Questions tagged [ordinary-differential-equations]
For questions about ordinary differential equations, which are differential equations involving ordinary derivatives of one or more dependent variables with respect to a single independent variable. For questions specifically concerning partial differential equations, use the [tag:pde] instead.
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Global uniqueness of the solution to a specific nonlinear ODE system with "global" initial condition
I get stuck trying to prove the uniqueness of the solution $y=\begin{pmatrix}y_1\\y_2\end{pmatrix}$ to the following system of nonlinear ODE's
\begin{align*}
y'_1&=\frac{y_1-y_2}{\xi(t)-y_1}\xi'(t)...
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Does probability flow ODE trajectory (in the context of diffusion models) represents a bijective mapping between any distribution to a gaussian? [closed]
I have read several papers about diffusion models in the context of deep learning.
especially this one
As explained in the paper, by learning the score function $(\nabla \log(p_t(x)))$, probability ...
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How do I approach this differential equation?
I am an undergraduate physics student. I have been assigned this equation by my professor, all he has told me that the solution must be elliptical. I have no idea how to go about solving it. Here's ...
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Is Trajectories of a system same for different initial conditions on the trajectories?
I have a system of differential equation $\frac{d\rm x}{dt}=f(\rm x, t)$, where $\text{x}\in A \subset \mathbb{R}^n$ and $t\geq 0$. The function $f$ is a smooth function. The set $A$ is the unit cube. ...
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How to solve integrating factor PDE
So I was trying to find a solution to this ODE,
$$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{y^2}{(x^2+y^2)^{\frac{5}{2}}}\iff (x^2+y^2)^{\frac{5}{2}}\,\mathrm dy-y^2\,\mathrm dx=0,$$
following the OP's ...
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Determining $t(x)$ from $\frac{dx}{dt}$?
I have a question that "feels" basic/stupid, but I've been really struggling with it. The basic question is: is there an easy way to determine $t(x)$ from $x(t)$ or $\frac{dx}{dt}$?
To ...
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Analytical method for a non-linear differential equation: $y' =(\frac{2 + 3 \cos(2x)}{y}) + 4$. [closed]
Is there any analytical or numerical solution for this equation: $y' = \left(\frac{2 + 3 \cos(2x)}{y}\right) + 4$.
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Geometric interpretation of controlled differential equations
A controlled differential equation is a differential equation $$dy(t)=F(y(t))dx(t),$$ where $x(t)$ is an input curve of bounded variation in a (finite-dimensional) vector space $V$, $y(t)$ is a ...
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Can you help me for prove this Elzaki transform? [closed]
a) proof
$$ E[tf'(t)]=v^2 \frac{d}{dv} [\frac{T(v)}{v}-vf(0)]-v[\frac{T(v)}{v}-vf(0)]$$
Using Elzaki transform
$$E[tf'(t)]=v^2 \frac{d}{dv} [E(f'(t))]-vE(f'(t)) $$
using$$ E[f'(t)]=\frac{T(v)}{v}-vf(...
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Finding an explicit optimal map using Brenier's Theorem
I have the following set up:
$$
X, Y = [-1,1] \\
\text{Probability Measures } \mu, \nu \text{ with densities:} \\
f(x)= \frac{15}{4}x^2(1-x^2) \\
g(y)=\frac{3}{4}(1-y^2) \\
\text{Cost function: } c(x,...
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Difficult set of differential equations for epidemiology help needed (three equations in total) [closed]
The equations are mostly simple operations, but I can’t solve it. I tried a bit of substitution, but I think I am doing it wrong.
$I’(t)=Sr_0$
$R(t)=N-I(t)-S(t)$
$S(t)=N-I(t)-R(t)$
$I(0)=I_0$
Please ...
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Mismatch between impulse response gotten from differentiation of unit step response vs solving directly with input $\delta(t)$ for operator $ 2D + 1 $
To find the unit step for the operator $ 2D + 1 $, where $ D $ is the differential operator, we proceed as follows:
Unit Step Response
The unit step function $ u(t) $ is defined as:
$$ u(t) = \begin{...
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An integral identity involving a generalized hypergeometric function.
Let $\theta \ge 0$, $S\ge 0$, $\zeta \ge 0$ and $x \ge 0$ be real numbers and let $q \ge 1$ be an integer. Then the following identity below holds true:
\begin{equation}
G_\zeta(x) := \int\limits_0^\...
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Annihilator method for solving systems of ODEs
According to Chapter 6 of the book titled "Fundamentals of Differential Equations and Boundary Value Problems" writen by Nagel, Saff and Snider, the classical "method of undetermined ...
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Excitability of the FitzHugh-Nagumo model
I have the following variant of the FitzHugh-Nagumo model:
$$\dot{u} = u - u^3 - v \\ \dot{v} = \epsilon(u-a)$$
Where $\epsilon>0$ and $a$ is a constant.
I need to give a value for $a$ such that ...
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