# 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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### A continuous differential equation with no solution

I am studying the following counterexample. But I don't understand where the fact that space is infinite dimensional is used. What fails in an infinite dimensional space?
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### A first order linear differential question that may have only the trivial solution $y = 0$.

Below is a problem I made up. I expected the differential equation to have a unique non-trivial solution. However, it did not. Is my solution wrong? Problem: Solve the following differential equation:...
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### find numerical solution of an integral equation

Let $f(u)=\int_0^u f(x)g(u-x)dx+h(u)$ where $g$ and $h$ are known, continuous function and $f$ is unknown and let $a=u_0<\dots<u_n=b$. How can I calculate $f(u_i)$ numerically? I thought about ...
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### Intuition and Timescale Analysis for a Nonlinear Differential Equation Dependent on Parameter $K$

I am exploring the behavior of the following first-order nonlinear differential equation: $$\tau \frac{dy}{dt} = 2Ky^K(a - y^Kb)$$ where $y, a, b \in \mathbb{R}$, $K \in \mathbb{N}$, and $\tau$ ...
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### Solution to Stochastic Differential Equation Including Heaviside Step Function

There has been several entries on solving deterministic differential equations that include indicator functions. Stochastic differential equations may introduce new difficulties. Namely, does the ...
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### Showing a function satisfying a certain differential inequality must always be positive

Suppose that $$f''(t)+cf(t) \geq |f(t)|^p$$ for $t\geq0$, where $c>0$ and $p>1$, $f \in C^2([0, \infty))$, and we know also that $f(0)>0$ and $f'(0)>0$. Is it true that $f(t)\geq 0$ for ...
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### Is there a source out there which works out a Hartman-Grobman-type theorem on a manifold?

My general goal is to understand more aspects of dynamical systems in the framework and language and differential geometry, and while for most things I can make out some sources which do that, I cant ...
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### Why do we linearize about critical points?

I'm a beginner who's just learned about autonomous equations as an intro to non-linear diff. eqs. I've noticed that in the few sources I've used, on the topic of linearizing first order autonomous ...
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### Seeking General Solution for Nonlinear ODE Without Initial Conditions

I am working on solving the following nonlinear second-order ordinary differential equation (ODE): $$f''(x) + f'(x) f''(x) + x f'(x) - f(x) = 0$$ I am looking for a ...
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### Optimal bidding and the value of the game

I migrated from this post, where the problem is a bidding game between two players on the sum of two fair die. Specifically, Player 1 rolls one die and sees the outcome and so does Player 2. They don'...
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### Checking if a polynomial is odd or even

I am reading a paper: 'Center of mass distribution of the Jacobi unitary ensembles: Painleve V, asymptotic expansions' by Zhan, Blower, Chen, Zhu. On page 14 we have a differential equation (4.29) for ...
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### Jack Hales 'Ordinary Differential Equations' as a first text on ODE's?

A few months ago I asked a question on here about comparing Arnold's text and Birkhoff/Rota's text, and the responses pushed me towards Arnold's. However after reading this question ... What is a good ...
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### Solution to heat equation on disk with boundary condition oscillating in time

I am trying to solve analytically the heat equation $\partial_t u = \Delta u$ on the unit disk $D_1\subset\mathbb{R}^2$ with Dirichlet boundary condition $e^{ikt}$. That is, I want a solution ...
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### How should I find the equilibrium points and the general equation for the phase paths?

Find the equilibrium points and the general equation for the phase paths of $\ddot{x}+\cos(x)=0$. Obtain the equation of the phase path joining two adjacent saddles. Above is the problem statement and ...
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### Assistance Needed on Invariant Curves in a $C^1$ Vector Field with Two Parameters
I've been exploring a problem for the past two months, and I would greatly appreciate your insights. I'm working with a $C^1$ vector field defined by the equation \$\dot{\mathbf{x}} = f(\mathbf{x}; \...