1
vote
1answer
65 views

Losing all races by the same margin of time

Suppose two cars are racing along a (straight) road at a constant speed $v_{0}$ m/s. At time $t = 0$, Car 2 is ahead of Car 1 by $d_{0}$ meters; or, one could say, Car 1 is losing by $d_{0} / v_{0}$ ...
0
votes
2answers
50 views

Solving a differential equation numerically to plot particle path

I'm trying to plot the evolution of a particle in an accretion disk by solving the equation $$2X\frac{\partial X}{\partial\tau}=V_R(X,\tau)$$ where I have found $V_R$ numerically to be ...
7
votes
1answer
102 views

Derivation of Schrödinger's equation

I recall a famous quote of the late physicist Richard Feynman: Where did we get that from? It's not possible to derive it from anything you know. It came out of the mind of Schrödinger. This ...
3
votes
1answer
42 views

How to use the Mehler kernel to get the solution of the Quantum harmonic oscillator with a given initial condition

In this wiki-article http://en.wikipedia.org/wiki/Mehler_kernel the fundamental solution of the differential equation for the Quantum harmonic oscillator is provided by the Mehler Kernel: ...
1
vote
0answers
21 views

Mathieu equation solution with non-periodic boundary conditions

I need to solve the Mathieu equation: $y''(x)+(a-2q \cos(2x)) y(x) = 0$ but with the unusal boundary condition: $y(x+\pi) = e^{i \alpha}y(x) \quad , \quad \alpha \in R$ if $\alpha = 0$ than the ...
0
votes
0answers
54 views

How do you solve this differential equation? $\tfrac{dx}{dz} = i (M x)$

How do you solve this differential equation : $\tfrac{dx}{ dz} = i (M x)$ where $M$ is a tridiagonal matrix with elements $100$. That is, $M$ is an array with $100$ elements in triagonal form, ...
2
votes
0answers
85 views

Computation of the Frenet-Serret trihedron in $\Bbb L^3$ (Lorentz-Minkowski space)

Consider $\Bbb L^3 = (\Bbb R^3, \langle , \rangle)$, with the convention $$\langle (x_1,y_1,z_1), (x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ and $\| v \| = \sqrt{|\langle v, v \rangle|}$. Let ...
1
vote
1answer
65 views

Calculating a double pendulum

consider the following situation of a double pendulum. We found the moving equations as $$ \ddot{\theta_1}=-L_1\sin\theta_1 + \frac{m_2}{m_1}\cos\theta_2\sin(\theta_2-\theta_1),\\ ...
0
votes
0answers
71 views

Solution to second order differential equation Quantum harmonic oscillator

Hi I am reviewing some quantum mechanics and and have come across a solution to a differential equation that I do not understand in the derivation of the quantum harmonic oscillator. the Schrodinger ...
0
votes
2answers
31 views

Inquiry about operator algebra

I've just began studying some quantum mechanics, and I'm not sure why certain rules in operator algebra are correct. For instance, in this book it is stated that ...
1
vote
1answer
65 views

The mathematics of anyons

I've recently learnt about particles called anyons which exist within a two dimensional framework. Which I find quite strange since we, well live in a three dimensional world. I've also found out that ...
1
vote
0answers
42 views

Eigenvalues for the double-well potential

I am trying to find eigen-values, $E$, for the following differential operator: $$\left[ -\frac{1}{2}\frac{d^2}{dx^2} +L\left(x^2-a^2\right)^2\right]y(x) = E\,y(x) $$ where $L,a$ are two positive ...
3
votes
0answers
29 views

Green's function the way George Green defined it

This is a curious question about the way George Green could have defined his Green's function. All the definitions I see have only Dirac-delta $\delta(x-x')$ function as their source on the RHS. But ...
1
vote
1answer
38 views

Solutions to differential equation

Let $\left\{a,\lambda\right\}\subset\mathbb{R}$. Let the following differential equation for a function $x\left(t\right)\in\mathbb{R}^{\mathbb{R}}$ be given: $$ \boxed ...
1
vote
0answers
57 views

Decoupling system of two partial differential equations

If I have the following systems of PDE $$ u_t+x^2u_{xx}-\dfrac{h_1(t)}{h_0(t)}e^{-(v-u)}-\dfrac{h_0'(t)}{h_0(t)} = 0,\\ v_t-\dfrac{h_0(t)}{h_1(t)}e^{-(u-v)}-\dfrac{h_1'(t)}{h_1(t)} = 0, $$ where ...
1
vote
1answer
27 views

Wronskian Bessel Equations

I need to compute the wronskian of $J_n$ and $Y_n$ (the Bessel functions of the first and second kinds). I've been able to find in many sources that it is $$W(J_n,Y_n)=\frac{\pi}{2x}$$, but I haven't ...
0
votes
0answers
35 views

Integral-Differential Equation Modeling Banked Turn

Solve this equation for the function $y(x)$: $y' = \alpha \left(\int\sqrt{1 + y'^2} dx \right)^2$ Of course this must first be solved for $y'$ and then integrated to get $y$. The following is not ...
2
votes
0answers
29 views

Solution to the “cubic” Helmholtz equation

What is known about the solutions of the differential equation in three-dimensions $$ \nabla^2 \phi = -\kappa^2 (\phi + (1/3!)\phi^3) $$ Without the cubic term, this gives a linear operator ...
2
votes
2answers
145 views

Integral calculus question relating to particle motion

"A particle of mass m is attracted toward a fixed point 0 with a force inversely proportional to its instantaneous distance from 0. If the particle is released from rest, at distance L, from 0, find ...
6
votes
3answers
166 views

What went wrong?

Intrigued by this question, one-dimensional inverse square laws, I started to try to find an answer and came up with what follows. However, I calculated the derivatives to double check myself, and ...
0
votes
1answer
40 views

Determine the motion for all time

In the frame $F=[0,\hat{k}]$, a particle of mass $m$, whose trajectory $[0,\infty)\xrightarrow{\rm r}\mathbb{R}$ is $r=z\hat{k}$ moves in response to a force ...
3
votes
1answer
54 views

Hermite Differential Equation - Non-integer values of $\lambda$

The Hermite differential equation, given by : $$ \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + \lambda y = 0 $$ has solutions of the $$ y(x) = \mathcal{H_n(x)} $$ when $ \lambda \: \epsilon \:\mathcal{Z_+} ...
1
vote
2answers
106 views

Particle Motion

So this is a simple problem but I'm just getting stumped. The question is: A particle not connected to a spring, moving in a straight line, is subject to a retardation force of magnitude ...
3
votes
2answers
73 views

Obtaining explicit solutions of the differential equation $\left(\frac{dy}{dx}\right)^{2}=\frac{1}{ay^2+by+c}$

I'm trying to see if it is possible to obtain an explicit form of the following differential equation $$\left(\frac{dy}{dx}\right)^{2}=\frac{1}{ay^2+by+c}$$ where $a,b$ and $c\in\mathbb{R}$\{$0$}
3
votes
1answer
127 views

Lowercase delta in differential-like equation

Preface: The following question comes from an expression seen in a biophysics paper published in Nature protocols. I'm aware that in pure mathematical notation $\delta$ is never used in the context ...
0
votes
2answers
83 views

Solve separation of variables problem

Originally I had $\frac{d^2y}{dt^2}=-A e^{y/B} (\frac{dy}{dt})^2$. Using a given hint: $\frac{dx}{dy}=\frac{dx}{dt}\frac{dt}{dy}=\frac{d^2y}{dt^2}\frac{1}{x}$ and $x=\frac{dy}{dt}$ I got: ...
1
vote
1answer
91 views

A theorem about oscillation in Arnold's mathematical methods of classical mechanics

There is a theorem in page 100 of Arnold's Mathematical Methods of Classical Mechanics, which says that: If $\cfrac{dx}{dt} = f(x) = Ax + R_2(x)$, where $A = \cfrac{\partial f}{\partial x}|_{x = ...
3
votes
1answer
82 views

When is it possible to construct ladder operators for a given Hamiltonian?

It is pretty cool (in my opinion) that one can solve Schrödinger's equation for the harmonic oscillator by using ladder operators, rather than just integrating it. In particular, it is possible to ...
2
votes
2answers
102 views

closed form solution to the heat equation

Let smooth functions $f(x) , g(t)$ are given solve the heat equation on the semi infinite domain $(a,\infty) \times (0,T)$. for simplicity, we can let $a = 0$. \begin{eqnarray} &&u_t(x,t) = ...
2
votes
1answer
97 views

Choice of the First Term in Legendre Polynomials

The two solutions of the Legendre's Differential Equation obtained by series solution method are : and Now according to my textbook, for the useful polynomial for n equal to a positive integer, ...
3
votes
1answer
118 views

Boundary Value Problem with Robin condition

How to solve the problem: $\left(3\right)$ \begin{cases} u_{tt}-a^{2}u_{xx}=f\left(x,t\right)\\ u_{x}\left(0,t\right)-h_{0}u\left(0,t\right)=g_{0}\left(t\right)\\ ...
0
votes
1answer
40 views

Periodicity on System of Equations

$$ y(t) = \begin{bmatrix} cos\sqrt\omega & -sin\sqrt3\omega & 0 & 0 \\ sin\sqrt3\omega & cos\sqrt3\omega & 0 & 0 \\ 0 & 0 & cos\omega & -sin\omega \\ 0 & 0 ...
0
votes
1answer
127 views

If $f(x,y,t):= u(r) \cos ( \omega t)$, use the multivariable chain rule to obtain an ODE for $u$ from the PDE for $f$.

Let $f(x,y,t) :=u(r)\cos \omega t$, where $r= \sqrt{x^2 +y^2}$. Physics tells us the following: For $f(x,y,t)$ to describe a vibrating membrane, with $f(x,y,t)$ telling how high the mem- brane is ...
1
vote
2answers
251 views

Easy way to find the streamlines

In a textbook, this problem appears: Find the streamlines of the vector field $\mathbf{F}=(x^2+y^2)^{-1}(-y\hat{x}+x\hat{y})$. The system we need to solve, I suppose, is: ...
1
vote
1answer
69 views

Need help with boundary conditions of a differential equation.

QUESTION: A particle $A$ is moving along the $X$ axis at a constant horizontal velocity $u\hat{i}$. Another particle $B$ is moving such that its velocity vector always points towards the particle ...
5
votes
1answer
85 views

Decomposite a vector field into two parts

Let A be a region in $\mathbb R^3$, and suppose $ \vec {\mathbf F}$ is a smooth vector field on A. I was asked to show that I can write $\vec {\mathbf F}=\vec {\mathbf F_1}+\vec {\mathbf F_2}$, s.t. ...
4
votes
1answer
168 views

Nonlinear equation (oscillon) comparison

Lagrangian for a spherically-symmetric, real scalar field in d spatial dimensions, $$L=c_d \int r^{d-1}dr\left[ \frac{1}{2} \dot\phi^2 - \frac{1}{2} \left(\frac{\partial \phi}{\partial r} \right)^2 ...
3
votes
2answers
280 views

Examples of applications of Linear differential equations to physics.

I wonder which other real life applications do exist for linear differential equations, besides harmonic oscillators and pendulums. I'm looking for examples to include in a document that talks about ...
5
votes
1answer
245 views

Conditions for Unique Solution for this PDE

$$ U_{xy}+\frac{2}{x+y}\left(U_{x}-U_{y}\right)=0 $$ with the boundary conditions $$ U(x_{0},y)=k(x_{0}-y)^{3}\\ U(x,y_{0})=k(x-y_{0})^{3} $$ where $k$ is a constant given by ...
1
vote
1answer
59 views

On one representation of Green's function

The Green's function for heat equation on finite interval is well known (with Dirichlet conditions): $$ G(x,x', t) = \frac{2}{l}\sum\limits_{n=1}^{\infty} ...
9
votes
1answer
204 views

Adding small correction term to ODE solution

Let $\mathbf{r}(t) = [x(t), y(t), z(t)]$ and $\mathbf{v}(t) = \frac{d}{dt}\mathbf{r}(t)$. I'm trying to solve $$ \frac{d}{dt}\mathbf{v}=\frac{q}{m}(\mathbf{v}\times\mathbf{B}) \tag{1} $$ where $q$ and ...
3
votes
1answer
266 views

1D Green's function: from interval to infinite line

Let's consider two problems for diffusion equation. The first one: $$ u_t = a^2u_{xx},\qquad 0<x<l,\quad 0<t\leq T $$ $$ u(x,0) = \phi(x), \qquad 0 \leq x \leq l $$ \begin{equation} ...
0
votes
1answer
70 views

Green's function. Basic

Can anyone give some advice about books where I could find introductory information about Green's function. What are the methods of constructing Green's function. Actually, Green's function for 3D ...
0
votes
1answer
151 views

Finding the Extremals of a Functional J.

The functional $J$ is defined on smooth functions $y \colon [a,b] \to \mathbb{R}$ satisfying $y(a) = u$, $y(b) = v$ and is given by $$J[y]=\int_a^b \sqrt{y} \sqrt{1+(y')^2}\, dx.$$ I have found ...
1
vote
1answer
330 views

How to solve a tensor differential equation?

Essentially, How does one solve the tensorial differential equation $$\frac{dx^a}{d\tau}=A^a{}_bx^b$$ where $x^a$ is a 4-vector and $A^a{}_b$ is a $(1,1)$ tensor. The original Problem How does ...
0
votes
1answer
327 views

Legendre Polynomial as Infinite Series

We've been covering Special Functions such as Legendre Functions, Bessel Functions, and Confluent Hypergeometric Functions For: $$ f(x)=\left\{\begin{matrix} +1 & 0<x<1\\ -1 & ...
0
votes
1answer
129 views

Nonlinear Second-order ODE BVP with 4 boundary conditions

My Lagrangian comes out in this form when I impose spherical symmetry: $$ φ''(ρ)+{3\overρ} φ'(ρ)+{4μ^4\over M^2} φ(ρ)-{4μ^4\over M^4} φ^{3}(ρ)-{μ^4\over2M} ϵ=0 $$ The following boundary conditions ...
0
votes
1answer
41 views

Are there solutions when the boundary conditions are particle positions at 2 different times instead of positions and speeds at an initial time?

Is it possible to find solutions for a dynamic system when the boundary conditions are particle positions at 2 different times instead of positions and speeds at an initial time? The question is ...
5
votes
1answer
91 views

Is any Newton equation an Euler-Lagrange equation?

Let $$ r'' = \mathrm{F}(r', r)$$ be Newton equation in one variable whith $\mathrm{F}$ locally Lipschitz. Is there a function $\mathcal{L}(r',r)$ such that the Newton equation is in fact ...
1
vote
1answer
82 views

Solutions of $\varphi'' = \alpha + \beta \varphi + \gamma \varphi^2$

Is there a general method to find the solutions of the equation $$\varphi'' = \alpha + \beta \varphi + \gamma \varphi^2$$ And more generally, is there a method to find solutions of a polynomial ...