# Questions tagged [differential-equations]

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### How the eigenvalue problem was solved?

In Gasiorowicz 3rd edition Chapter 3, I've tried to solve this problem I checked the solution's manual, When I tried to integrate it, the answer I got is $$\psi(x)=Ce^{x^2/2\lambda}$$ Can you ...
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### Does existence of an analytic solution to an equation of motion given by Newton's second law depend on coordinates?

Newton's second law is a coordinate agnostic statement, we can use it to calculate the forces in a coordinate system, and hence, the motion of the body in that coordinate system. However, depending on ...
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### Kepler problem in cartesian coordinates

I'm trying to solve the Kepler problem in Cartesian coordinates, that is, I want to show that the trajectory is an ellipse using Cartesian coordinates instead of using polar coordinates, as is usually ...
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### Second linearly independent solution of Airy Differential equation

The Airy differential equation is $$\frac{d^2y}{dx^2}=xy.$$ After Fourier transforming the equation, we get $$y=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i\left(kx+\frac{k^3}{3}\right)}dk.$$ Here $k$...
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1 vote
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### Application of Helmholtz equation

I have the Helmholtz equation $$\nabla^2f = -k^2f$$ I am trying to solve it as a second order differential equation using a numerical method. However, I am unable to find an application of it, other ...
1 vote
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### Expression for Steady state of Forced vibration [duplicate]

In my book under the topic Steady state of the forced oscillator, they started with the equation: $$\frac{d^2x}{dt^2}+γ\frac{dx}{dt}+ω_0^2x=fe^{jωt}$$ I know the equation for damped oscillation but it ...
1 vote
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### Integrating Hamilton's equations: is there any difference between these two integration methods?

Consider a general set of Hamilton's equations \begin{align} \dot{q}(q, p) &= \partial_p H(q, p) \\ \dot{p}(q, p) &= -\partial_q H(q, p) \end{align} A first-order integrator one could ...
1 vote
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### Hand damping of a vibrating string or membrane

Problem is the following: If I have a guitar string or a drum membrane which is vibrating (and thus creating sound), when I place my hand or finger on it looses energy quickly, and eventually ...
1 vote
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### Derivative of operator with respect to parameters

From Shankar's QM book pg. 56: For an operator $\theta(\lambda)$ that depends on a parameter $\lambda$ defined by $$\theta(\lambda)=e^{\lambda\Omega}$$ where $\Omega$ is also a constant operator, we ...
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### Initial value formulation of Yang-Mills equation

In Wald Chapter 10, he discusses the initial value formalism of electromagnetism - how Maxwell's equations are actually a system of three equations plus an initial value constraint, and how we can ...
1 vote
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### What areas of research use linear algebra the most? [closed]

I'm in my 4th semester as a physics and recently added a minor in math. I'm in my second linear algebra course, and am finding it extremely interesting and fun to understand. To complete my minor, I ...
1 vote
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### Analytic solution to time-dependent Schrödinger equation

Is there a known analytical solution for the following Schrödinger equation $$i \partial_t \psi=-\frac{1}{2}\partial^2_{x} \psi + \psi x.$$
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### How to obtain Green function for the Helmholtz equation?

all. I am following Jackson's Classical Electrodynamics. At Chapter 6.4, the book introduces how to obtain Green functions for the wave equation and the Helmholtz equation. I have a problem in fully ...
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### Logistic population equation and exponential model

In Verhulst's model of population growth and control, Let $K$ represent the carrying capacity for a particular organism in a given environment, and let $r$ be a real number that represents the growth ...
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### Differential equation of a series $RLC$ circuit driven by a DC voltage source?

From math below it seems no oscillations are possible and the steady state reaches instantly. I know this is wrong but I'm new to differential equations and don't see my mistake. Summary: For the ...
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### Wave equation - three questions

there are three question that I've never thought about the homogeneous wave Equation. For the Cylindrical wave, here it is possible to find the derivation (see paragraph 5.9.3). Is the cylindrical ...
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### Speed of heat through an object

According to the Heat equation (the PDE), heat can travel infinitely fast, which doesn't seem right to me. So I was wondering, at what speed does heat actually propogate through an object? For example,...
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### Inhomogeneous Wave Equation

In some literature that I am consulting this is called inhomogeneous wave: $$\sigma=\sigma_{0}e^{-i\omega t}e^{-ikx}e^{\gamma x}$$ where $\gamma$ is an attenuation parameter. I imagine the term ...
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### Intuition for group property of the flow of differential equations

Consider the IVP \left\lbrace\begin{aligned}x' &= f(t, x), \\ x(0) &= x_0, \end{aligned}\right. with complete flow $\phi(t, x_0)$. If $f$ is smooth, then the flow has the group property, ...
I am thinking continuously regarding the additive constant in Hamilton-Jacobi theory. But I didn't get a good idea. Why only one additive constant, can we take 2 or 3 additive constants? S=S'+\...