All Questions
34 questions
1
vote
0
answers
52
views
Stability of Euler discretization
I am looking at the discretization of an ODE:
$$x_{n+1} = x_n + \alpha f(x_n),$$
where $x_n\in R^d$ and $f$ is continuously differentiable and such that $f(0)=0$ and $f'(0)$ is Hurwitz (i.e., the real ...
- 562
1
vote
1
answer
109
views
Bound on $L^1$ norm of solution of two-point boundary value problem
This has to be known, but I have not been able to find it in the literature (probably due to not being too familiar with two-point boundary value problems). I have a function $u:[0,1]\to\mathbb{R}$ ...
- 3,065
3
votes
1
answer
252
views
Reference request: analysis of a nonlinear Fokker-Planck type equation
It is well-known that the linear Fokker-Planck equation (written in one space dimension for simplicity) $$\partial_t \rho = \partial_x \left(\rho_\infty \partial_x\left(\frac{\rho}{\rho_\infty}\right)\...
- 730
0
votes
2
answers
163
views
Convergence of solutions to parametrized ODE when no limiting ODE exists
There is plenty of literature on the convergence of the solutions to the real ODE, parametrized by $N \in (0;\infty)$,
\begin{equation}
f_N' (x)
=
a_N (x) \cdot f_N (x)
+ b_N (x)
\end{equation}
to the ...
- 335
2
votes
1
answer
231
views
Solution of Riccati system of ODEs
We have following equation:
$$
w(t,v) = \exp\Bigl(-\phi (t) \frac{v^2}{2}-\psi (t) v -\chi (t)\Bigr),\quad (t,v)\in [0,T]\times \mathbb{R},
$$
where $(\phi, \psi ,\chi)$ are solutions of the Riccati ...
- 43
3
votes
2
answers
552
views
Second order differential equation with oscillating behavior
I consider a differential equation $y^{\prime \prime} (x) + V(x) y(x) = 0$ in the interval $[0,\infty)$, where $C_1 \leq V(x) \leq C_2$ for all $x \in [0,\infty)$ for some constants $C_2 > C_1 >...
- 5,562
2
votes
0
answers
55
views
References for generalized confluent hypergeometric differential equation
According to Wolfram, a generalization of the confluent hypergeometric differential equation is given by:
$$y''+\left(\frac{2R}{x}+2F'+p\frac{H'}{H}-H'-\frac{H''}{H'}\right) y'+\left[\left(p\frac{H'}{...
user161091
5
votes
1
answer
1k
views
Continuous dependence on initial parameters of an ODE for non-Lipschitz functions?
For ODEs, the standard theorem of continuous dependence of initial parameters deals only with functions that are Lipschitz. Do there exist more general results holding for non-Lipschitz functions? If ...
- 215
0
votes
1
answer
117
views
Lorenz ODEs with negative parameters
Consider the Lorenz system
$$\dot{x}(t) = \sigma(y-x) \, ,$$
$$\dot{y}(t) = x(\rho-z) - y \, ,$$
$$\dot{z}(t) = xy-\beta z \, .$$
Usually one considers the parameters $\sigma, \rho,$ and $\beta$ to be ...
- 3,574
7
votes
2
answers
593
views
Dependence of a solution of a linear ODE on parameter
Is the following theorem known, or can be easily derived from known results?
Consider the differential equation
$$w''-kz^{-1}w'=(\lambda+\phi(z))w,$$
where $k>0$ is fixed, $\lambda$ is a large (...
- 91.8k
13
votes
1
answer
661
views
Poincaré on analytic dependence on parameters of solutions of linear differential equations
There is the following important General Principle: if a parameter enters
in a linear differential equation additively, for example
$$\frac{d^2w}{dx^2}+(q(x)+\lambda)w=0,$$
where the parameter is $\...
- 91.8k
12
votes
4
answers
2k
views
History of ODE and PDE reference request
Is there any reference (book or articles) which made the history (up to the modern times) and the conceptual development of Ordinary Differential Equations and Partial Differential Equations? It will ...
- 1,759
6
votes
0
answers
227
views
Origins of the generalized shift operator exp(t*g(z)d/dz)
Charles Graves in the 1850s investigated iterated operators of the form $g(x) \frac {d}{dx}$ (see page 13 in The Theory of Linear Operators ... (Principia Press, 1936) by Harold T. Davis). Graves ...
- 10.5k
3
votes
1
answer
202
views
Stability of nonsmooth, Lipschitz continuous, autonomous system of differential equations
Consider the following autonomous system of differential equations:
$$\frac{\mathrm d\mathbf x}{\mathrm dt} = \mathbf v(\mathbf x)$$
where $\mathbf x, \mathbf v \in \mathbb R^n$. Assume that $\...
- 884
43
votes
3
answers
7k
views
Could the Riemann zeta function be a solution for a known differential equation?
Riemann zeta function is a function of complex variable $s$ that analytically continous the sum of Dirichlet series .defined as :$$\zeta(s)=\sum_{n=1}^{\infty}\displaystyle \frac{1}{n^s} $$ for when ...
5
votes
3
answers
478
views
Nonlinear ODE: $y'=(1+axy)/(1+bxy)$
Consider the first order nonlinear ODE problem:
$$
y'(x)=\frac{1+ay(x)x}{1+by(x)x}, \quad x>0
$$
where $a, b>0$ are some constants. I would like to know if these kind of equations were ...
- 3,946
2
votes
1
answer
455
views
Smooth dependence on the initial condition of the integral of an ODE
I am considering an ODE $\dot{x}=f(x)$, with $x\in\mathbb{R}^d$ and $d<\infty$. $f$ is a $C^k$ function and I denote by $\Phi_t x$ be the value of the solution at time $t$.
I assume that my ODE ...
- 562
2
votes
0
answers
79
views
Point Spectrum of a Second Order System of Differential Equations
Consider the following operator acting on $H^1(\mathbb{R})$
$$
\mathcal{L} \left(\begin{array}{c} \phi \\ \psi \end{array}\right) = -\left(\begin{array}{c} \phi \\ \psi \end{array}\right)'' + V(x) \...
- 516
3
votes
0
answers
135
views
Motivation for the existence of periodic solutions [closed]
I have been reading the book Critical Point Theory and Hamiltonian System by Mawhin and Willem, as well as several other papers on the existence of periodic solutions for equations of the form
$$\ddot{...
- 41
2
votes
0
answers
280
views
Uniqueness of analytic center manifold
In a book, i have read a remark which says that the center manifold of an equilibrium point of a differential equation is not unique in general but is unique in the class of analytic manifold. The ...
- 21
4
votes
1
answer
354
views
Reference request: Invariant sets of dynamical systems
(I should preface this with the disclaimer that this is a slightly elaborated version of a question that I posted onto math se recently to which I received no responses, and have since deleted the ...
- 185
7
votes
2
answers
697
views
Local positivity of solutions to linear differential inequalities (Chaplygin's theorem)
According to the entry "Differential inequality" of the Encyclopedia of Mathematics
http://www.encyclopediaofmath.org/index.php/Differential_inequality
the following result is due to Chaplygin (1919)...
- 1,371
1
vote
1
answer
207
views
Number of solutions of a system of equation!
Let $\Theta =(\theta_1,\ldots, \theta_n)\in {\mathbb T}^n$. I want to show that the system of equations
$$
\sum_j 2\sin(\theta_i -\theta_j)+\sin(2\theta_i -2\theta_j) =0,\ \ i=1,\ldots, n,
$$
has ...
2
votes
1
answer
255
views
Parameter dependent differential equation in a Lie group
It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference ...
- 5,824
4
votes
1
answer
902
views
Exact Differential Equations of Order n via Pfaffian Differential Equations?
I'm wondering if somebody could both shed some light on, & offer references for more details about, this interesting quote:
The derivation of the conditions of exact integrability of an ...
- 93
12
votes
2
answers
2k
views
Reference for a nice proof of "undetermined coefficients"
I'm teaching an honors differential equations class and have been using linear algebra heavily. I thought it would be interesting to include a proof of the method of undetermined coefficients along ...
- 7,273
26
votes
9
answers
14k
views
Graduate ODE textbook
Suppose that a hypothetical math grad student was pretty comfortable with first-year real variables and algebra, and had even studied some other things (algebraic geometry, Riemannian geometry, ...
2
votes
3
answers
3k
views
Power series solutions for nonlinear ordinary differential equations - references
I'm having a hard time finding some references on series solutions for "nonlinear" ODE's, the most I could find was a small excerpt on Wikipedia.
https://en.wikipedia.org/wiki/...
- 579
4
votes
5
answers
891
views
Analytic hypoellipticity of linear ordinary differential operators
Let $P = a_n(x) D_x^n + a_{n-1}(x) D_x^{n-1} + \ldots + a_0(x)$ be a linear ordinary differential operator with polynomial (or real analytic) coefficients $a_j(x)$. Suppose that $a_n(x)$ doesn't ...
- 1,412
4
votes
1
answer
1k
views
Limit of a discrete time dynamical system
I have the following discrete time dynamical system
$$ y(t+1) = y(t) + \frac{1}{1+\exp(z+ u f y(t))} ,\quad y(0)=0,$$
where $z$ is a real number $f$ and $u$ are non-negative reals. I know I have ...
- 53
54
votes
13
answers
90k
views
Good differential equations text for undergraduates who want to become pure mathematicians
Alright, so I have been taking a while to soak in as much advanced mathematics as an undergraduate as possible, taking courses in algebra, topology, complex analysis (a less rigorous undergraduate ...
3
votes
1
answer
2k
views
A formula for the Jacobian of a flow
Let $U : \mathbb R^d \to \mathbb R^d$ be a smooth vector field, and let $F_t : \mathbb R \times \mathbb R^d \to \mathbb R^d$ be the corresponding smooth flow, defined by the differential equation $$\...
- 8,512
5
votes
2
answers
952
views
Good references for analytic solutions to nonlinear ordinary differential equations?
I am faced with a non-autonomous initial value problem for a function $x:[0,\infty) \to \mathbb{R}^2$ of the form
$$ x'(t) = f(t,x(t)) $$
for $f: [0,\infty) \times \mathbb{R}^2 \to \mathbb{R}^2$ with ...
- 33.3k
4
votes
2
answers
735
views
Analyzing the solution to a second-order, non-linear ODE
Let $\psi : [0,\infty] \to \mathbb R$ be a strictly positive, continuously differentiable function, and consider the non-linear ODE $$\ddot x = - \frac{1}{4} \frac{\psi'(x)}{\psi(x)} \left( \dot x^2 - ...
- 8,512