All Questions
1,964 questions
11
votes
0
answers
100
views
When could a diligent calculus student compute all Picard iterates algebraically?
As is well known, in the typical proof of the Picard–Lindelöf theorem, one shows the existence of a solution of the initial value problem $y'(t) = f(t,y(t))$, $y(t_0) = y_0$ by considering the Picard ...
- 13k
0
votes
0
answers
32
views
question about some algebraic simplifications performed as we solve differential equations with Laplace transform
I am trying to follow this discussion of Laplace transforms on youtube:
https://www.youtube.com/watch?v=ofvkZXgbIxE&t=610s
The relevant portion is 10 minutes in to the video.
There is some algebra ...
- 101
13
votes
4
answers
2k
views
The ten most fundamental topics in geometric group theory
What are the ten most fundamental topics in geometric group theory?
This is a pedagogical question prompted by the fact that I am teaching geometric group theory to undergraduates. They are expected ...
2
votes
1
answer
111
views
Second order differential equation with non constant coefficient
Is it possible to solve the differential equation for $y(t)$ the following ODE?
$$
y^{\prime \prime}(t)+ \frac{f^{\prime}(t)}{2 f(t)} y^{\prime}(t) + k^{2} y(t) = 0
$$
It can also be rewritten as
$$
\...
- 45
2
votes
0
answers
74
views
When is a first-order delay differential equation equivalent to a higher-order ordinary differential equation?
The proportional delay differential equation
$$
xf'(x)+2xf'(x/2)+C+4f(x/2)-5f(x)=0
$$
with initial condition $f(0)=C$ expresses that Simpson's rule exactly integrates $f$ over any interval $[0,x]$ and ...
- 3,065
7
votes
1
answer
161
views
When is a non-linear first-order ODE equivalent to a linear second-order ODE?
The Riccati equation $y'(x)+y(x)^2=f(x)$ is non-linear, but can be transformed into the linear equation $-u''(x)+f(x)u(x)=0$ via $y(x)=\frac{u'(x)}{u(x)}$.
Is there a general statement known about ...
- 3,065
1
vote
0
answers
44
views
Differential system of equations I would like to simplify
I have 2 functions of time $f(t),g(t)$ and a condition for the time-derivative of a third function $h(t)$, say $$\dot{h}(t)=\dot{g}(t)\cos{f(t)},$$ so $h$ is defined provided a value for $h(0)$ (as $h(...
- 11
2
votes
1
answer
146
views
How to integrate this differential equation?
I am trying to solve the following differential equation:
\[
\frac{1}{\sqrt{1+y}} \frac{dx}{dy} - \frac{2\sqrt{1+y}}{x} = 2(x+5).
\]
After performing the substitution:
\[
p = \sqrt{1+y}, \quad y = p^2 ...
- 21
16
votes
1
answer
977
views
Pedagogically intuitive reformulation of Zorn's Lemma for functional analysis
While teaching an applied functional analysis class, I’ve noticed that students often struggle to develop an intuitive understanding of Zorn’s lemma. It’s relatively straightforward to explain why ...
- 5,824
2
votes
2
answers
426
views
Questions about some parallel between polynomial and differential equation
Do the relations between Galois groups and solutions to polynomial equations with one variable have a counterpart between Lie groups and solutions to differential equations ?
Do the relations between ...
- 3,094
0
votes
1
answer
74
views
Closed form solutions for the differential equation $f'(x)=B\left(\frac{1}{2},a\right) \sqrt{f(x)} (1-f(x))^{1-a},f(0)=0$
I wonder if there are more closed form (preferably, in elementary functions or basic special functions, like Zeta, Gamma and Polygamma) solutions for the differential equation
$f'(x)=B\left(\frac{1}{2}...
- 10.1k
0
votes
0
answers
76
views
Existence solutions of the system of equations on Riemannian manifold
Is there a way to show that the following system of two equations has a solution? I don't want to find an explicit solution, but just verify its existence.
$$f''(r) + \beta \coth(r) f'(r) = \rho_0 e^{-...
- 272
2
votes
2
answers
136
views
Non-linear recursion relation with fractional exponent
I'm trying to solve the following non-linear recursion relation:
$a_{n+1} = a_n + c\cdot a_n^b, \quad n \geq 1$,
where $a_1 > 0$, and $c > 0$ and $0 < b < 1$ are constants. Mostly I'd like ...
-4
votes
1
answer
139
views
The solution of Green’s function in Dirac Delta ODE
I’m asking about the solution of the 2nd order Green’s function ODE:
$$\left( \dfrac{d^2}{d\eta^2}+ q^2 - 1 \right) g(\eta) = \delta(\eta-\tilde{\eta}) $$
Which is given by:
$$ g(\eta) = c_1~ e^{t\...
- 117
3
votes
1
answer
80
views
Solution of $d Y_t/dt = A(t) Y_t, Y_0 = I_d$ is positive definite?
Let $\{A(t)\}_{t \in [0,1]}$ be time-varying symmetric matrices in $\mathbb{R}^{d\times d}$. We consider the following ODE for $Y_t \in \mathbb{R}^{d \times d}$
$$
\tag{1}
\frac{d Y_t}{dt} = A(t) Y_t, ...
- 399
0
votes
0
answers
51
views
Time periodic Euler flows
What are some examples of solutions to the incompressible Euler equation on the torus $u:\mathbb{R}\times \mathbb{T}^d\rightarrow \mathbb{R}$ (with $d\in \{2,3\}$)
$$\partial_t u+u\cdot \nabla u +\...
- 94
-1
votes
1
answer
86
views
How to solve this 2nd order Dirac Delta ODE
Any help how to solve this ODE to get $g(t)$:
$$\left( \dfrac{d^2}{dt^2}+ q^2 - \frac{2}{(2+3 t)^2} \right) g(t) = \delta(t-z) $$
I wonder if the delta function requires any techniques in solving. I ...
- 117
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
5
votes
1
answer
212
views
Stability of ODEs with polynomial nonlinearity
Consider the following ODE system:
$$
x′=f(x)\iff
\begin{pmatrix}
x_1^\prime \\
\vdots\\
x_k^\prime\\
\vdots\\
x_n ^\prime
\end{pmatrix} =
\begin{pmatrix}
f_1(x) \\
\vdots\\
f_k(x)\\
\vdots\\
f_n(x)
\...
- 807
4
votes
1
answer
167
views
Pressureless explicit solutions to incompressible Euler
What are some examples of (semi-)explicit solutions of the incompressible Euler equations which satisfy the following
they are pressureless
they are periodic in space
they have nontrivial time ...
- 94
0
votes
0
answers
44
views
Classical solution to a semi-linear parabolic PDE
Let $\Omega\subset \mathbb R^d$ be convex and compact with regular boundary $\partial\Omega$, where $d\ge 2$. Consider the boundary problem on $I\times \Omega$, for $I=(0,T)$ or $I=\mathbb R_+$,
$$\...
- 101
1
vote
0
answers
92
views
Possible error in Panayotounakos & Zarmpoutis's 2011 general solution to the Abel equation of the first kind $y'_x=y^3+F(x)$
When working through the solution method found in Panayotounakos & Zarmpoutis's paper [1] on the Abel equation of the first kind I have come across a possible mistake which leaves the resulting ...
- 176
0
votes
0
answers
40
views
Energy estimation of density operator to von Neumann equation
Consider the Schrödinger equation on $\mathbb R_+\times\mathbb R^n$ as follows:
$$i\partial_t\varphi(t,x)=-\frac12\Delta_x\varphi(t,x),\quad \varphi(0,x)=\varphi_0(x).$$
Denote by $\varphi$ its ...
- 1,399
2
votes
0
answers
74
views
Approximate solutions to $x''(t)=-cx + f(t)x$
I recently studied a problem which involved two particles joined by a harmonic spring moving in a potential and through some manipulation, I obtained the equation
$x''(t) = -\omega^2x + f(t)x$,
where $...
- 3,738
1
vote
0
answers
106
views
The proposition associated with a set
Given a set $U$ and a set $A \subseteq U$, is there an accepted symbol for the proposition $p$ over the universe $U$ such that for each $x \in U$, $p(x)$ is the assertion that $x \in A$? (The symbol $...
- 19.7k
0
votes
0
answers
65
views
Rotations and bi-analytic functions
Are the bi-analytic functions $\partial^2_{\overline{z}} f=0$
invariant under rotations?
- 41
0
votes
0
answers
32
views
Simple functional particular solution for the Riccati equation
In deriving the generating function for permutation statistics, I encountered a Riccati equation that the generating function satisfies, but I am unable to find any simple functional particular ...
3
votes
1
answer
240
views
Solutions and asymptotics of the ODE $ f''=f^{-\alpha} $
Consider the ODE $ f''=f^{-\alpha} $, where $ \alpha>1 $ and $ f>0 $ in $ \mathbb{R} $. Assume that for $ [f]_{\frac{2}{\alpha+1}}\leq A $, where $ A>0 $ is a constant and
$$
[f]_{\frac{2}{\...
- 1,219
1
vote
0
answers
50
views
type of solutions of $-u^{\prime\prime}=\lambda e^{u}$ based on the value of the parameter $\lambda$. (Gelfand problem)
My question comes from the book
Stable solutions of elliptic partial differential equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 143. Boca Raton, FL: CRC ...
- 11
1
vote
0
answers
50
views
Solvability of a Riccati Equation with Periodic Coefficients
Consider the Riccati equation
\begin{equation}
y'(x)=P+\left(A\cos(fx-\phi)+Q\right)y(x)^2,
\end{equation}
for $A,Q\in\mathbb{R}$ and $P,f,\phi\in\mathbb{R}_{>0}$ subject to the initial condition $...
- 41
5
votes
0
answers
204
views
When is a Function a Flow
Let $f:\mathbb{R}^d\to \mathbb{R}^d$ be a continuous injective function. Is there a way to verify if $f$ is a flow of a time homogeneous ODE? That is, if there is a Lipschitz time independent vector ...
- 5,405
3
votes
0
answers
66
views
A combinatorial Dyson-Schwinger equation, tree diagrams, and compositional inversion of a Laurent series
In "Tree hook length formulae, Feynman rules and B-series", Bradley Jones and Karen Yeats state on pg. 9:
Combinatorial Dyson-Schwinger equations are functional equations with solutions in
$...
- 10.5k
5
votes
0
answers
99
views
Differential equations analogue of fundamental theorem of symmetric functions
In Gian-Carlo Rota's article "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations", at the end of the third lesson he states a theorem:
"Every differential ...
- 226
3
votes
0
answers
108
views
A question on essentially self-adjoint differential operators of the type $\Delta=D^{\ast}D$
Let $(M,g)$ be a smooth (connected, complete, oriented) Riemannian manifold and let $D:C^{\infty}(M)\to C^{\infty}(M)$ be a linear partial differential operator, which I view as an operator in $L^{2}(...
- 1,171
2
votes
1
answer
92
views
Behavior of ODE with time : Does not cross boundary
Let $a>0$, and for any time point $t\in [0,1]$, define $\sigma_t^2:= t^2 + (1-t)^2$. Next,
we define the following ODE:
$$
\frac{d X_t}{dt} = \frac{2t-1}{\sigma_t^2} X_t + \frac{a(1-t)}{\sigma_t^2} ...
- 399
0
votes
0
answers
64
views
The shape of water drop on smooth solid surface
We know that the shape of a water drop satisfies the Young-Laplace equation: $\Delta p = 2\gamma H$, where $\Delta p$ is the pressure difference between the inside and outside surface of the water ...
2
votes
0
answers
90
views
Positivity for a kinetic PDE
Let us consider the following kinetic equation:
$$ \partial_t f + v \cdot \partial_x f = \rho[f] \, M[T] - f $$
for a the phase space density $f=f(x,v,t)$ on $\mathbb{T}^1 \times \mathbb{R} \times (0, ...
- 185
1
vote
0
answers
124
views
Solve coupled ODEs analytically in the limit of a small parameter
I have the following set of coupled second order non-linear ODEs :
$$ x^2 a''(x) + x a'(x) - \Big(\frac{1}{\epsilon^2}\Big)b^2(x) a(x) = 0 \\
x b''(x) - b'(x) - 2x b(x)a^2(x) = 0$$
with boundary ...
- 11
2
votes
1
answer
74
views
Eigenvectors of matrices and solutions of (finite dimensional) Schroedinger equation
I am trying to understand certain statement in physical literature (a reference is given below). My question is a finite dimensional version of what is really necessary.
Let $A,B$ be Hermitian $n\...
- 21.8k
0
votes
0
answers
66
views
Parametrization of elliptic curve with differential equation $(x,y)=(f(x),f'(x))$ involving Lambert $W$ function
For non-zero complex $A$, define the curve over the complex numbers
$C: x^2 y^2-A x-y=0$. $C$ is an elliptic curve.
$C$ has the differential equation parametrization $(x,y)=(f(x),f'(x))$
where
$$ f(x)=...
- 25.4k
4
votes
3
answers
288
views
A polynomial vector field on $\mathbb{R}^3$ which has a knot periodic orbit
Is there a polynomial vector field
$$P(x,y,z)\partial_x+Q(x,y,z)\partial_y+R(x,y,z)\partial_z$$
which has a closed orbit $K$ such that $K$ is a non trivial knot?
- 366
0
votes
0
answers
51
views
Given a recurrence relation and its closed form, how can I get the same result by solving its ODE?
Assume the following three-term recurrence relation
\begin{equation}
[\gamma+x(2n+1)]a_n-x(n+1)a_{n+1}-x n a_{n-1}=x (\delta_{n,0}-\delta_{n-1,0})
\end{equation}
for $n\ge 0$ and $\gamma$ and $x$ ...
- 19
1
vote
0
answers
54
views
Are total curvature and the unknoting number of closed orbits of algebraic vector fields bounded uniformly by the degree of vector field?
I am interested in this question since 1999 when I heared the definition of a knot and I read the definition of unknoting and the total curvature of a knot.
To what extent can closed ...
- 366
0
votes
0
answers
49
views
Derivative in recurrence relation of Slevinsky's formula for derivatives of function composition
I'm working on an efficient Taylor series solution to an ODE along the lines of
$$ \vec{y}'(t) = f(\vec{y}(t)) $$
Differentiation of the equation gives
$$ y^{(k+1)} = \frac{\partial^k f(y(t))}{\...
- 151
4
votes
1
answer
407
views
Inverse relationship between Stirling numbers of the first and second kind via generating functions
In combinatorics, a well-known result is that the matrix formed by the Stirling numbers of the second kind $\left(S(n,k)\right)_{n,k\geq 0}$ and the matrix of the signed Stirling numbers of the first ...
- 151
2
votes
1
answer
154
views
What conditions on the rate matrix $Q$ ensure unique convergence in continuous-time Markov chains?
In the study of discrete-time Markov chains, the conditions under which all initial distributions converge to a unique stationary distribution are well-understood. Specifically, if the transition ...
- 807
2
votes
0
answers
94
views
A surprisingly simple and difficult problem on sums of upper bounds
Let $T$ be a large integer, and $C$ be a positive real constant.
Consider a sequence $\{p_t\}_{T\geq t\geq 1}$ of real numbers in $[0,1]$. The sequence $\{b_t\}_{T\geq t\geq 1}$ can be defined as ...
- 353
0
votes
0
answers
75
views
Does nice behavior near a singular point force solution to be in Frobenius series?
I have a pair of partial differential operators $\Delta_1$ and $\Delta_2$ in $y_1, y_2$ formed from constants, multiplication by $y_1$ or $y_2$ and derivatives in the form $y_1 \frac{\partial}{\...
- 151
0
votes
0
answers
22
views
Compartment models with infinite compartments
Compartment models are used in a number of fields. While studying the literature (and especially E. Allen‘s book on Modeling with Itô Stochastic Differential Equations) I have noticed that all the ...
- 309
7
votes
0
answers
162
views
A differential equation and recurrence related to P-partitions
I am interested in polynomials $G_n(z)$ defined by the recurrence
$$G_{n+1}(z) - 2G_n(z) + (1-nz)G_{n-1}(z)=0$$
for $n\ge1$ with the initial values $G_0(z) = 1$ and $G_1(z) = 1$.
The next few values ...
- 17k