1
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2answers
50 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
92 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
61 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
21 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
61 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
122 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
74 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 ...
3
votes
0answers
197 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
109 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
43 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
68 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
41 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
58 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
17 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
88 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
61 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
73 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
51 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
111 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
313 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
156 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
169 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
87 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
105 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
81 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
41 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
148 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
33 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
57 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
367 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
45 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
72 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
106 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
79 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'' + ...
1
vote
1answer
159 views

Runge-Kutta method for multiple springs

If we have a spring attached to a wall with an object on the other side, the differential equations describing the system are: $$x'=v$$ $$v'=-\frac{k}{m}x-\frac{b}{m}v$$ Where: x is position of the ...
4
votes
2answers
558 views

Solution to over-damped harmonic spring

(A kind soul at physics.stackexchange suggested I post here as well, sorry if out of bounds.) I'm trying to programmatically model a damped harmonic spring for use in mobile UI animations (physics ...
1
vote
0answers
19 views

Solution to over damped harmonic spring [duplicate]

I'm trying to programmatically model a damped harmonic spring for use in mobile UI animations (physics isn't my background, please pardon any misconceptions). Having derived the parameters for the ...
1
vote
2answers
299 views

Differential equation word problem water leaking $y=x^2$

A tank has the shape of a parabola $y=x^2$ revolved around the y-axis. Water leaks from a hole area $B= .0005 m^2$ at the bottom, let $y(t)$ be the water level at time $t$. How long does it take for ...
0
votes
1answer
127 views

Velocity depending on position

Let $v(x)$ be the velocity of a point on a line. Find the acceleration $a(x)$. I found the following relations: $$x(t)=\int v(t) dt$$ $$v'(t)=v'(x)v(t)$$ but now I'm stuck.
1
vote
1answer
142 views

A finite difference method for robust convergence despite large time steps in first order ODE

Suppose we're looking at a first order ODE of the form $$ \frac{dx}{dt}=-\lambda x+ b u $$ where $\lambda$ and b are functions of $x$ and $u$ is an 'energy generating' term which is a function of $x$ ...
2
votes
0answers
398 views

Hard Differential Equation. Please help.

first of all I'm not a mathematician, so I apologize if any of my understanding and terminology isn't up to par. Also, I've never used this website (or any of these kind of question/answer) websites ...
0
votes
0answers
72 views

Close to giving up on this differential equation! Please help! [duplicate]

first of all I'm not a mathematician, so I apologize if any of my understanding and terminology isn't up to par. Also, I've never used this website (or any of these kind of question/answer) websites ...
3
votes
2answers
106 views

pressure in earth's atmosphere as a function of height above sea level

While I was studying the measurements of pressure at earth's atmosphere,I found the barometric formula which is more complex equation ($P'=Pe^{-mgh/kT}$) than what I used so far ($p=h\rho g$). So I ...
3
votes
2answers
269 views

Examples of applications of Linear differential equations to physics.

I wonder which other real life applications do exist for linear differential equations, besides harmonic oscillators and pendulums. I'm looking for examples to include in a document that talks about ...
3
votes
2answers
441 views

Integrating velocity field to get position

I feel silly for simply being brainstuck, but consider the following integral, physically it would be the solution of $\mathbf{p} = \tfrac{d\mathbf{v}}{dt}$ - the position of a given particle in ...
2
votes
1answer
204 views

How to solve the differential equation for the motion equation of a body in a gravitational field from one fixed source

I want to develop the motion equation of a body in a classic gravitational field ($F=\frac{Gm_1m_2}{r^2}$). Starting by creating the lagrangian of a body under gravitational force, in polar ...
0
votes
1answer
130 views

Finding the Extremals of a Functional J.

The functional $J$ is defined on smooth functions $y \colon [a,b] \to \mathbb{R}$ satisfying $y(a) = u$, $y(b) = v$ and is given by $$J[y]=\int_a^b \sqrt{y} \sqrt{1+(y')^2}\, dx.$$ I have found ...
1
vote
1answer
79 views

Differential equations basic problem

I know this is a basic Physics problems but somehow I can't solve it. We have the differential equation: $2x''x^2 - 4 x^2x' - 2 x^3 = 0$ We have to conclude that the system: $x' = y $ $y' = 2y + ...