0
votes
2answers
25 views

Why write the solution of the harmonic oscillator form 1 is equal to writing form 2?

Why write the solution of the harmonic oscillator form $$\psi=A\cos\omega_0 t+B\sin \omega_0t$$ is equal to writing form $$\psi=C_1e^{i\omega_0t}+C_2e^{-i\omega_0t}$$? I would like to see how one ...
0
votes
0answers
21 views

Using differential equations vs general solutions

I was looking at Newton's Cooling Law with a student when he asked me why, for this particular case, do we use the ODE and not just the general solution to analyse data. I couldn't come up with an ...
1
vote
0answers
28 views

Forced oscillation in a pendulum and resonances

In a pendulum without the small angles approximation the equation describing the motion of the mass is: $$\ddot{\phi}(t)=-\dfrac{g}{l}\sin\left(\phi(t)\right)$$ Applying a sinusoidal force ...
1
vote
1answer
64 views

Solving the equation of damped oscillator

I'm asked to prove that any solution of the equation $$\ddot\Phi+\Gamma\dot\Phi+\omega_0^2\Phi=0;\qquad \omega_0>\frac\Gamma 2$$ is $$\Phi=A_0e^{-\frac{\Gamma}{2} t}e^{i(\omega t-\beta)};\qquad ...
0
votes
5answers
99 views

Assumptions in Word Problems (Calculus)

I just had a small question about assumptions in mathematical word problems. Suppose you are given a calculus problem (related-rates), "A spherical balloon is inflated with gas at the rate of 800 ...
38
votes
15answers
6k views

Why learn to solve differential equations when computers can do it?

I'm getting started learning engineering math. I'm really interested in physics especially quantum mechanics, and I'm coming from a strong CS background. One question is haunting me. Why do I need ...
2
votes
1answer
60 views

Solving weak 2 body problem

I tried to solve a physics problem about two body problem where the masses $M$ and $m$ are $M \gg m$. The body $m$ is at radius $R$ from the mass $M$ and is falling down with initial speed $v(0) = 0$. ...
2
votes
0answers
50 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
106 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
168 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 ...
7
votes
0answers
145 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
65 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
1answer
52 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
95 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
30 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
105 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
75 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
68 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
88 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 ...
5
votes
2answers
128 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
73 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
30 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
96 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
172 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
88 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
236 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
132 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
113 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
79 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
71 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
106 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
69 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
84 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
61 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
91 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
612 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 ...
8
votes
2answers
187 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
373 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
97 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
137 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
113 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
44 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
167 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
46 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 ...