# 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 variables. For questions specifically concerning partial differential equations, use the [tag:pde] instead.

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### On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
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### How to solve $\dot{x} = \frac{f(x)}{\|f(x)\|}$?

How to solve the following ODE? $$\dot{x} = \frac{f(x)}{\|f(x)\|},$$ where $x : \mathbb{R} \to \mathbb{R}^n$, i.e., $x(t)$ is the trajectory. The right-hand side $f : \mathbb{R}^n \to \mathbb{R}^n$ ...
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### Witt's proof of Gelfand-Mazur / Ostrowski's theorem

Now asked on MathOverflow. Background: It seems that, after his groundbreaking work on quadratic forms and inventing Witt vectors, Ernst Witt developed the hobby of giving extremely short proofs to ...
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### Modelling a Water Rocket. Requires Some Validation and Help. ( WARNING : Extremely Long but Interesting Post )

Good day people of math.stackexchange.com UPDATE: Version 2 can be found here: https://physics.stackexchange.com/questions/275284/modelling-a-water-bottle-rocket-version-2-long-post-warning. This is ...
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### Separable non-linear ODE (with radicals)

I am trying to solve the equation $$\frac{dy}{dt}=\sqrt{\left(\gamma-1+\frac{2\alpha\beta}{2\alpha-1}\right)e^{-2\alpha y}-\frac{2\alpha\beta}{2\alpha-1}e^{-y}+1}\tag{1}$$ $y(0) = 0$; $t_{0}=0$; ...
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### When do differential equations induce maps between algebraic varieties and how to find these varieties

The intuition: Consider a single variable polynomial differential equation with integer-polynomial coefficients, for example $$y '' = -y$$ Then consider a pair of algebraic varieties (that is ...
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### Numerically solving a non-linear PDE by an ODE on the Fourier coefficients

I need to solve numerically a PDE of the form $$u_t(x,t)=u_{xx}(x,t)+u_x(x,t)^2-a(x)u_x(x,t)-a_x(x)$$ with initial condition $u(x,0)=u_0(x)$. I can assume that both $u(\cdot,t)$ and $a(\cdot)$ are ...
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### Solving a dual integral equation involving a zeroth-order Bessel function

Consider the following dual integral equations \begin{align} \int_0^\infty q^3 f_0(q) J_0 (qr) \, \mathrm{d} q &= g(r) \qquad\qquad\quad (0<r<1) , \\ \int_0^\infty f_0(q) J_0 (qr) \, \...
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### Recommendation for intro to geometric integrators?

Explicit Request Looking for book or lecture note recommendations on numerical optimization that (ideally) have the following: Emphasis on geometric and physical intuition Emphasis on symplectic ...
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### Region of attraction of simple ODE with perturbation

There are a few nice discussions about ROA covering a few subtopics: Region of attraction of : $x'=-y-x^3,y'=x-y^3$ via Lyapunov Function Region of attraction and stability via liapunov&#...
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