# Questions tagged [differential-equations]

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### Physical systems satisfying the differential equation $y' = {}$ rotation of $y$ by $t$?

Denote by $R(t)$ the matrix that rotates by $t$, i.e. $$R(t)=\begin{pmatrix} \cos(t) & -\sin(t) \\ \sin(t) & \cos(t) \end{pmatrix}.$$ Are there any (realistic?) physical systems which ...
0answers
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### Kinetic energy of point $B$ which has $0$ mass

In the problem, I came to find the kinetic energy of point $B$ which has $0$ mass as the problem states. But the force $\vec F$ is acting on it as in the figure below: How can I find the energy? ...
0answers
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### Find an optimal shape analytically

This is better described with an example: Imagine we posses our current understanding of physics, but we do not know how to make an airplane because we do not know about airfoils. To make this more ...
0answers
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### How to rewrite a differential equation into another form of differential equation? [closed]

How to write the differential equation $$x^2 \ddot y +2x\dot y - y = x^2$$ into the form $$\ddot y +\dot y - y = x^2$$ where $y$ is being integrated in terms of x? I tried manipulating the above ...
1answer
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### From the initial condition problem of the Euler-Lagrange equation to the principle of least action

I browsed through many similar questions about the Initial Condition Problem (ICP) and Boundary Value Problem (BVP) for Euler-Lagrange equations, here some interesting but (in my opinion) incomplete ...
1answer
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### Is there a differential equation describing the wavefunction of a hadron?

In Newtonian Physics there's a differential equation describing the motion of multiple bodies in orbit around each other. In non relativistic quantum mechanics there's a differential equation ...
2answers
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### Equation and boundary conditions for the temperature problem in a cylinder with thermal power on its axis

I'm having trouble trying to establish the equation for the following problem: On the axis of a long cylinder, whose radius is r = r0, there is a conductor with a current that releases the thermal ...
1answer
176 views

### Perturbation theory on perihelion advance

I'm trying to get a relativistic solution to Kepler's equation starting with $$\frac{d^2 u}{d\phi^2}+u = \frac{M}{L^2}+3Mu^2$$ by ignoring the higher order terms; $$u(\phi, \epsilon)=u_0+\epsilon u_1$$...
0answers
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### ODEs with rational first integrals [closed]

I would like some examples of ODEs (i.e., $\dot{x}=f(x)$, where $x\in\mathbb{R}^n$) that possess one or more rational first-integrals of the form $$H(x)=\frac{a_1^Tx+\alpha_1}{a_2^Tx+\alpha_2},$$ ...
4answers
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### Heat Diffusion Equation with extra terms

$$\frac{\partial U}{\partial t}=\alpha \frac{\partial^2 U}{\partial x^2}+\beta U$$ I have been given this partial differential equation and am asked to find an application for it. I can see that the ...
0answers
31 views

### Converting a partial derivative from Lagrangian to Eulerian coordinates

This question pertains to section 2.3 of https://www.whoi.edu/science/PO/people/jprice/class/ELreps.pdf Lagrangian and Eulerian Representations of Fluid Flow:Kinematics and the Equations of Motion (...
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1answer
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### How to solve a stochastic differential equation

This is something I’ve been puzzled about over the past week or so. How do you go about getting the equilibrium probability distribution solution for a stochastic differential equation, like ...
2answers
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### The first and second form of Euler-(Lagrange) equation with explicit time dependence

I have learned the first and second form of Euler-(Lagrange) equation with no explicit time dependence (the time dependence only implicit on the function to be solved, say $y\left(t\right)$), from ...
1answer
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### Integration of Bloch equation in magnetic resonance

From Bloch equation we have \begin{equation}\label{bloch_01} \tag{1} \frac{d M_z}{dt} = \frac{M_0-M_z}{T_1} \end{equation} from there we can integrate and we get \begin{equation}\label{bloch_02} \tag{...
2answers
106 views

### What do we exactly mean when we say that a problem has an analytical solution?

What do we actually mean when we say that a certain problem has or does not have an analytical solution? I ask this because some systems that are said to have an analytical solution actually are no ...
0answers
26 views

### Poincare section for the duffing Oscillator

I have used the 4th Order Runge-Kutta method in order to estimate the values in which the Duffing Oscillator is chaotic. According to Wikipedia, the Duffing Oscillator is chaotic for values of $\alpha$...
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### Solving Laplace equation on a triangle with mixed boundary conditions

From sources [Ref: 1 to 5], one learns that there is a class of boundary conditions (see Fig 1) on a triangle (in this case the 30-60-90 triangle) that allow for closed form solution for eigensystems ...
1answer
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### Phase plots: The exact particular solution is a function of time, can't find fixed points. Now, in this situation, how to draw phase plots?

I want to draw phase plots. The differential equations are two coupled second-order non-linear differential equations. I have the exact particular analytic solutions. However, the solutions are a ...
1answer
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### Special Conformal Transformation Acting on Spinor Variables

I'm working in 3,1 Minkowski spacetime, representing null vectors as a product of two commuting spinors so that eg. $$p_i^{\dot{\alpha}\alpha} = |i]^{\dot{\alpha}}\langle i|^\alpha.$$ I know that ...
1answer
29 views

### 2D Incompressible Fluid Simulation solving / diffusion factor

I've been reading about fluid simulations - specifically, incompressible fluid dynamics using the incompressible Navier-Stokes equations. Every resource I've looked at has two key components that I ...
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### Set of first order ODE [duplicate]

What's the condition for $\dot{x} = f(x,y)\\ \dot{y} = g(x,y)$ To be rewritable as $\dot{x} = \frac{\partial F(x,y)}{\partial y}\\ \dot{y} = -\frac{\partial F(x,y)}{\partial x}$ Can I always find a ...
1answer
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### Separation of variables for a function of 3 variables $V(x, y, z)$ [closed]

I'm trying to find the solutions for $V(x,y,z)$ by separation of variables. Is it correct to say: $$\frac{1}{X}\frac{d^2x}{dx^2} + \frac{1}{Y}\frac{d^2y}{dy^2} + \frac{1}{Z}\frac{d^2z}{dz^2} = 0$$ ...
1answer
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### Differential form of Massieu’s function [closed]

Massieu’s function is given by: $$F_{M}=-\frac{U}{T}+S$$ And its differential form is given by: $$dF_{M}=\frac{U}{T^{2}}dT+\frac{P}{T}dV$$ Well, it seems that: $$\frac{\partial S}{\partial T}=0$$ How ...
0answers
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### Boundary conditions for equation of motion of a chain under small vertical motion of its support

I would like to find the boundary conditions for the of motion of a chain under small vertical motion of its support endpoint. I also assume displacement of the chain in the vertical $y$ direction is ...
2answers
72 views