Questions tagged [differential-equations]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
41 views

Physical systems satisfying the differential equation $ y' = {}$ rotation of $y$ by $t$?

Denote by $R(t)$ the matrix that rotates by $t$, i.e. $$ R(t)=\begin{pmatrix} \cos(t) & -\sin(t) \\ \sin(t) & \cos(t) \end{pmatrix}. $$ Are there any (realistic?) physical systems which ...
0
votes
0answers
20 views

Kinetic energy of point $B$ which has $0$ mass

In the problem, I came to find the kinetic energy of point $B$ which has $0$ mass as the problem states. But the force $\vec F$ is acting on it as in the figure below: How can I find the energy? ...
1
vote
0answers
19 views

Find an optimal shape analytically

This is better described with an example: Imagine we posses our current understanding of physics, but we do not know how to make an airplane because we do not know about airfoils. To make this more ...
0
votes
0answers
5 views

How to rewrite a differential equation into another form of differential equation? [closed]

How to write the differential equation $$ x^2 \ddot y +2x\dot y - y = x^2 $$ into the form $$ \ddot y +\dot y - y = x^2 $$ where $y$ is being integrated in terms of x? I tried manipulating the above ...
1
vote
1answer
35 views

From the initial condition problem of the Euler-Lagrange equation to the principle of least action

I browsed through many similar questions about the Initial Condition Problem (ICP) and Boundary Value Problem (BVP) for Euler-Lagrange equations, here some interesting but (in my opinion) incomplete ...
0
votes
1answer
30 views

Is there a differential equation describing the wavefunction of a hadron?

In Newtonian Physics there's a differential equation describing the motion of multiple bodies in orbit around each other. In non relativistic quantum mechanics there's a differential equation ...
-1
votes
2answers
41 views

Green's function for a linear differential equation [closed]

Given the definition of Green's function $G(x,s)$ by Wikipedia as the solution of $L ~G(x,s) = \delta(x-s)$. Consider the following equation $$\Big( \frac{d}{dt} + i A + B \Big) f(t) = C$$ where $A$, $...
2
votes
2answers
215 views

Is there any shortcoming of the Langevin equation which is solved by its generalization?

The ordinary Langevin equation describing the velocity $v(t)$ of a Brownian particle of mass $M$ in a fluid bath in equilibrium at a fixed temperature reads $$M\frac{dv}{dt}=-M\gamma v(t)+\zeta(t)+F_{\...
1
vote
1answer
58 views

Solution to electromagnetic wave equation

The Helmholtz wave equation is given as : $$\nabla^2 \vec E =\mu\epsilon \frac{\partial^2 \vec E}{\partial t^2}$$ Considering $\vec E=E_x(z) e^{j \omega t}$ the Helmholtz wave equation now takes the ...
0
votes
2answers
76 views

Solution to differential equation

If I have a differential equations of the form $$\frac {d^2y}{dt^2}=\alpha^2y$$ Assuming the roots of the characteristic equation is complex the solution to the differential equation is: $$y=C_1e^{j\...
0
votes
1answer
37 views

Understanding mathematical concept behind phase space and phase portait

I'd like to expose the problem through an example, which was what made me think about it. It's a rational mechanics problem. Consider the one dimensional Cauchy problem $\begin{cases}m\ddot{x} = F(x,\...
0
votes
0answers
13 views

Equation and boundary conditions for the temperature problem in a cylinder with thermal power on its axis

I'm having trouble trying to establish the equation for the following problem: On the axis of a long cylinder, whose radius is r = r0, there is a conductor with a current that releases the thermal ...
3
votes
1answer
176 views

Perturbation theory on perihelion advance

I'm trying to get a relativistic solution to Kepler's equation starting with $$\frac{d^2 u}{d\phi^2}+u = \frac{M}{L^2}+3Mu^2$$ by ignoring the higher order terms; $$u(\phi, \epsilon)=u_0+\epsilon u_1$$...
1
vote
0answers
18 views

ODEs with rational first integrals [closed]

I would like some examples of ODEs (i.e., $\dot{x}=f(x)$, where $x\in\mathbb{R}^n$) that possess one or more rational first-integrals of the form $$H(x)=\frac{a_1^Tx+\alpha_1}{a_2^Tx+\alpha_2},$$ ...
0
votes
4answers
54 views

Heat Diffusion Equation with extra terms

$$\frac{\partial U}{\partial t}=\alpha \frac{\partial^2 U}{\partial x^2}+\beta U$$ I have been given this partial differential equation and am asked to find an application for it. I can see that the ...
0
votes
0answers
31 views

Converting a partial derivative from Lagrangian to Eulerian coordinates

This question pertains to section 2.3 of https://www.whoi.edu/science/PO/people/jprice/class/ELreps.pdf Lagrangian and Eulerian Representations of Fluid Flow:Kinematics and the Equations of Motion (...
-3
votes
0answers
68 views

How to solve the Schrödinger equation for the hydrogen atom?

We know that the time-independent Schrödinger equation, if we are using Cartesian Coordinates, is given by: $\begin{aligned}-\frac{\hbar^{2}}{2 m}\left(\frac{\partial^{2} \psi(x, y, z)}{\partial x^{2}}...
2
votes
2answers
57 views

When is a system of equations 'closed'?

In Sean M Carroll's Introduction to General Relativity: Spacetime and Geometry, after deriving the Tolman-Oppenheimer-Volkoff equation on page 233, Chapter 5, he says: "To get a closed system of ...
0
votes
1answer
29 views

Why do you add independent solutions when finding a general equation for SHM?

In my physics class, we have an assignment based on simple harmonic motion with the differential equation: $$ \frac{d^2x}{dt^2} + a\frac{dx}{dt} + a^2x = 0 $$ Different parts of the question help us ...
1
vote
0answers
38 views

Bidirectional Benjamin-Ono vs Benjamin-ono

I am a mathematician and not a physicist, and I am trying to physically understand the difference between the model described by the Benjamin-Ono equation and the model described by the bidirectional ...
1
vote
0answers
25 views

Betti number of sphere and vector-Laplacian

I am having trouble understanding the relationship between Laplacian and Euler numbers through concrete calculations. In my understanding, the "dimension" of the scalar field satisfying $\...
0
votes
1answer
42 views

How to solve a stochastic differential equation

This is something I’ve been puzzled about over the past week or so. How do you go about getting the equilibrium probability distribution solution for a stochastic differential equation, like ...
2
votes
2answers
112 views

The first and second form of Euler-(Lagrange) equation with explicit time dependence

I have learned the first and second form of Euler-(Lagrange) equation with no explicit time dependence (the time dependence only implicit on the function to be solved, say $y\left(t\right)$), from ...
0
votes
1answer
15 views

Integration of Bloch equation in magnetic resonance

From Bloch equation we have \begin{equation}\label{bloch_01} \tag{1} \frac{d M_z}{dt} = \frac{M_0-M_z}{T_1} \end{equation} from there we can integrate and we get \begin{equation}\label{bloch_02} \tag{...
5
votes
2answers
106 views

What do we exactly mean when we say that a problem has an analytical solution?

What do we actually mean when we say that a certain problem has or does not have an analytical solution? I ask this because some systems that are said to have an analytical solution actually are no ...
1
vote
0answers
26 views

Poincare section for the duffing Oscillator

I have used the 4th Order Runge-Kutta method in order to estimate the values in which the Duffing Oscillator is chaotic. According to Wikipedia, the Duffing Oscillator is chaotic for values of $\alpha$...
2
votes
0answers
76 views

Solving Laplace equation on a triangle with mixed boundary conditions

From sources [Ref: 1 to 5], one learns that there is a class of boundary conditions (see Fig 1) on a triangle (in this case the 30-60-90 triangle) that allow for closed form solution for eigensystems ...
1
vote
1answer
30 views

Phase plots: The exact particular solution is a function of time, can't find fixed points. Now, in this situation, how to draw phase plots?

I want to draw phase plots. The differential equations are two coupled second-order non-linear differential equations. I have the exact particular analytic solutions. However, the solutions are a ...
2
votes
1answer
33 views

Special Conformal Transformation Acting on Spinor Variables

I'm working in 3,1 Minkowski spacetime, representing null vectors as a product of two commuting spinors so that eg. $$p_i^{\dot{\alpha}\alpha} = |i]^{\dot{\alpha}}\langle i|^\alpha.$$ I know that ...
1
vote
1answer
29 views

2D Incompressible Fluid Simulation solving / diffusion factor

I've been reading about fluid simulations - specifically, incompressible fluid dynamics using the incompressible Navier-Stokes equations. Every resource I've looked at has two key components that I ...
0
votes
0answers
37 views

Can we find eigenvalues of an operator without knowing the boundary conditions?

Imagine I have a space spanned by the two basis vectors $e_1 = sin x$ and $e_2=cos x$ and I define some inner product using $\langle f|g\rangle = \frac{1}{π}\int_0^{2\pi} f^*g dx$ and some operator $D ...
0
votes
0answers
10 views

Solving coupled propogation equations for EM waves

I have recently come across a set of partial differentials that describe the propogation of two coupled EM fields, in a 2D system currently being investigated. In their most general form they are $\...
0
votes
2answers
73 views

In series solution what form of the series should be assumed?

As reading Quantum Mechanics I encountered some differential equations which can be solved by series solution method. In case of Harmonic Oscillator we assume the series as "$u(y)=\sum_{n}c_{n}y^{...
2
votes
0answers
72 views

How to solve general wave equation and dispersion relation using Fourier series?

In this paper (open access), the authors used Fourier series with most general wave equation to find the dispersion relation. I am presenting some main equations as snippets to depict their solution. ...
0
votes
1answer
62 views

Complex exponential method of solving differential equations

In the twenty third Feynman lecture, the solution of the following differential equation is discussed: $$ \frac{d^2 x}{dt^2} + \frac{kx}{m} = \frac{F}{m}$$ AFter 'complexifying' this differential ...
2
votes
1answer
87 views

Why is the separation constant (used in obtaining the electron's wave function in hydrogen) a constant and not a function of position (and time)?

The Schrödinger equation for the hydrogen atom in polar coordinates is: $$ -\frac{\hbar^2}{2\mu}\left[\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial \psi}{\partial r}\right) + \frac{1}...
2
votes
1answer
40 views

Are there any results about regularity of the generalized eigenfunctions used in spectral representations of unbounded differential operators?

I am using the "Direct integral" version of the spectral theorem, as given e.g. in Hall. It says that a diagonal representation for an unbounded operator on the Hilbert space $L^2(\mathbb{R}^...
0
votes
0answers
57 views

Angular part solutions of Schrodinger's equation

Hello Physics Community i'm trying to solve the angular part of Schrodinger's equation, specifically the $\theta$ part , $$ \left[\sin \theta \frac{d}{d \theta}\left(\sin \theta \frac{d}{d \theta}\...
0
votes
1answer
27 views

Attempting to numerically integrate two nonlinear ODEs to generate a model for oscillations of a fluid column

This question is related to the paper located at https://doi.org/10.1063/1.1476670. I am attempting to model the oscillations of a fluid column. I have 2 ODEs that describe the oscillations, namely: $...
1
vote
0answers
19 views

Set of first order ODE [duplicate]

What's the condition for $\dot{x} = f(x,y)\\ \dot{y} = g(x,y)$ To be rewritable as $\dot{x} = \frac{\partial F(x,y)}{\partial y}\\ \dot{y} = -\frac{\partial F(x,y)}{\partial x}$ Can I always find a ...
0
votes
1answer
93 views

Separation of variables for a function of 3 variables $V(x, y, z)$ [closed]

I'm trying to find the solutions for $V(x,y,z)$ by separation of variables. Is it correct to say: $$\frac{1}{X}\frac{d^2x}{dx^2} + \frac{1}{Y}\frac{d^2y}{dy^2} + \frac{1}{Z}\frac{d^2z}{dz^2} = 0$$ ...
0
votes
1answer
76 views

Weak solution of Schrödinger Equation

Consider a particle in a box $\Lambda = [0, L]$. The wavefunction $\psi \in L_D^2(\Lambda)$ where $D$ denotes a Dirichlet Condition $\psi(0)=0=\psi(L)$. We have, then $$ - \frac{\hbar^{2}}{2m} \frac{d^...
2
votes
2answers
45 views

Why does general homogenous solution of differential equations modelling circuits die off after a long time?

I was reading this answer in Elecronic engineering stack exchange which said that when solving the linear second order differential equation modelling circuits having ac source. We only need to ...
0
votes
2answers
30 views

SIR model in terms of fluids

Let $s_i (t)$ be the fraction of people who are susceptible of getting the disease in community (node) $i$ at time $t$, $x_i (t)$ is the fraction of infected people in community $i$ at time $t$, and $...
1
vote
0answers
27 views

Smoluchowski diffusion equation for 2D diffusion

I am trying to solve the Smoluchowski diffusion equation, for 50 particles in the x/y plane. I have derived the equations $$ X(t+\delta t) = X(t) +\sqrt{2D\delta t}\phi_x \\ Y(t+\delta t) = Y(t) +\...
1
vote
1answer
54 views

Differential form of Massieu’s function [closed]

Massieu’s function is given by: $$F_{M}=-\frac{U}{T}+S$$ And its differential form is given by: $$dF_{M}=\frac{U}{T^{2}}dT+\frac{P}{T}dV$$ Well, it seems that: $$\frac{\partial S}{\partial T}=0$$ How ...
2
votes
0answers
45 views

Boundary conditions for equation of motion of a chain under small vertical motion of its support

I would like to find the boundary conditions for the of motion of a chain under small vertical motion of its support endpoint. I also assume displacement of the chain in the vertical $y$ direction is ...
0
votes
2answers
72 views

Is there a way to solve the following differential equation for a sphere rising in a fluid?

Given the boundary conditions, how do I find the analytical solution (for the velocity) of the following expression: $$ \left(\frac{2}{3} \pi \rho_f a^3 + \frac{4}{3} \pi \rho_s a^3\right) \frac{d ^2 ...
2
votes
0answers
26 views

What does it mean for resistance, as it appears in Ohm's law, to be an integral, evaluated over the body as a whole?

In literature I read: The three linear flux laws mentioned are: As seen, a correspondence exist between the hydraulic conductivity $K$, thermal conductivity $\lambda$, and electrical conductivity $\...
1
vote
1answer
49 views

Why are interacting field theories called nonlinear? Explanation for interacting EM field, in particular

The classical equation of motion for the electromagnetic field interacting with a charged fermion field $\psi$ of charge $eq$ is given by $$\Box A^\mu(x)=j^\mu(x)$$ where $j^\mu(x)=eq\bar{\psi}(x)\...

1
2 3 4 5
13