1
vote
1answer
16 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
26 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
21 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
94 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 ...
0
votes
0answers
40 views

Find sufficient and necessary conditions such that the solution $u$ of this PDE is unique

Let us consider the following PDE probem: $$Δu(x,y)=\frac{\partial^2u(x,y)}{\partial x^2}+\frac{\partial^2u(x,y)}{\partial y^2}=0, (x,y)∈(0,1)\times\mathbb{R}$$ $$u\left(\frac{1}{2},y\right)=0, ...
6
votes
3answers
160 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
38 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 ...
1
vote
0answers
28 views

An algebra associated with an important function

In the paper here the authors make a claim that the Natanzon potential (an implicit potential very important in mathematical physics) follows an $SO(2,2)$ algebra. This potential defined as : $$ ...
3
votes
1answer
41 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
88 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
66 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 $c\in\mathbb{R}$\{$0$} I ...
3
votes
1answer
82 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
68 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
88 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 = ...
2
votes
1answer
63 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
91 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) = ...
1
vote
0answers
38 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
106 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
33 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
99 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
165 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
66 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
83 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
165 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
269 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
188 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
50 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
196 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
210 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
67 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
130 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 ...
0
votes
0answers
71 views

Geodesic equation for a 2D manifold

I am having trouble understanding how the following statement (taken from some old notes) is true: For a 2D manifold such that $$ds^2=\frac{1}{u^2}(-du^2+dv^2)$$ If we assume that $$\dot x^a\dot ...
1
vote
1answer
251 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
228 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
128 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
40 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
89 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
81 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 ...
4
votes
3answers
165 views

Physics notation justified

Sometimes in physics they do things like this one: If $dq=f\left(x\right)\cdot dr$ then $\frac{dq}{dt}=f\left(x\right)\cdot \frac{dr}{dt}$ Which mathematically is a wrong deduction. Is there any ...
4
votes
1answer
106 views

How do I squeeze a $\theta(t)$ and $\varphi(t)$ out of this?

A ball attached to a fixed-length massless rod swings about under gravity. Mathematically: $$L=T-U=\frac{MR^2}{2}(\sin^2(\theta)\dot{\varphi}^2+\dot{\theta}^2)+MgR \cos(\theta)$$ ...
4
votes
1answer
132 views

Existence Energy of Wave Equation

I was just going trhough some properties of the wave equation, including the energy of the wave equation given by $E(t)=\int_{-\infty}^{\infty}u_t^2+c^2u_x^2 dx$, i.e the sum of kinetic and potential ...
2
votes
0answers
172 views

How to convert a hologram into an image?

Suppose one knows in full detail the phase and intensity of monochromatic light in a plane. This is basically what a hologram records, at least for some section of a plane. By using this as the ...
1
vote
0answers
84 views

limit of a tricky multi-variable function

I am reading a paper on the sine-gordon model. The solution for a two kink solution is given as: ...
0
votes
1answer
104 views

Finding the spectrum of the Schrodinger operator

Let $H(f) = -f'' + V(x) f$ be the Schrodinger operator on $\mathbb R$. I am trying to calculate the spectrum (eigenvalues) of the operator $H$ in $L^2(\mathbb R)$ for various choices of $V$. In ...
0
votes
2answers
207 views

Show that the following cycle has a limit cycle

By direct calculation show that (using polar coordianted) that $$ \dot x=x-y-x(x²+y²) $$ $$ \dot y=x+y-y(x²+y²) $$ Show that this has a limit cycle I need help understanding how to test whether it ...
0
votes
1answer
522 views

Which of the following is gradient/Hamiltonian( Conservative) system

The question that I have to solve is found below. However, I do not know how to start the solution since I am unsure about the defintion of a Gradient/Hamiltonian System. What must I check first to ...
4
votes
0answers
205 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 ...
3
votes
1answer
1k views

Frobenius Method to solve $x(1 - x)y'' - 3xy' - y = 0$

So, Im trying to self-learn method of frobenius, and I would like to ask if someone can explain to me how can we solve the following DE about $ x = 0$ using this method. $$ x(1 - x)y'' - 3xy' - y = 0 ...
2
votes
2answers
303 views

A differential equation of Buckling Rod.

I tried to solve a differential equation, but unfortunately got stuck at some point. The problem is to solve the differentail equation of hard clamped on both ends rod. And the force compresses the ...
1
vote
2answers
555 views

Proving a property of Legendre polynomials containing its derivatives

I am trying to prove the following property of Legendre polynomials. $$nP_n(x)=x{P_n^\prime(x)} - P^\prime_{n-1}(x)$$ My guess is that I somehow have to use the Bonnets recursion formula ...