# Tagged Questions

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### How the Jacobian is connected to the movement of particle from one domain to another? [on hold]

I am dealing with the proof of Reynold-Transport Theorem. There the Jacobian is used for the changing position of particles from one domain to another. Can anyone help me to understand what does ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) = ...
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### 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, ...
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### 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)\\ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...
### 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 ...