2
votes
0answers
43 views

ODE particular solution (physics)

I have to do this exercise: ($Z(t)=I(t)$, it's printed wrong). I have a doubt about the first item. To find all resonance when $R=1$, I found the particular solution $I_{p}(t)=A\sin(\omega ...
5
votes
2answers
83 views

Solution of differential equation with Dirac Delta

Is it possible to solve a differential equation of the following form? $\partial_x^2y + \delta(x) \partial_x y = 0$ where $\delta(x)$ is the dirac delta function. I need the solution for periodic ...
0
votes
3answers
112 views

Is speed a function of position?

Let $x$ be a smooth function from $[0,\infty)$ to $\mathbb{R}^n$ satisfying the following differential equation $x''(t) = f(x(t))$, where $f$ is a smooth function from $\mathbb{R}^n$ to itself. Then ...
0
votes
1answer
22 views

Differential problem in Gausses law

From the Gauss law we know, $\nabla \cdot \vec{E} = \rho / \epsilon_0 $. We have given that, $\vec{E}= kr^3 \hat {r}$ Now I have problem to get the identified part. Can you please elaborate ...
6
votes
0answers
115 views

How to solve a time-dependent Schrodinger equation in periodic Dirac delta potential

I'm trying to solve a 1D time-dependent Schrodinger equation: $$ i\frac{\partial \psi(x,t)}{\partial t}=\left[-\frac{1}{2} \frac{\partial^2}{\partial x^2}+V(x)+F(t)*x\right]\psi(x,t) $$ where $V(x)$ ...
1
vote
1answer
52 views

Calculating a double pendulum

consider the following situation of a double pendulum. We found the moving equations as $$ \ddot{\theta_1}=-L_1\sin\theta_1 + \frac{m_2}{m_1}\cos\theta_2\sin(\theta_2-\theta_1),\\ ...
0
votes
3answers
43 views

Solving 2nd-order ODE for SHO

In physics for a Simple Harmonic Oscillator, we have the differential equation $$ {\frac {d^2x}{dt^2}} + \frac kmx = 0 $$ from the balance of forces, which has a solution $$ x(t) = {x_o}\cos(\omega ...
0
votes
1answer
92 views

numerical update rule for discretized hawkes excitation process

So I think I am just misunderstanding some simple notation or something and would appreciate some help. I am trying to replicate this model in an agent based model, but I cannot seem to figure out the ...
0
votes
0answers
25 views

Newton's differential equation

As we all know one of Issac Newton's many achievement was to use his theory of gravitation and his law of motion to determine the way the planets move. I am looking for a not too deep resource in ...
2
votes
1answer
102 views

How were the solutions to these differential equations found?

These two very strange differential equations came up yesterday while I was doing a physics problem that I made up: EQ 1) $y'^{2} = k \sin(y)$ EQ 2) $y'' = k\cos(y)$ where $y'$ means ...
0
votes
2answers
25 views

Find the units of measurement of constant from formula

$$m\frac{dv}{dt}=mg-kv^2$$ $v=\ms^{-1} $m=kg$ $g=ms^{-2}$ $v^2=(ms^{-1})^2 I re-arrange the formula to isolate K $$K=-\frac{m\frac{dv}{dt}}{v^2}+\frac{mg}{v^2}$$ Sub in the units ...
0
votes
1answer
51 views

derive an equation for this mass spring damper

derive an equation to represent this mass spring damper in terms of input fore $F$ and relates to output displacement $(x)$ when springs $K_1=3$ , $K_2=5$ damper $C=6$ and mass $M=1$ , $F$ is a step ...
0
votes
1answer
25 views

Finding units of measurement of coefficients in ODE's

If we have a question where we have to find the coefficient's units such as K in this case. The actual formula contains more parts but it is simply the derivatives that I am unsure about. ...
0
votes
1answer
63 views

PDE from London's Equation with Cylindrical Symmetry

The question is from ISSP by Kittel and as follows: (a)Find a solution of the London equation that has cylindrical symmetry and applies outside a line core. In cylindrical polar coordinates, we want ...
1
vote
2answers
66 views

Solving a differential equation?

I'm trying to analyze the transient state of a RC circuit. My book gives me the following differential equation: $$\frac{d(v(t))}{dt} + av(t) = c$$ for some constants $a$ and $c$. The book thens ...
4
votes
2answers
124 views

What are integrating factors, really?

I can follow the rationale for integrating factors well enough, but they still feel like voodoo to me. Every single description of integrating factors I've seen (and I've seen quite a few, including ...
2
votes
0answers
72 views

How to maximize speed of rest position approach of nonlinearly damped spring oscillator?

Inspired by comments to answer for this question: Suppose we have a system which is described by the equation $$\ddot x=-x+g(\dot x),$$ with initial conditions $x(0)=1$, $\dot x(0)=0$. If ...
2
votes
0answers
27 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
3answers
86 views

Coupled mass spring system with damping, I need help with the equation

I know that the equation $mx''+cx'+kx=f(t)$ is used for a normal mass spring system, but I don't know how to express the differential equation for a coupled mass spring system with damping. These are ...
1
vote
2answers
138 views

Free fall with resistance: solution to the ODE

I'm having trouble solving this ODE: $$\ddot x = \mu \dot x^2 - g, \space \space x(0)=x_0$$ This is the ODE that determines the equation of motion of an object with air resistance. $\mu$ is a ...
0
votes
1answer
82 views

Solving an ODE with Mathematica, using Lagrangian mechanics

Question: The velocity of light above a hot surface decreases with the height from that surface. The velocity is given by $v=v_0(\frac{1-y}{\alpha})$ where $y$ is the vertical distance above the ...
4
votes
0answers
220 views

Modelling a Water Rocket. Requires Some Validation and Help. ( WARNING : Extremely Long but Interesting Post )

Good day people of math.stackexchange.com This is a pet project that I plan to use to convince my Prof that I would rather try something similar to this than to do the prescribed project. Edit : ...
4
votes
2answers
126 views

one-dimensional inverse square laws

I suddenly became curious about the following differential equation: \begin{align*} \frac{d^2x}{dt^2} = \frac{k}{x(t)^2} && x(0) = x_0 > 0 && \frac{dx}{dt}(0) = v_0 \end{align*} ...
0
votes
1answer
44 views

Initial conditions of a second-order ODE

This may look like a physics problem, but everything physical is explained below and I am looking for a mathematical solution. An RLC circuit (pictured above) is governed by two equations: $$ ...
0
votes
1answer
105 views

Pendulum with angular velocity

I have another pendulum problem again but this time it's with angular velocity. My question is: If a pendulum is initially at its unstable equilibrium position, then how large an initial angular ...
-1
votes
2answers
53 views

Modeling with First Order Equations [closed]

A ball with mass 015kg is thrown upward with initial velocity 20m/s from the roof of a building 30m high. There is air resistance of magnitude v^2/1325 directed opposite to the velocity , where the ...
5
votes
0answers
64 views

Earnshaw's theorem

Proposition Suppose $U\colon\Omega\to\mathbb R$ is a non-constant harmonic function, i.e. $U\in\mathcal C^\omega$, i.e. analytic, and $\Delta U=0$, where $\Omega\subseteq\mathbb R^n$ is a region. ...
0
votes
0answers
19 views

Plotting motion from velocity equations

I have limited knowledge of physics but enough to frustrate myself. On this webpage there are two equations for Vx dot and Vy dot. The equations are in the background of this page: ...
1
vote
2answers
95 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 ...
1
vote
1answer
65 views

How do derive equivalent complex versions of linear differential equations.

I've done this before and have forgotten some of the details. I will try my best to re-derive it. Please help fill in the blanks. In my Acoustics book it says: An alternating force may be ...
2
votes
3answers
75 views

How do you solve this differential equation?

Though I've read questions on this site and really appreciate the quality of the answers, this is my first question, so I hope it follows the site's guidelines. When working with potential energy ...
0
votes
1answer
59 views

Physics problem - about a shell. Differential equation

Can you help me please, to write the differential equation for this problem, and give me an idea how to solve this equation. A shell of mass $2$ kg is shot upward with an initial velocity of $200$ ...
3
votes
1answer
114 views

Non-Linear ODE Strategy

I encountered the following $2^{nd}$-order, non-linear ODE while working on a classical mechanics problem: $$ \frac{d^2r}{dt^2}-\frac{\alpha^2}{r^3}+\beta=0 $$ where $\alpha \ \text{and}\ \beta$ are ...
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
460 views

Solving differential equation regarding temperature change (Newton's law)

I'm solving a differential equation problem set and I bumped into the following DE problem where I got few question marks: The temperature of a body at time $t$ is $T(t)$ and the temperature of ...
7
votes
2answers
174 views

Wave-Particle Duality in PDE?

I am reading Arnold's Lectures on Partial Differential Equations. It is definitely a good book, yet sometimes I am a little bit confused. One theme of the first chapter seems to be From the ...
2
votes
2answers
282 views

Free-fall according to Newton's gravitation law

Most analysis of free-fall assume that bodies fall with constant acceleration. If however one analyses free-fall according to Newton's gravitation law, one is lead to a differential equation which I ...
5
votes
0answers
96 views

Solving numerically the equation of motion of D7 brane perturbation

I want to solve this equation $$ \partial_{\rho}^{2}\phi+\frac{3}{\rho}\partial_{\rho}\phi+\left(\frac{M^{2}}{(1+\rho^{2})^{2}}-\frac{l(l+2)}{\rho^{2}}\right)\phi=0 $$ numerically. I know that ...
1
vote
1answer
120 views

Solving for Asteroid Orbit with Respect to Time

I am trying to create a differential equation with which I am can numerically solve to plot the orbit of an asteroid around Jupiter so far I have assumed the mass of jupiter is 0.001 of the mass of ...
2
votes
1answer
105 views

Proof: Gradient of a Hamiltonian System

I am trying to prove the following: Given that $f \in C^1(E) $ where E is a open simply connected subsets of the plane. Show that the system $\dot x=f(x)$ is a hamiltonian if and only if $\nabla ...
0
votes
1answer
43 views

Two Body Problem Equations

Let $X_1$ and $X_2$ be particles of mass $m_1$ and $m_1$, where $X_1$=$(x_1^1,x_2^1,x_3^1)$ and $X_2$=$(x_1^2,x_2^2,x_3^2)$. The potential energy of this system is $U=gm_1m_2/|X_2-X_1|)$ and ...
3
votes
2answers
160 views

How to solve coupled linear ODE?

I wan to solve the following ODE's:- $$L_1 q''(t)+R_1q'(t)+\frac 1C_1 q(t)-Mq_2''(t)=V\sin(\omega t)$$ $$L_2 q_2''(t)+R_2q_2'(t)+\frac 1C_2 q_2(t)-Mq''(t)=V\sin(\omega t)$$ $L,C,R,V>0$, I already ...
1
vote
0answers
42 views

Particle in a Polya Vector field

For a given analytic function $H$ from $\mathbb{C}$ to $\mathbb{C}$, we define the Polya Vector Field to be $\bar{H}$. This then corresponds to a irrotational, conservative vector field on ...
1
vote
1answer
59 views

find the error in a harmonic motion problem

I was going over the HW solutions I got back from a prof. And most of it I am ok with, But there is one bit that is sort of bothering me. It has to do with solving the equation of motion for a ...
4
votes
1answer
432 views

Damped Harmonic Oscillator and Response Function

This is another one of those questions that I feel like I am almost there, but not quite, and it's the math that gets me. But here goes: For a driven damped harmonic oscillator, show that the full ...
0
votes
1answer
48 views

Finding time t for a body with air resistance k to reach to location x

Since gravity for this problem is irrelevant I started from the following equation: $$ma = -kv$$ From here I integrated both sides in order to find an expression of v as a function of t: V stands ...
1
vote
0answers
75 views

Is the one demential time-independent Schrödinger equation solvable in potential (1+Tanh(x+1))(-1+Tanh(x-1))

The one dimensional time-independent Schrödinger equation reads: \begin{equation} -\frac{h^2}{2m}\frac{d^2\psi}{dx^2}+U(x)~\psi=E~\psi \end{equation} where $\psi(x)$ is the wavefunction, U(x) is the ...
0
votes
1answer
136 views

Understanding quaternions & gradient descent in a paper on inertial / magnetic sensor arrays

I hope this question is appropriate here! I and a friend at work are trying to understand Sebastian Madgwick's paper, "An efficient orientation for inertial and inertial/magnetic sensor arrays" ...
0
votes
1answer
87 views

Does integration over one complete cycle equals to 4 times integration over quarter-cyle?

From the article pendulum(mathmetics) from wikipedia. There is a demonstration that this equation: $$\dfrac{dt}{d\theta } = ...
1
vote
2answers
65 views

When can we make a change of variables $f'$ for $f$?

In my applied math class, my instructor introduced the example of two point masses, both with mass $m$, with positions $x_1(t)$ and $x_2(t)$. Newton's law gives us the differential equation $$r'' + ...