Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.
Questions tagged [differential-equations]
1651 questions
19
votes
4 answers
ODE's without a Lipschitz condition
When teaching ODE's earlier this semester, one of my students asked the following question for which I didn't know the answer (and none of the textbooks I consulted seem to discuss it). It is standard that if $f(x,y)$ is Lipschitz, then the ODE…
Arthur
- 191
12
votes
2 answers
curvature flow for loops in S^2
Consider the unit 2-sphere and a smooth simple closed curve c embedded in it. I would guess that under the well studied parabolic equation which evolves the curve according to its curvature vector, that c shrinks through embedded sccs to a "round"…
Michael Freedman
- 735
- 4
- 7
9
votes
2 answers
Simplest proof of dimension of solution space for linear ODEs
Given a general n-th degree linear ODE, what's the easiest way to prove that there are precisely n linearly independent solutions?
ahh
- 99
9
votes
1 answer
Higher Airy functions: an exponentially decreasing solution of $f^{(n)}(x) = (-1)^n \, x^a \, f(x)$
Consider the differential equation
$$
f^{(n)}(x) = (-1)^n\, x^a \, f(x),
$$
where $a > 0$ is a real number.
My numerical experiments suggest that this equation has a unique solution on $\mathbb{R}_+$ satisfying
$$
f(x) \sim_{x \to \infty} x^{-\beta}…
Dimitri Zvonkine
- 363
8
votes
1 answer
An upper bound for the solution of an integro-differential equation
For those who are interested in "motivations" this has something to do with modeling flames in turbulent jets. However the question itself is irritatingly elementary and requires no mathematical or applied background whatsoever. Here is the simplest…
fedja
- 59,730
7
votes
0 answers
What subspaces of n-tuples of rational functions can be the solution space to a system of differential equations?
Let $K=\mathbb{C}(x)$ denote the field of rational functions (in 1 variable), and let $K^n$ denote an $n$-dimensional $k$-vector space (with basis). For some integer $m$, let
$$\delta_{11},…
Greg Muller
- 12,679
7
votes
3 answers
General systems of linear differential equations with variable coefficients
I am wondering what can be said in general about the fundamental matrix of a system of linear differential equations. For simplicity, let $A(t)\in\mathbb{C}^{n\times n}$ be a time dependent matrix, smooth on some interval $I$, say $[0,t_I]$.
Is is…
PLG
- 81
6
votes
1 answer
If an initial value problem has a solution on $[0,a)$ for all $a>0$, will it have a solution on the whole $[0,\infty)$?
Consider the initial value problem on $[0,\infty)$:
$$x'(t)=f(t, x(t)) \qquad x(0)=0,\label{1}\tag{$*$}$$
where $f:(0,\infty)\times\mathbb R\to\mathbb R$ is a continuous function. Assume that for every $a>0$, there exists a $C^1$ solution $x(t)$ to…
Feng
- 517
5
votes
3 answers
Any help on one ODE
Some weeks ago I was asked to solve one ODE. I tried all methods I know, but couldn't crack this equation. Also I tried to use Matlab's dsolve function - without any result.
$y' + \frac{2x}{y} = x^2$
Does anyone have any suggestions on this?
P.S.…
5
votes
2 answers
Existence of Solution for a Differential Matrix Equation
I have some problems proving that a Differential Matrix Equation has a solution. I apologize if the question is too elementary, but I've found this theorem stated everywhere on the web without any reference or clue about how to prove it.
What I…
4
votes
3 answers
Looking for the solution of first order non-linear differential equation ($y ′+y^{2}=f(x)$) without knowing a particular solution
I have been working on Riccati Equation. I have tried many different methods to find a closed form for the solution of first order non-linear differential equation ($y'+y^{2}=f(x)$) without knowing a particular solution. My aim is to open a topic…
Mathlover
- 302
4
votes
1 answer
How to solve an ODE with an unknown parameter but given y values at three points
I have a second-order ODE with an unknown parameter $p$,
$$\frac{y''}{(1+y'^{2})^{\frac{3}{2}}} -p - A(x-B)^2 =0,$$
where $x$ is the independent variable, $y$ is the unknown function, $p$ is unknown parameter to be determined, and $A$ and $B$ are…
Jilong Yin
- 161
4
votes
1 answer
Sobolev density of smooth functions which are zero on a measure zero subset
Suppose that $\Omega$ is a bounded domain and Let $A\subseteq \Omega$ is a subset of measure zero. Is the set of smooth functions which are zero on $A$ dense in Sobolev space? For instance $W^{1,2}(\Omega)$?
Ryan Vaughn
- 409
4
votes
1 answer
Solve differential system of equations
Consider the following system:
$$
\begin{cases}
x_1 + 3 x_3 = 4a, \\
f(x_1) + 3 f(x_3) = 8 f(a), \\
f'(x_1) = 3 f'(x_3).
\end{cases}
$$
I want to find all functions (or at least learn some properties that hold for all of them) $f : [0,1] \to [0,1]$…
Yauhen Yakimenka
- 325
4
votes
1 answer
Criterion for finite time blowup of an ODE
Hi I'm trying to understand the most general conditions under which I can conclude finite time blow up of an ODE of the form $\dot{x} = f(x)$ with initial condition $x_0 > 0$ and $f(x) \geq 0$ for all $x \geq 0$.
If I re-write this in a separable…
Dorian
- 2,601