# Tagged Questions

DO NOT USE THIS TAG just because the question contains a differential equation!

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### Physical background to the ODE $y'(x) + \frac{1}{x} = y(x)$

Most books on asymptotic methods start with a discussion on the ODE $y'(x) + \frac{1}{x} = y(x)$, which has solution $$y(x) = \int_0^{\infty} \frac{e^{-xt}}{1+t} \,dt.$$ A discussion on the ...
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### Can one classify partial differential equations according to the causality properties of their solutions (and if yes, then how)?

Recently, I bumped into this interesting comment by Valter Moretti which made me wonder about the following, more general question (to which I suspect the answer is affirmative): Can we easily tell, ...
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### Solving a first order non linear differential equation in the specific case of the orbit of a particle moving under a central conservative force

Hello it's my first time posting in here and I hope someone in here can help me as my teachers quite dislike questions that aren't specifically in the curriculum. I'm reading About classical mechanics ...
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### Differential equation for velocity regarded as a function of distance [closed]

Given a differential equation for velocity, $dv/dt + v = 1$, as well as its solution, is it possible to derive a differential equation for velocity with respect to distance? I found a solution to the ...
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### Spring-mass system with variable stifness $m \ddot{x}+k(x)x=0$, time period is known, stifness is unknown

This question is somewhat related to my previous question: What is the time period of an oscillator with varying spring constant? In that question, time period of mass-spring system with variable ...
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### Initial conditions for second order ODE with complex stiffness

I tried this on Math Stack Exchange. I'm trying to find initial conditions to ensure systems of the form stay bounded $$\ddot{x}_i+\sum_{j=1}^N k_{ij} x_j = 0, \quad k_{ij} \in \mathbb{C}.$$ For ...
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### Non-Linear O.D.E

I have reached a set of ODE as \begin{align} &\ddot{\vec{a}}(t)+\omega_0^2\frac{\cos{(b(t))}\sin{(a(t))}}{a(t)}\vec{a}(t)=0\\ &\ddot{b}(t)+\omega_0^2\cos{(a(t))}\sin{(b(t))}=0 \end{align} ...
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### How do I correctly introduce time into this equation?

So, for the past few years it's been my goal to create an equation that would give me the position of an object in a gravitational field at time $t$, given it's initial position and velocity. At first ...
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### Schrodinger equation, commutative operators, and Symmetry

When solving Schrodinger's equation in 3D with a spherical laplacian you reach a point at which you introduce a separation constant and can see that the same eigenvalue satisfies the radial and ...
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### Poisson equation in 2D and 3D: geometrical reason for the difference

The Poisson equation in 3D shows a fundamental solution in 3D which decays with $\sim 1/r$, whilst in 2D it shows a much different decay $\sim -\ln r$. While in 3D not only the solution, but also its ...
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### Laplace equation between circles [closed]

I need to solve the simple Laplace equation $$\nabla^2f(r,\theta)=0$$ with boundary conditions: $$f(a,0)=g(\theta)$$ $$\lim_{A\rightarrow\infty}f(A,\theta)=1$$ what would be a straightforward way to ...
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### What dynamical system could this $\dot y = \alpha(y-\lambda), y\geq \lambda$ equation describe?

Just out of curiosity, can anyone identify electrical, mechanical, chemical, etc process that is governed by a differential equation of the form $$\dot y = \alpha(y-\lambda), y\geq \lambda$$ where ...
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### General boundary condition for 1D heat equation

I'm studying from Numerical Solution of Partial Differential Equations by K.W.Morton and D.F.Mayers (Amazon link). I'm confused with general boundary conditions. Could someone give me a clue? For ...
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### How to determine sign of coefficients in simple spring, damper, mass system?

For a system of the sort shown below: I have come to realize that I continuously make mistakes when it comes to determining the signs (or specifically the direction of the forces) of the ...
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### Concentration distribution in a phase separated mixture. Can't get the correct ODEs and boundary conditions

I wish to compute the equilibrium concentration distribution of a binary mixture that has phase separated. I start with writing the free energy as a functional depending of the concentration. I use ...
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### Integrating Factor Solution for Plasma Wave Equation

As part of a derivation in Bernstein '58 [1] a linear first-order (eqn. (9) in the image) appears: But the general solution I would usually take (as appears in Gradshteyn and Ryzhik and checked in ...
I am interested in modeling the time evolution of phase-space $\rho(\vec{q},\vec{p},t)$. I have attempted to use Liouville's theorem \$\partial_t\rho=-\sum_{i=1}^{3}(\partial_{q_i}\rho)\dot q_i+(\...