DO NOT USE THIS TAG just because the question contains a differential equation!

learn more… | top users | synonyms

1
vote
2answers
111 views

Does spatial coupling prohibit resonances due to an external source field?

The harmonic oscillator coupled to a sinodial external source $$\tfrac{\partial^2 x(t)}{\partial t^2}+\omega_0^2 x(t)=F_0\sin(\omega_\text{ext}\ t),$$ has the solution $$x(t)=x(0)\cos(\omega_0 t)+C ...
9
votes
6answers
947 views

Why is it natural to look for solutions involving dimensionless quantities?

While studying the Heat Equation, I got stuck in a statement in my book. It says: We have seen that the combination of variables $\displaystyle \frac{x}{\sqrt{Dt}}$ is not only invariant with ...
13
votes
2answers
467 views

Conservation of Mathematical Constraints when deriving Energy and Momentum from $F=ma$

Background: Starting from $F = ma$, integrating with respect to time, and using basic calc, one can derive $\int Fdt = m (v_f - v_i)$ Starting from $F = ma$, integrating with respect to distance, ...
2
votes
0answers
34 views

Physical interpretation related to a non-linear partial differential equation

I am doctoral student in pure mathematics working on a particular problem. My question is if this problem has applications to real world phenomena. I will try to explain the direct problem starting ...
0
votes
1answer
145 views

Heat equation : boundary conditions?

Say we have a bar centered at $x=0$, that is heated. We have the 1D Heat equation which we can solve to find a parabolic temperature profile : $$\kappa \frac{d^2 T}{d x^2}=-\frac{Q}{L S}$$ with ...
4
votes
1answer
90 views

WHY does the “order” of a differential equation = number of “energy storage” elements in a system?

OK. in all engineering courses there comes a point when they introduce you to systems theory and modeling of systems (for eg. via the impulse response) and then the Laplace transform. The modern ...
3
votes
1answer
131 views

Research problems in application of Lie groups to differential equations

Are there any open problems in physics involving Lie groups and differential equations for a phd theses. Some applications are say, Noether's theorem in classical or quantum field theory. But I am ...
1
vote
1answer
54 views

Is there any physically relevant example of constructing series solution about infinity of an ordinary differential equation?

I was reading about how to test if a given second order ordinary differential equation has singularity at infinity from Arfken and Weber. I understood the steps mathematically but I could not find its ...
1
vote
1answer
77 views

Help with Modeling a Liquid Vortex. (Related to General Fusion)

I want to model liquid lead swirling in a sphere. This is connected to General Fusion’s fusion machine. A 55 million dollar, Jeff Bezos funded, 60 person company trying to change the world with ...
3
votes
3answers
61 views

Proper and rigourous derivation of Gaussian beam?

Gaussian beams are known solutions to the Paraxial Wave Equation: $$ \frac{\partial^2 \Psi(x,y,t)}{\partial^2 x} + \frac{\partial^2 \Psi(x,y,t)}{\partial^2 y} = 2ik\frac{\partial ...
3
votes
0answers
70 views

Naive questions on the classical equations of motion from the Chern-Simons Lagrangian

Consider a Chern-Simons Lagrangian $\mathscr{L}=\mathbf{e}^2-b^2+g\epsilon^{\mu \nu \lambda} a_\mu\partial _\nu a_\lambda$ in 2+1 dimensions, where the 'electromagnetic' fields are $e_i=\partial ...
5
votes
1answer
116 views

Is the mathematical form of the acoustic diffusion equation present in other fields of physics?

We are working in the field of High Performance Computing and we have developed a very efficient parallel implementation for solving the Acoustic Diffusion Equation as described below: $$ ...
1
vote
1answer
61 views

Differential Equations for Block Diagram of Satellite Attitude Control System

I am trying to understand the procedure to setup differential equations from a block diagram. The enclosed example is about the attitude control of a satellite. The ultimate goal is to find a ...
0
votes
2answers
49 views

$R = dV/dI$ for varying temperature

I'm trying to do my prelab for an E&M course, and am asked if, for plotting $V$ vs $I$ with a varying temperature, I should expect a linear slope. I know that both $V$ and $I$ depend on $R$, and ...
5
votes
2answers
256 views

Time-dependent Schrödinger equation with $V=V(x,t)$

I was wondering about the following: If you have the time-dependent Schrödinger equation such that $$i \hbar \frac{\partial\psi(x,t)}{\partial t} = - \frac{\hbar^2}{2m} ...
1
vote
0answers
34 views

Is there a second order differential wave equation that only allows a finite set of discrete eigenvalues?

I tried constructing a second order differential wave equation that only allows a finite set of discrete eigenvalues by using the power series expansion such as \begin{align} A_{j+2} = ...
2
votes
2answers
323 views

What does it mean to “solve an equation”?

I don't understand what is meant by there being a "solution" to an equation. For example, what does a solution to the wave or heat equation represent, and what are we solving for? Of course, we can ...
8
votes
1answer
273 views

Black & Scholes and the Quantum Mechanics

I am interested in the link between the Black & Scholes equation and quantum mechanics. I start from the Black & Scholes PDE $$ \frac{\partial C}{\partial t} = -\frac{1}{2}\sigma^2 S^2 ...
1
vote
0answers
55 views

product solutions for PDEs, physical motivation

Given a boundary value problem with independent variables $x_1,x_2, \dots , x_n$ and a PDE say $U(x_i, y, \partial_j y,\partial_{ij} y, \dots )=0$ we typically begin constructing a general solution by ...
2
votes
2answers
85 views

Why fundamentally does classical mechanics lead to second order dynamics? [duplicate]

What's so special about second order equations in classical mechanics? I have a basic understanding of the Lagrangian and Hamiltonian formulations of classical mechanics, so I'm not looking for ...
1
vote
3answers
109 views

Solving the simplest coupled nonlinear ODES for chemical kinetics [closed]

I am just trying to get the integrated form for the kinetics of the reaction $A + B \rightarrow C + D$ characterized by: $$ -\dfrac{d[A]}{dt} = -\dfrac{d[B]}{dt} = k[A][B] \; . $$ As you note, ...
0
votes
0answers
21 views

characterising basins of attractions for master equations

I have a master equation for $(x,y,z)$ with the constraint $x+y+z=N$. $x$ can be regarded as the number of animal of a certain species in the whole system. In other words, I have a differential ...
0
votes
2answers
38 views

Curves satisfying this functional [closed]

This is a problem in Hartle's "GRAVITY": Consider the functional $$S[x(t)]= \int_{0}^{T} \left[\left(\frac{dx(t)}{dt}\right)^2 + x^2(t)\right]\text{ }dt$$ Find the curve $x(t)$ satisfying the ...
3
votes
1answer
67 views

solution of pendulum equation [closed]

I have the pendulum expression $$\ddot{\theta}+\omega_{o}^{2}\sin(\theta)=0,$$ where I used a Taylor expansion for the sine term: ...
10
votes
4answers
2k views

What is the physical meaning/concept behind Legendre polynomials?

In mathematical physics and other textbooks we find the Legendre polynomials are solutions of Legendre's differential equations. But I didn't understand where we encounter Legendre's differential ...
1
vote
1answer
36 views

Normal mode of a coupled pendulum: why the constant $\psi_1$, $\psi_2$

I need to solve a problem that tells me to find out the motion of both the pendulums that appear in the first 45 seconds of this video I think this kind of motion is described by a system of ...
2
votes
2answers
64 views

How do you integrate an expression over a variable in the limit of an integral?

I am trying to follow the steps to solve the integro-differential equation that arises from a plasma sheath problem given in this paper. This is the step I can't follow: ...
1
vote
0answers
93 views

A general solution to continuity equation

Let us write the standard continuity equation $$\frac{\partial \rho}{\partial t} + \vec{\nabla} \cdot \vec{\jmath} = 0.$$ Should the relation $\vec{\jmath} = \rho \vec{v}$ be considered as a general ...
0
votes
0answers
67 views

Compatibility between solutions of explicit Maxwell equations vs. wave equation?

When trying to solve for the allowed propagation frequencies in a cylindrical waveguide, I approached the problem by solving the wave equation for all three components of $\bar{E}$, and subsequently ...
0
votes
0answers
26 views

Looking for Solutions to Symmetric Potential

I'm a little confused on the basic method of finding a separable solution to a give potential distribution. If we have a symmetric potential, say it hits zero and $-a$ and $a$, constituting two sides ...
0
votes
0answers
34 views

Is there any general position function $x(t)$ that gives the solution to $x''(t) = k/x(t)^2$, where k is a constant? [duplicate]

In physics class, I often come across various inverse square law equations like the following: $F_G= G\frac{m_1m_2}{r^2}$ $F_E = k_e\frac{q_1q_2}{r^2}$ Specifically, we are typically given ...
11
votes
2answers
487 views

First-order wave equation: Why is its presence not common?

The (one-dimensional) wave equation is the second-order linear partial differential equation $$\frac{\partial^2 f}{\partial x^2}=\frac{1}{v^2}\frac{\partial^2 f}{\partial t^2}\tag{second order PDE}$$ ...
1
vote
1answer
102 views

The source of gravitation in a spacetime without matter

In a discussion concerning: Physical meaning of non-trivial solutions of vacuum Einstein's field equations there were a number of answers claiming that the flatness of the Ricci space (Rµv=0) ...
0
votes
0answers
32 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 ...
0
votes
0answers
209 views

Coupled mass spring system with damping and initial values

After researching through the web, I can't figure out how to express into a differential equation a coupled mass spring system with damping and initial values. Two masses and two springs, no external ...
1
vote
0answers
48 views

Derivation of cylindrical line heat source problem?

I have a line heat source embedded inside a cylinder and I am trying to find the temperature distribution T(r,t). By using the similarity variable, the solution to the differential equation is ...
4
votes
2answers
153 views

What information is lost in the symmetrization necessary to derive the BBGKY hierarchy?

The book on Kinetic theory I'm reading derives the BBGKY hierarchy after introducing the reduced distribution functions $f_s(q^1,p_1,q^2,p_2,\dots,q^s,p_s):=\int\ \rho\ \ \mathrm d q^{s+1} \mathrm d ...
17
votes
5answers
1k views

Any physical example of an “explosive” differential equation $ y' = ky^2$?

I was told that in physics (and in chemistry as well) there are processes that may be described by a differential equation of the form $$ y' = ky^2. $$ That is, the variation of a variable depends ...
0
votes
2answers
367 views

FWHM in resonance amplitude square derivation

Consider a linear harmonic oscillator subject to a periodic force: $$ \ddot x + 2 \beta \dot x + \omega _0 ^2x = f_0\cos \omega t$$ The solution tends to: $$A \cos (\omega t - \delta)$$ where: ...
2
votes
2answers
154 views

Stick and slip motion: mass and spring inside a box model

I am trying to determine a set of differential equation which can describe the motion of a mechanical system as below. Here, at the bottom we have a plate, and a box on top of it. Inside the box, ...
0
votes
1answer
72 views

Showing specififc internal energy $e$ is a perfect differential

I am reading a book on Gas dynamics and there is a small section on thermodynamics before the conservation laws of mass momentum and energy are introduced. The book says $$ p = R \rho T$$ where ...
0
votes
1answer
261 views

General Solution of Mechanics Problem

I had a homework problem that Given velocity, $v^2(t)=\frac{K}{x(t)}$, where $x(t)$ is distance, find $v$ as a function of $t$. Of course if we assume a positive root, it is easy but what if ...
3
votes
0answers
83 views

Solution of QM tasks by using asymptotics

When we solve QM tasks by solving Schrodinger equation, such as tasks about particle in Morse potential, Poschl-Teller potential and many others, we usually find an approximations (lets call them as ...
2
votes
1answer
81 views

Solving 2nd-order ODEs

I was reading this PDF REF per request: ...
0
votes
1answer
254 views

Problem involving 1st law of thermo and ideal gas law

Problem: $1.0 \text{ kg}$ of air at pressure $10^6 \text{ Pa}$ and temperature $398 \text{ K}$ expands to a five times greater volume. The expansion occurs such that in every instance the added ...
0
votes
0answers
72 views

Boundary conditions for 2D helical waveguide

I'm interested in looking at standing wave solutions for the wave equation on a 2D annulus, with the twist that the annulus is "streched" in to a helix in 3D, but so that the rings themselves are ...
4
votes
1answer
103 views

What exactly is the meaning of weak formulations and what is its purpose?

What is the purpose behind weak formulation of PDEs? I have read in the book by Zienkiewicz and Taylor that a weak formulation is more "permissive" that the original problem in the sense that it ...
1
vote
1answer
681 views

Finding a differential equation to for the amount of chemicals in a body of water

I am taking a differential equations course, and most of the problems relate to physical phenomenon. The calculus is not giving me trouble, but the way of approaching the problems is hanging me up. ...
2
votes
0answers
93 views

Differential Equations - Waves (Physics self-study suggestions) [closed]

I apologize ahead of time, in case this post is not allowed. After taking a few courses at a community college, I've taken the fall 2013 semester off (I was accepted into a university for the spring ...
1
vote
2answers
372 views

Geometrical interpretation of complex eigenvectors in a system of differential equations

Let's consider a system of differential equations in the form $$\dot{X} = M X$$ in two dimensions ($X = (x(t), y(t))$). In the case that $M$ has real values, it is easy to give a geometric ...