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Questions tagged [differential-equations]

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Derive Columb's Law from Gauss's Law by solving PDEs?

The standard way of deriving Columb's law is given in this post. I am just wondering if I can do it in another way. Let a point charge $q$ be located at the origin. The problem is then essentially ...
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Why are Cauchy boundary conditions an over-specification of boundary conditions for solving Poisson’s equation?

I was referred to Physics.SE by the following content published in Jackson’s Classical Electrodynamics: This rather surprising result [the fact that the potential within a charge-free volume is ...
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Cosmology - Demonstration for equation of the evolution of the density contrast

In a context of cosmology, I need help about a differential equation that I can't get to demonstrate: The growth of density fluctuations obeys a second order differential equation. At early enough ...
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Confusion on Quantum Harmonic Oscillator Eigenvalues

In standard PDE theory, one generates eigenvalues to Sturm-Liouville problems over a finite domain. So, for a wave equation, we have an infinite number of eigenvalues $\lambda_n$ for a Dirichlet ...
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Solution to a differential equation in terms of Bessel function [closed]

I am looking after the solution of the following differential equation: $\partial_y \partial_y \psi + \frac{\partial_y \psi}{y} + \left( \frac{\omega^2 }{y^2} - \frac{\Lambda}{y} \right) \psi = 0$ ...
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Is separation of variables in the heat equation dimensionally consistent?

This may be a trivial question but is about the statement that the function $U(x,t)$ in the heat equation may be expressed in the form $X(x)\cdot T(t)$. It's that $X$ and $T$ both are functions ...
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Extracting solution from driven SHM

I guess maybe I should rather ask at the math stack exchange? I have a simple harmonic undamped oscillator driven by a cosinusoidal force: $$\ddot{x}+\omega_o^2x=f\cos(\Omega t).$$ I've managed to ...
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How do we find the equation for the gyrating motion of a particle in a uniform magnetic field and a non-uniform Electric field? [closed]

Considering the gyrating motion is not negligible and also retaining the guiding center drift, how do we get the trajectories x(t),y(t),z(t) of the particle? In this case is the variation in the ...
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Choosing initial condition for Hamilton-Jacobi PDE from initial $x$ and $p$

For separable solutions to Hamilton-Jacobi PDE (say in 2D), we treat the Hamilton's principal function $S$ as $$S= W(x) + W(y) - E*t$$ and treat the separate parts as constants and find $W(x)$, $W(y)$...
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More than two linearly independent solutions to the (linear second order) radial wave equation?

I'm puzzled by the following radial wave equation: $$\left(\frac{\hbar^2}{2m_r}\left(-\frac{d^2}{dr^2} -\frac{2}{r}\frac{d}{dr} + \frac{l(l+1)}{r^2}\right) + V(r)\right)R_{nl}(r) = ER_{nl}(r)$$ ...
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Master equation for a dynamical system on networks

I am trying to mathematically model the following idea that describes the dynamical evolution of a quantity over a graph. Let us imagine we have a directed graph, with $n$ nodes and $m$ edges. Each ...
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Initial Condition in Spaghetti Cracking

In this Paper B. Audoly, S. Neukirch - Fragmentation of Rods by Cascading Cracks: Why Spaghetti Does Not Break in Half on Page 2 (bottom), the author argues that using an integral of motion, the ...
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What numerical methods are being used here to solve the differential equation?

I am having trouble following and understanding the numerical approach to solving for $h(r,t)$ in the following text: How is the integral force balance (10) used to calculate the new pressure ...
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Elastic wave in isotropic medium

The 2D elastic wave equation is given by:  \begin{align} \rho \frac{\partial^2 u}{\partial t^2} &= C_{11} \frac{\partial^2 u}{\partial x^2} + C_{44} \frac{\partial^2 u}{\partial y^2} + (C_{12} ...
If we have a catenary line with a mass per unit length of m with one end fixed and the other end moving in sinusoidal motion like $A.sin(\omega.t)$, how would I ...