# Tagged Questions

Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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### Center Manifold Exercise (small solution for small changes of the parameter)

Hi I'm stuck with this problem at first I didn't know how to begin so I copy an argument from [Carr, Application of Centre Manifold Theory]. But I don't know how can I find the coefficient from a, b ...
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### Differential Equations: Jordan Form of a Matrix

I am using Lawrence Perko's book Differential Equations and Dynamical Systems, for my Differential Equations course. At the moment we are going over Jordan Forms of a linear system $x^{'}(t) = Ax$, ...
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### Proofs from Ch. 1 of Arnold's ODEs

I've started reading Vladimir Arnold's Ordinary Differential Equations on my own. I like it so far, the only problem is that all of the exercises (as yet) are of the type "prove $X$" and without an ...
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### Squares and constants in the dynamical system

I have $$\begin{eqnarray} x'&=& x^2 - y^2 -1 \\ y'&=& 2y \end{eqnarray}$$ How can I solve such a system? I have tried the substitution $X= x^2 - 1$ but I still get constants in ...
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### Existence of Periodic Solution

I'm working with the system of equations below that represents a Pendulum with constant forcing. \begin{align*} \theta'&=v\\ v'&=-bv-\sin(\theta)+k \end{align*} Where $\theta$ gives the ...
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### How to adapt the discrete-time to continuous, $(A) \Rightarrow (B)$?

in class was proved oseledets theorem for discrete time, following guidelines Ricardo Mañe book. Theorem discrete Oseledets (A) : Let $M ^ n$ be a Riemannian manifold, $f: M \rightarrow M$ be ...
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### How can one derive Stokes lines of the Stokes phenomenon of asymptotics from a differential equation?

Is there a standard technique to calculate Stokes lines and anti-Stokes lines of the Stokes phenomenon of asymptotics for a function defined as the general solution to a differential equation without ...
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### Second order DE question

I am looking for tips for this equation: $4xy''+y'+xy'+\frac{3}{2}y=0$. I am solving with the substitution y=a(x)b(x), but it is getting messy..
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### Solution to Schrödinger equation $\partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t)V(x)f(x,t).$

I want to solve $$i\partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t-t_0)V(x)f(x,t),$$ for any $V \in C^{\infty}[-1,1]$ and $f: [-1,1] \times \mathbb{R_{\ge 0}} \rightarrow \mathbb{C}$. I would ...
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### Finding the exact solution of a differential equation

Let $y=f(x)$. Is it possible to find an exact solution of the following differential equation?: $$\ddot y+2\dot y-5xy=e^{-2x}\nonumber$$ Many thanks in advance, -- Cesar
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### Let $Z=Z(x,y)$ be a solution of $\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}$ = 1

Let $Z=Z(x,y)$ be a solution of $$\frac{\partial z}{∂x}\frac{\partial z}{\partial y} = 1$$ passing through $(0,0,0)$. Then $Z(0,1)$ is 0 1 2 4 By Charpit Method I get the solution ...
The Thomas Fermi model of atoms and nuclei is used in many applications of atomic and nuclear physics. The ODE related to this model is: $$\frac{d^2}{dx^2}\phi(x)=x^{-\frac{1}{2}}\phi(x)^{3/2}$$ with ...
This is a curious question about the way George Green could have defined his Green's function. All the definitions I see have only Dirac-delta $\delta(x-x')$ function as their source on the RHS. But ...