# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $200$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...