

Let $\gamma$ be an arc-length parametrized curve in $\mathbb{R}^3$. Let say I add a constant to the torsion of $\gamma$ and let $\widetilde{\gamma}$ be the curve associated to the curvature of ...
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Show that any closed disk in $\mathbb{R²}$ containing a limited semi-orbit of $x' = f(x)$ necessarily contains one equilibrium point.

Show that any closed disk in $\mathbb{R²}$ containing a limited semi-orbit of $x' = f(x)$ necessarily contains one equilibrium point. Sorry for the mistakes in the translation, I am Brazilian and ...
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Is there some relationship between algebraic curves and partial differential equations that goes beyond classifying different PDE's

I ask primarily because despite not having taken that many math classes (up to two semesters of a PDE class in college), it would be very interesting if maybe we could gain intuition regarding ...
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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 ...
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How do I find Green Function for this BVP

I saw this question: Find the Green function for the problem: $$y''(x)+y(x) = h(x)$$ $$y(0)=y(\pi), y'(0)=y'(\pi)$$ My attempt: First I should consider the homogeneous case, in that case: $y''=0 ...
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clarification of a doubt over a defined result in ODE

I was going through the topic of Wronskian in ODE came up with the following result: I have a little doubt. Can we say the same if we interchange $y_1$ and $y_2$ i.e. between consecutive zeroes of ...
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Analytic approximation of $\ddot x+\gamma sign(\dot x)+x=0$

I am trying to find an analytic approximation to this non-linear differential equation. $$ \ddot x+\gamma sign(\dot x)+x=0 $$ $\gamma$ is a very small parameter. The solution I am getting is $$ ...
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Did I correctly derive the scheme for this PDE using the Crank Nicolson Method?

I'm taking an Applied Numerical Methods course this semester, and I was given the following homework problem: Basically, before I begin writing any sort of code, I would like to ensure that I have ...
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What does affine invariance mean in the context of the Newton's method?

The textbook Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (by Ascher, Mattheij, and Russell) states on page 329: [W]e observe that Newton's method is affine ...
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Is there an analytic solution for this Fokker-Planck equation?

The Fokker-Planck equation for a probability distribution $P(\theta,t)$: \begin{align} \frac{\partial P(\theta,t)}{\partial ...
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How can one prove the existence and uniqueness of solutions to linear differential equations?

It is a theorem (I think) that the equation: $$\mathbf{x}'(t) = A(t)\mathbf{x}(t) + \mathbf{b}(t); \qquad \qquad \mathbf{x}(t_0) = \mathbf{x}_0$$ Has a unique global solution for any matrix ...
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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 ...
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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) = ...
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102 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 ...
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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 ...
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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 ...
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Coefficients of spherical solution to Laplace's equation with difficult Robin boundary conditions

I'm trying to solve Laplace's equation in an (axisymmetric) external spherical domain. The controlling equation is: $$\nabla^2 f = 0$$ $f$ must dissappear at infinity, and at the surface of the ...
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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 $$ ...
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Arnold ODE Problem

Problem 1 of Section 1.2.4 of Arnold's ODE book asks, "Can the integral curves of a smooth (continuously differentiable) equation $\frac{dx}{dt} = v(x)$ approach each other faster than exponentially ...
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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 ...
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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} ...
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Confusion about superposition principle of the PDE and Boundary Condition of an ODE.

I want to solve a PDE like this: $\frac{\partial y}{\partial t}=a\frac{\partial ^2y}{\partial x^2}-b\frac{\partial y}{\partial x}-c y,(a,b,c\in \mathbb{R})\tag{1}$ with the boundary conditions: $ ...
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What allows us to break up $dy/dx$ in solving a separable differential equation?

Suppose you had a separable first order differential equation that can be written as $$ \frac{dy}{dx}=f(x,y)=g(x)h(y) $$ Rigourously, what allows us to rearrange this as $$ \frac{1}{h(y)}dy=g(x)dx? ...
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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.
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A second order differential equation

How does one solve the following differential equation $y^{"}+xy^{'}+(1-x^2)y=y\sin x$? I don't know how to proceed?
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Solution for $\frac{a}{x} = \int_0^1 \frac{f(z)}{\left(f(x)+f(z)\right)^2} dz$

I am looking for the function $f(x)$ that solves $\frac{a}{x} = \int_0^1 \frac{f(z)}{\left(f(x)+f(z)\right)^2} dz$ such that $f(0)=0$. Even hints how to approach to this question would be very ...
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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 ...
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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?
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Fundamental Solution of a Nonlinear ODE (using Riccati Transformation & Wronskian)

I am given the differential equation: \begin{equation*} y^{\prime}(t) = y(t)^{2} + 2\sin(t)\cos(t) - \sin^{4}(t) \end{equation*} and one solution $y_{1}(t) = \sin^{2}(t)$. I wish to find a second ...
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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 ...
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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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Determining whether the origin is an attracting fixed point for a scalar system

I have been asked to determine and prove the attraction properties of a continuous-time dynamical system, generated by the ODE \begin{equation} \frac{dx}{dt} =-x \end{equation} which gives the system ...
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Quadratic system of ODEs

I have a quadratic ODE system that looks like this: $\dot{x}=Ax+diag(x)Nx$ where $x \in R^n$ and $A,N \in R^{n \times n}$ and $diag(x) \in R^{n \times n}$ is a diagonal matrix in which $x$ is its ...
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Prove that all the solutions of (2): $\frac{dy}{dt}=A(t)y+f(t)$ are bounded in $ \left[t_0,+\infty \right )$

I have a problem: Assume that system (1): $$\dfrac{dx}{dt}=A(t)x$$ is stable, where $A(t) \in C\left [t_0,+\infty \right )$, when $t \to \infty$ and $$\begin{cases} & \mathrm{ } ...
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Can Fredholm integral equation of the first type be represented as a differential equation?

Can Fredholm integral equation of the first type be represented as a differential equation? In other words, given a Fredholm integral equation of the second type does there exist a differential ...
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120 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 ...