Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.
9
votes
0answers
91 views
Ramanujan style nested differential Equation
So I was exploring some math the other day... and I came across the following neat identity:
Given $y$ is a function of $x$ ($y(x)$) and
$$
y = 1 + \frac{\mathrm{d}}{\mathrm{d}x} \left(1 + ...
9
votes
0answers
275 views
Osgood condition
Let $h$ and $g$ be continuous, non-decreasing and concave functions in the interval $[0,\infty)$ with $h(0)=g(0)=0$ and $h(x)>0$ and $g(x)>0$ for $x>0$ such that both satisfy the Osgood ...
8
votes
0answers
203 views
Fourth Order Nonlinear ODE
I was looking at an ode $w^{(4)} + w^3 = 0$ with initial conditions $[w'''(0),w''(0),w'(0),w(0)]=[1,0,0,0]$. I can see via maple that there is a blowup around 3.7. I was wondering if there was a way ...
7
votes
0answers
154 views
Addition formula for $f_n(x+y)$ in closed form.
$n$ is a positive integer.
$$f_n(x)^n+\left(\frac{df_n(x)}{dx}\right)^n=1$$
$f_n(0)=0$,
$f_n'(0)=1$ then
I am looking for the addition formula for $f_n(x+y)$ in closed form.
if $n=1$ then
...
6
votes
0answers
75 views
Measure-driven differential equations
Background: I need some help to understand the concept behind measure-driven differential equations. The solution of an ordinary differential equation is continuous. In order to describe discontinuous ...
5
votes
0answers
62 views
Invariant submanifolds
Let $M$ be a smooth manifold, and let $N$ be a submanifold. Let $V$ be a smooth vector field on $M$ which generates a flow $\Phi_t$ on $M$. My intuition tells me (perhaps modulo some technical ...
5
votes
0answers
88 views
Looking for a Lyapunov function for the next system
I am really stuck looking for a Lypaunov candidate for the next system (which in simulation is stable).
$$ \dot{x} = -(A+A^T)x + Ay \\
\dot{y} = K(x-y)
$$
where x and y are vectors in R^3, A is a ...
5
votes
0answers
84 views
Uniqueness of an infinite system of linear ODEs
How to prove that $\dot{x}=ax,\space x(0)=1$ has a unique solution if $a,x$ are infinite dimensional matrices?
More specifically, let $Q$ be a bounded infinitesimal generator, i.e. ...
5
votes
0answers
630 views
Restricted Three-Body Problem
The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity ...
4
votes
0answers
62 views
Optimizing a functional with a differential equation as a constraint
I am working on solving the following optimization problem. I think it is well-poised but, if not, please give me some pointers that could make the question make more sense.
We have a parametric ...
4
votes
0answers
77 views
Clarification in a paper
This is regarding a clarification in page 384 of a paper published in Annals of Statistics by Amari.
In page no. 384, he defines $$R_i(t)=\frac{\partial}{\partial \theta_i} ...
4
votes
0answers
51 views
Heat Kernel Asymptotics on Manifold with Boundary
On a closed Riemannian manifold $M$, the heat kernel $k_t(x, y)$ of the Laplace-Beltrami operator (or more general of any generalized symmetric Laplace-type operator acting on sections of a vector ...
4
votes
0answers
95 views
Hints/Help studying an Abel Differential Equation
I want to know more than qualitative information about the Abel differential equation
$\frac{dy}{dx}+y^3+x=0$. $\qquad ... \;(1)$
Since I don´t know how to solve this and as far as could see, this ...
4
votes
0answers
154 views
How to solve this differential equation for $y$ in terms of $x$ and $k$
$$yy'+\frac yx+k=0$$
How to solve this differential equation for $y$ in terms of $x$ and $k$ where $k$ is a parameter of $x$?
$y(x)=y$ is a function and $x(k)=x$ is a gamma function
4
votes
0answers
55 views
Solution to differential equation $f^{(n)}-(n+1)f^{(n-1)}-(n+1)nf^{(n-2)}-\dotsc-(n+1)!f=g$
Let $n$ be a given positive integer and $g$ be a continuous function. We are looking for a function $f \in C^n(\mathbb{R})$ such that
$$f^{(n)}-(n+1)f^{(n-1)}-(n+1)nf^{(n-2)}-\dotsc-(n+1)!f=g.$$
It ...
4
votes
0answers
114 views
Solving inhomogenous bessel equation
I have the following differential equation to be solved $\dfrac{d^2\psi}{dr^2}+\dfrac{d\psi}{rdr}+4\left(\omega^2-k_0^2-\dfrac{n^2}{r^2}\right)\psi=AJ_n^2(kr)+\dfrac{k}{r}J_n(kr)J_{n+1}(kr)-\omega ...
4
votes
0answers
78 views
Uniqueness result in linear differential equation of degree $n$.
Suppose that $f$ is such that
$$f^{(n)}=\sum_{j=0}^{n-1}a_jf^{(j)}$$
Some little work is needed to get to ($a_j=0$ if $j<0$)
$${f^{(n + 1)}} = \sum\limits_{j = 0}^{n - 1} {\left( {{a_{j - 1}} + ...
4
votes
0answers
215 views
Confused by a proof in Rudin *Functional Analysis*
I am reading Rudin's Functional Analysis and got quite confused by his proof of Thm 8.5, that is, the existence of fundamental solutions for differential operator $P(D)$, where $P$ is a polynomial.
...
4
votes
0answers
140 views
“Algebraic multiplicity” for eigenvalues of a Sturm-Liouville-like problem?
Following Coddington-Levinson's book Theory of ordinary differential equations, chapter 7: "Self-adjoint problems on finite intervals", let us consider the eigenvalue problem
$$\pi(l):\begin{cases} ...
4
votes
0answers
196 views
A solution of $-y'' + q(x)y= \lambda y$
Could you help me with the following problem (from Poschel and Trubowitz)?
I am looking for a solution of the differential equation $-y'' + q(x)y= \lambda y$, for $0 \leq x \leq 1$ with ...
4
votes
0answers
136 views
How to solve a differential equation associated with square wheels?
I'm looking for a general solution for $f(t)$ given an unrelated function $g(t)$ in
$$f(t)^2 - 2g(t)f(t)\sin(t) - 2f'(t) + g(t)^2 - 2g(t)\cos(t) + 1 = 0$$
Is it possible to solve without knowing ...
4
votes
0answers
135 views
How to analysis the stability of these ODE?
Study whether the null solution of the system:
$$\begin{cases}
\frac{dx_1}{dt}=x_2(t)\\
\frac{dx_2}{dt}=-w(t)^2 x_1(t)\\
\end{cases}
$$
is Lyapunov stable, where
$$ w(t)=
\begin{cases} 0.4 ...
3
votes
0answers
64 views
Differential equation $y'(t) = 1-y(t) e^{y(t)-1}$
I am interested in finding a clean explicit solution (if possible) to the differential equation
$$
y'(t) = 1-y(t) e^{y(t)-1},
$$
where $0 \le t < 1$ and $0 \le y \le 1$.
This can obviously be ...
3
votes
0answers
60 views
Kähler Geodesics
Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric
$$\begin{pmatrix} ...
3
votes
0answers
41 views
Is any Newton equation an Euler-Lagrange equation?
Let $$ r'' = \mathrm{F}(r', r)$$
be Newton equation in one variable whith $\mathrm{F}$ locally Lipschitz.
Is there a function $\mathcal{L}(r',r)$ such that the Newton equation is in fact ...
3
votes
0answers
196 views
Solve a differential equation using Fourier series
Assume I have a second order differential equation $\ddot{x} = F(x,\dot{x})$ (or an equivalent equation of first order) and that I know there is a periodic solution to it (for simplicity's sake, ...
3
votes
0answers
47 views
Nice corollaries to Poincaré-Bendixson theorem
I am interested in the applications of Poincaré-Bendixson theorem not (explicitely) related to differential equations. For convenience, recall the theorem:
Let $X : \mathbb{R}^2 \to \mathbb{R}^2$ be ...
3
votes
0answers
26 views
Links to pdf-articles or books where there is an information on some linear integral operator
Please write me links to pdf-articles or books where there is some information on properties of operators like these:
$$
(Af)(x,y)=\int_{D}\frac{f(z) \, dz}{|x-z| |z-y|}
$$
or
$$
(Bf)(x,y)=\int_D ...
3
votes
0answers
23 views
Find $\alpha$ such that $y'=\sqrt{1+y^4}-|y|^\alpha$ has global solutions
How do I find $\alpha$ such that $y'=\sqrt{1+y^4}-|y|^\alpha$ has global solutions?
For example, imposing $y'=0$ for $\alpha=4$ we get that for solutions with starting point in ...
3
votes
0answers
58 views
Existence Theorem for Geodesics
The text I am reading now defined geodesics to be those curves that satisfy the following differential equation:
$\ddot{\gamma}^k(t)+\dot{\gamma}^i(t)\dot{\gamma}^j(t)\Gamma^k_{ij}(\gamma(t)) = 0$
...
3
votes
0answers
70 views
Distributional differential equation, somehow related to compact support distributions
I've been mulling over a problem from Friedlander's Introduction to Distribution Theory for a few days now: in Chapter 3 (on distributions with compact support), it asks to solve the differential ...
3
votes
0answers
71 views
Behaviour of $r'=r-r^3 , \theta'=(\sin\theta)^2+a$
What are the local and global behavior of solutions of
$r'=r-r^3$
$\theta'=(\sin\theta)^2+a$
at the bifurcation value $a=-1$?
3
votes
0answers
124 views
Integrating angular velocity to obtain orientation
Suppose that $\gamma:[0,1]\to \operatorname{SO}(3)$ is a path in the space of orientation preserving rotations of $\mathbb R^3$. It is classical that we can find a corresponding $\omega:[0,1]\to ...
3
votes
0answers
49 views
Complex nonlinear differential equation
I have the following nonlinear differential equation:
$$\ddot z(t)-\sin(z(t))=0$$
where $z(t)$ is a complex variable.
The solution of the same equation with $z(t)$ real, is a function of Jacobi ...
3
votes
0answers
114 views
system of implicit nonlinear differential equations
Here I have a system of nonlinear differential equations:
$
(M+2m)\ddot{x} + m(l_1 \ddot{\theta}_1\cos\theta_1 - l_1\dot{\theta}_1^2\sin\theta_1) + ...
3
votes
0answers
121 views
Chebyshev Diff EQ
Find a power series solution about $x_0=0$ for the Chebyshev differential equation
$$(1-x^2)y''-xy'+n^2 y=0,$$
as a function of of the integer $n$. Show that the solutions form a terminating ...
3
votes
0answers
67 views
The linearization of a gradient vector field along a heteroclinic connection
A gradient vector field $X$ in $\mathbb{R}^n$ has two equilibria $x_1, x_2$. The vector field defines a cooperative dynamical system. The linearization about $x_1$ has one positive eigenvalues and ...
3
votes
0answers
82 views
3
votes
0answers
140 views
Harmonic oscillator with stochastic forcing
It's well known that the solution of the differential equation:
$$\ddot x(t)+\omega^2x(t)=\sin(\psi t)$$
has the form:
$$x(t)=C_1 \sin(\omega t)+C_2 \cos(\omega t)-\frac{\sin(\psi ...
3
votes
0answers
92 views
$\frac{dy}{dx}=1+\frac{2}{x+y}$ solution by an “integrating term”
I though about this trick and then found an example to apply it to:
$$\frac{dy}{dx}=1+\frac{2}{x+y}$$
This is the trick: add $\frac{dx}{dx}=1$ to both parts
...
3
votes
0answers
355 views
Is it possible to have Wronskian=0 with independent solutions to a linear differential equation?
In Wikipedia it says that if the Wronskian of two function is 0 everywhere it does not imply they are linearly dependent.
However, in books treating differential equations it seems that, if the two ...
3
votes
0answers
95 views
Prove there are at least two periodic solutions
Could anyone comment on the following ODE problem? Thank you.
Given a 2-d system in polar coordinates:
$$\dot{r}=r+r^{5}-r^{3}(1+\sin^{2}\theta)$$
$$\dot{\theta}=1$$
Prove that there are at least ...
3
votes
0answers
238 views
Gompertz growth equation
:)
Hi!
I'm almost finished with a homework problem, but I cannot quite finish it. The problem is as follows:
Given the Gompertz growth equation
$$\frac{dN}{dt}=K(t)N(t),\ N(0)=N_0 \\ ...
3
votes
0answers
100 views
Satisfying a Differential Equation and complex Laguerre
I have the following problem
Show that $$L_n(x)=\frac{e^x}{2 \pi i}\oint \frac{t^n e^{-t}}{(t-x)^{n+1}}dt$$ satisfies $$x\, L_n^{\prime\prime}+(1-x)L_n^\prime+n\, L_n=0$$ where the contour is ...
3
votes
0answers
58 views
Steady-state of `degenerate' delayed differential equation
Consider the simple delayed differential equation:
$X'(t) = -a X(t) + a X(t - d)$
where $d$ and $a$ are positive constants. I'm interested in the possible steady-state (stationary) solutions of ...
3
votes
0answers
90 views
Trace of BV function
Let $\Omega$ be a bounded open set in ${\mathbb R}^n$ with smooth boundary. Let $t > 0$ be small enough so that for every $x \in \partial \Omega$, there exists a unique $y \in \Omega$ with $|x-y| ...
3
votes
0answers
157 views
Approximating a system of differential equations as a Bézier curve
I am looking for a general transform to approximate the solution to an n-dimensional system of differential equations and initial conditions as a cubic or quadratic Bézier curve.
Sorry if my ...
3
votes
0answers
83 views
Finding $\mathbf r(t)$ for the parameterized two-body equations of motion
I'm trying to understand the equations of two-body motion. Namely, given the position, velocity and mass of two orbiting bodies at time $t$, how can I explicitly find their position and velocity for ...
3
votes
0answers
70 views
Special forms of ODEs
In my previous question, @Gerben suggested that it is more likely that WA recognizes an ODE in"Sturm-Liouville" form. Is there a reason for this particular form being preferred to the usual ...
3
votes
0answers
77 views
Inequality of ODE solutions
Says I have two (scalar) ODE: $u' = f(u,t)$ and $v' = g(v,t)$ where
Both $f$ and $g$ are piecewise-continuous and locally Lipschitz, for existence & uniqueness of solutions $u(t)$ and ...
